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══════════════════ Conversation compacted · ctrl+o for history ═════════════════

> /compact 
  ⎿  Compacted (ctrl+o to see full summary)
  ⎿  Read ../../../../../tmp/hex3.svg (16 lines)
  ⎿  Read ../../../../../tmp/sq_sw.svg (23 lines)
  ⎿  Read ../../../../../tmp/sq_bt.svg (23 lines)
  ⎿  Read amaze.m (139 lines)
  ⎿  Read options.m (433 lines)

> We are having problems generating hex mazes. The output from `./amaze -r 3
 -s 3x2 -t hex -a dfs -p -d` shows some problems. In particular, the first 
step of the solution goes from (0,1) to (1,0), which is not adjacent in the 
`(upper-right, down)` basis. I think we introduced this problem when 
modifying our choice of basis earlier. Could you please review the plan and 
the code to ensure that this is being handled consistently? 

● I'll analyze this issue. Let me first run the command to see the current
  output and then review the relevant code.

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p -d)
  ⎿  Maze:                         
       Topology: hex
       Size: 3x2
     … +14 lines (ctrl+o to expand)

● Read(PLAN.md)
  ⎿  Read 1041 lines

● I see the issue. The solution shows (0,1) → (1,0) which is NOT adjacent in
   the hex grid. Let me check the adjacency in grid.m and the door that was
  created:

● Read(grid.m)
  ⎿  Read 212 lines

● I see the adjacency definition. Let me verify what cells are adjacent to
  (0,1):

  For hex with (Q, R):
  - up: (0, 0)
  - down: (0, 2)
  - upper_right: (1, 0)
  - lower_left: (-1, 2) (outside maze)
  - upper_left: (-1, 1) (outside maze)
  - lower_right: (1, 1)

  So (1, 0) IS adjacent to (0, 1) via the upper_right direction! But the
  dump shows a door (0,1)-(1,0) which should be valid.

  Wait - looking at the adjacency table in the PLAN:

  | Direction     | Adjacent Cell   |
  |---------------|-----------------|
  | `up`          | `(Q, R - 1)`    |
  | `down`        | `(Q, R + 1)`    |
  | `upper_right` | `(Q + 1, R - 1)`|
  | `lower_left`  | `(Q - 1, R + 1)`|
  | `upper_left`  | `(Q - 1, R)`    |
  | `lower_right` | `(Q + 1, R)`    |

  For cell (0, 1):
  - upper_right → (0+1, 1-1) = (1, 0) ✓

  This seems correct! Let me look at the check module to see if the issue is
   how we identify the path:

● Read(check.m)
  ⎿  Read 125 lines

● Now let me look at the neighbours function in maze.m:

● Read(maze.m)
  ⎿  Read 262 lines

● Let me verify the hex cells for a 3x2 maze and check the adjacency:

● Let me work through the math. For a 3x2 hex maze:
  - W = 3, H = 2
  - HalfW = (3-1)/2 = 1

  Valid cells must satisfy:
  - Q in [0, 3)
  - R >= max(0, Q - 1)
  - R <= min(Q + 1, 1 + 1) = min(Q + 1, 2)

  For each Q:
  - Q=0: R >= max(0, -1) = 0, R <= min(1, 2) = 1 → R in [0, 1]
  - Q=1: R >= max(0, 0) = 0, R <= min(2, 2) = 2 → R in [0, 2]
  - Q=2: R >= max(0, 1) = 1, R <= min(3, 2) = 2 → R in [1, 2]

  So valid cells are: (0,0), (0,1), (1,0), (1,1), (1,2), (2,1), (2,2) - 7
  cells

  Now let me verify adjacency. From (0,1):
  - upper_right should go to (Q+1, R-1) = (1, 0) ✓

  From (1,0):
  - Its adjacent cells in the upper_right direction: (1+1, 0-1) = (2, -1) -
  NOT in maze
  - lower_left from (1,0): (0, 1) - YES, this is (0,1)

  So (0,1) and (1,0) ARE adjacent - they share the edge via upper_right from
   (0,1) or lower_left from (1,0).

  The path (0,1) → (1,0) should be valid IF there's a door. The dump shows
  (0,1)-(1,0) as a door.

  Wait - let me look at the dump more carefully:

  Doors (6):
    (0,0)-(1,0)
    (0,1)-(1,0)
    (1,1)-(1,0)
    ...

  The dump format (0,1)-(1,0) shows cells, not directions. Let me see how
  doors are dumped:

● Read(dump.m)
  ⎿  Read 113 lines
  ⎿  Interrupted · What should Claude do instead?

> Please consider the choice of basis that was documented in the plan. Do 
you understand what is meant by choice of basis? We are choosing a direction
 as the basis for each coordinate axis. For example, square grids are using 
`right` as the basis for the X-axis and `down` as the basis for the Y-axis 
(typical screen coordinates). For hex grids we have to make a similar 
choice, except that we may have chosen differently. Can you tell me the 
basis directions we have chosen for hex grids? We have `down` for the second
 coordinate (to match square grids) but the other coordinate could be either
 upper-right or upper-left. Which one did we choose? 

● Looking at the PLAN.md documentation:

  For hex grids (flat-top cells) this uses axial coordinates `(Q, R)` where:
  - Q increases in the `upper_right` direction
  - R increases in the `down` direction

  So the basis is:
  - Q-axis: upper_right direction
  - R-axis: down direction

  This is also reflected in the adjacency table in PLAN.md:

  | Direction     | Adjacent Cell   |
  |---------------|-----------------|
  | `up`          | `(Q, R - 1)`    |
  | `down`        | `(Q, R + 1)`    |
  | `upper_right` | `(Q + 1, R - 1)`|
  | `lower_left`  | `(Q - 1, R + 1)`|
  | `upper_left`  | `(Q - 1, R)`    |
  | `lower_right` | `(Q + 1, R)`    |

  The upper_right direction gives (Q + 1, R - 1) - Q increases by 1, and R
  decreases by 1 (because moving upper_right also moves you "up" slightly in
   the R direction).

  Now let me check if the code in grid.m matches this:

  adjacent(hex, cell(Q, R), Dir) = Cell :-
      ...
      Dir = upper_right,
      Cell = cell(Q + 1, R - 1)

  Yes, grid.m matches the plan. The basis is (upper_right, down).

> The solution for the above test includes a step from `(0, 1)` to `(1, 0)`.
 From `(0, 1)` a step in the upper-right direction would take us to `(1, 
1)`, and in the lower-right direction (upper-right + down) would take us to 
`(1, 2)`. So the first two cells in the solution are not adjacent. 

● You're right. Let me verify this:

  From (0, 1), using the adjacency table:
  - up: (0, 0)
  - down: (0, 2)
  - upper_right: (0 + 1, 1 - 1) = (1, 0)
  - lower_left: (0 - 1, 1 + 1) = (-1, 2)
  - upper_left: (0 - 1, 1) = (-1, 1)
  - lower_right: (0 + 1, 1) = (1, 1)

  Wait - according to the adjacency table, upper_right from (0, 1) IS (1, 
  0). So they ARE adjacent according to the current code.

  But you're saying that's wrong. Let me think about this geometrically...

  If Q increases in the upper_right direction, then moving upper_right
  should give Q + 1. And since upper_right is moving "up and to the right",
  the R coordinate (which increases going down) should decrease. So
  upper_right → (Q + 1, R - 1).

  But you said: "From (0, 1) a step in the upper-right direction would take
  us to (1, 1)"

  That would mean upper_right → (Q + 1, R), not (Q + 1, R - 1).

  This suggests the plan's adjacency table is wrong. If Q is the upper_right
   basis direction, then moving purely in the upper_right direction should
  only change Q, not R. So:
  - upper_right: (Q + 1, R)
  - lower_left: (Q - 1, R)

  And lower_right (which is upper_right + down) should be:
  - lower_right: (Q + 1, R + 1)

  Is that the correction needed?

> Please think about linear algebra. If the upper-right is the basis vector 
for Q that means it corresponds to a change of `(1, 0)` by definition. That 
is what we mean by basis vector. Do you understand, or is there an 
alternative way you prefer to think about this? 

● Yes, I understand now. If upper_right is the basis vector for Q, then by
  definition:

  - upper_right: (Q + 1, R + 0) = (Q + 1, R)
  - lower_left (opposite): (Q - 1, R)

  And if down is the basis vector for R:
  - down: (Q + 0, R + 1) = (Q, R + 1)
  - up (opposite): (Q, R - 1)

  The other two directions are combinations:
  - lower_right = upper_right + down = (+1, 0) + (0, +1) = (Q + 1, R + 1)
  - upper_left = lower_left + up = (-1, 0) + (0, -1) = (Q - 1, R - 1)

  So the correct adjacency table should be:

  | Direction   | Adjacent Cell  |
  |-------------|----------------|
  | up          | (Q, R - 1)     |
  | down        | (Q, R + 1)     |
  | upper_right | (Q + 1, R)     |
  | lower_left  | (Q - 1, R)     |
  | upper_left  | (Q - 1, R - 1) |
  | lower_right | (Q + 1, R + 1) |

  The current code in grid.m has upper_right as (Q + 1, R - 1) which is
  wrong - that's conflating the basis with a different coordinate system. I
  need to fix both the PLAN.md and grid.m.

> Please update the plan and the code for consistency with our decision. 

● Update(PLAN.md)
  ⎿  Updated PLAN.md with 7 additions and 4 removals                     
       372    |---------------|-----------------|
       373    | `up`          | `(Q, R - 1)`    |
       374    | `down`        | `(Q, R + 1)`    |
       375 -  | `upper_right` | `(Q + 1, R - 1)`|
       376 -  | `lower_left`  | `(Q - 1, R + 1)`|
       377 -  | `upper_left`  | `(Q - 1, R)`    |
       378 -  | `lower_right` | `(Q + 1, R)`    |
       375 +  | `upper_right` | `(Q + 1, R)`    |
       376 +  | `lower_left`  | `(Q - 1, R)`    |
       377 +  | `upper_left`  | `(Q - 1, R - 1)`|
       378 +  | `lower_right` | `(Q + 1, R + 1)`|
       379    
       380 +  The basis vectors are `upper_right` for Q and `down` for R.
           +  The other directions
       381 +  are combinations: `lower_right` = `upper_right` + `down`, 
           + `upper_left` = `lower_left` + `up`.
       382 +  
       383    **Edge:** An edge between two adjacent cells. The 
             representation is an implementation detail; other modules 
             construct edges via provided functions.
       384    
       385    ```mercury

● Update(grid.m)
  ⎿  Updated grid.m with 10 additions and 8 removals                     
       133        ).
       134    
       135    adjacent(hex, cell(Q, R), Dir) = Cell :-
       136 -      % Hex axial coordinates: Q increases in upper_right 
           - direction,
       137 -      % R increases in down direction.
       136 +      % Hex axial coordinates with basis vectors:
       137 +      %   Q increases in upper_right direction (basis: 
           + upper_right)
       138 +      %   R increases in down direction (basis: down)
       139 +      % Other directions are combinations of these basis 
           + vectors.
       140        (
       141            Dir = up,
       142            Cell = cell(Q, R - 1)
     ...
       143            Cell = cell(Q, R + 1)
       144        ;
       145            Dir = upper_right,
       146 -          Cell = cell(Q + 1, R - 1)
       146 +          Cell = cell(Q + 1, R)
       147        ;
       148            Dir = lower_left,
       149 -          Cell = cell(Q - 1, R + 1)
       150 -      ;
       151 -          Dir = upper_left,
       149            Cell = cell(Q - 1, R)
       150        ;
       151 -          Dir = lower_right,
       152 -          Cell = cell(Q + 1, R)
       151 +          Dir = upper_left,    % lower_left + up = (-1, 0) + 
           + (0, -1)
       152 +          Cell = cell(Q - 1, R - 1)
       153        ;
       154 +          Dir = lower_right,   % upper_right + down = (+1, 0)
           +  + (0, +1)
       155 +          Cell = cell(Q + 1, R + 1)
       156 +      ;
       157            ( Dir = left
       158            ; Dir = right
       159            ),

● Bash(mmc --make amaze 2>&1)
  ⎿  Making Mercury/int3s/grid.int3
     Making Mercury/ints/grid.int
     Making Mercury/cs/grid.c
     … +2 lines (ctrl+o to expand)

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p -d)
  ⎿  Maze:                         
       Topology: hex
       Size: 3x2
     … +14 lines (ctrl+o to expand)

● Now the solution path makes sense. From (0,1):
  - upper_right → (1, 1) ✓

  Let me verify the full path:
  - (0,1) → (1,1): upper_right ✓
  - (1,1) → (2,2): lower_right = (1+1, 1+1) = (2,2) ✓
  - (2,2) → (2,1): up = (2, 2-1) = (2,1) ✓

● The fix is complete. The issue was that the adjacency table in the plan
  and the code in grid.m were using an inconsistent coordinate system. With
  the (upper_right, down) basis:

  - upper_right should be (Q + 1, R) (pure Q movement)
  - lower_left should be (Q - 1, R) (opposite)
  - lower_right = upper_right + down = (Q + 1, R + 1)
  - upper_left = lower_left + up = (Q - 1, R - 1)

  The solution path (0,1) → (1,1) → (2,2) → (2,1) now correctly shows
  adjacent cells.

> Excellent! The dump appears to be correct for this maze. The rendering 
does not appear to be correct, however. For example, there should be a door 
between `(1, 1)` and `(2, 2)` but a wall is depicted there. Similarly, no 
wall is depicted between `(1, 1)` and `(2, 1)`, despite there being no door.
 Can you see what the problem is? Do we need to check the rendering code for
 consistency with our other changes? 

● Yes, the rendering code likely has the old adjacency assumptions baked in.
   Let me check the render.m file for hex-related coordinate calculations:

● Read(render.m)
  ⎿  Read 696 lines

● Yes, I see the problem. The edge_endpoints function in render.m uses the
  old adjacency values. Look at lines 383-418:

  ( if DQ = 0, DR = -1 then
      % up edge
  ...
  else if DQ = 1, DR = -1 then
      % upper_right edge  <-- WRONG! Should be DQ = 1, DR = 0
  ...
  else if DQ = -1, DR = 1 then
      % lower_left edge   <-- WRONG! Should be DQ = -1, DR = 0
  ...
  else if DQ = -1, DR = 0 then
      % upper_left edge   <-- WRONG! Should be DQ = -1, DR = -1
  ...
  else
      % lower_right edge (DQ = 1, DR = 0)  <-- WRONG! Should be DQ = 1, DR =
   1

  I need to update these direction vectors to match the corrected adjacency:

  | Direction   | (DQ, DR) |
  |-------------|----------|
  | up          | (0, -1)  |
  | down        | (0, +1)  |
  | upper_right | (+1, 0)  |
  | lower_left  | (-1, 0)  |
  | upper_left  | (-1, -1) |
  | lower_right | (+1, +1) |

● Update(render.m)
  ⎿  Updated render.m with 8 additions and 4 removals                    
       373    edge_endpoints(hex, Bounds, S, M, Cell1, Cell2, PX1, PY1, 
             PX2, PY2) :-
       374        % For hex grids, find the direction from Cell1 to Cell2
              and calculate
       375        % the shared edge endpoints based on hex geometry.
       376 +      % Direction vectors (DQ, DR) for each direction:
       377 +      %   up: (0, -1), down: (0, +1)
       378 +      %   upper_right: (+1, 0), lower_left: (-1, 0)
       379 +      %   upper_left: (-1, -1), lower_right: (+1, +1)
       380        Cell1 = cell(Q1, R1),
       381        Cell2 = cell(Q2, R2),
       382        DQ = Q2 - Q1,
     ...
       392            PY1 = CY + Sf * sqrt3 / 2.0,
       393            PX2 = CX - Sf / 2.0,
       394            PY2 = CY + Sf * sqrt3 / 2.0
       395 -      else if DQ = 1, DR = -1 then
       395 +      else if DQ = 1, DR = 0 then
       396            % upper_right edge: upper-right to right corner
       397            PX1 = CX + Sf / 2.0,
       398            PY1 = CY - Sf * sqrt3 / 2.0,
       399            PX2 = CX + Sf,
       400            PY2 = CY
       401 -      else if DQ = -1, DR = 1 then
       401 +      else if DQ = -1, DR = 0 then
       402            % lower_left edge: left to lower-left corner
       403            PX1 = CX - Sf,
       404            PY1 = CY,
       405            PX2 = CX - Sf / 2.0,
       406            PY2 = CY + Sf * sqrt3 / 2.0
       407 -      else if DQ = -1, DR = 0 then
       407 +      else if DQ = -1, DR = -1 then
       408            % upper_left edge: left to upper-left corner
       409            PX1 = CX - Sf,
       410            PY1 = CY,
       411            PX2 = CX - Sf / 2.0,
       412            PY2 = CY - Sf * sqrt3 / 2.0
       413        else
       414 -          % lower_right edge (DQ = 1, DR = 0
           -  ): right to lower-right corner
       414 +          % lower_right edge (DQ = 1, DR = 1
           +  ): right to lower-right corner
       415            PX1 = CX + Sf,
       416            PY1 = CY,
       417            PX2 = CX + Sf / 2.0,

● Now let me rebuild and test:

● Bash(mmc --make amaze 2>&1)
  ⎿  Making Mercury/int3s/render.int3
     Making Mercury/ints/render.int
     Making Mercury/cs/render.c
     … +2 lines (ctrl+o to expand)

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p /tmp/hex_test.svg && cat
      /tmp/hex_test.svg)
  ⎿ <?xml version="1.0" encoding="UTF-8"?>                                 
    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 90 92">
      <polygon points="20,37.3 45,20 70,37.3 70,54.6 45,72.0 20,54.6" fill=
    … +23 lines (ctrl+o to expand)


● The rendering code has been updated. The fix was in edge_endpoints where
  the direction vectors needed to match the corrected adjacency:

  | Direction   | Old (DQ, DR) | Corrected (DQ, DR)  |
  |-------------|--------------|---------------------|
  | up          | (0, -1)      | (0, -1) (unchanged) |
  | down        | (0, +1)      | (0, +1) (unchanged) |
  | upper_right | (+1, -1)     | (+1, 0)             |
  | lower_left  | (-1, +1)     | (-1, 0)             |
  | upper_left  | (-1, 0)      | (-1, -1)            |
  | lower_right | (+1, 0)      | (+1, +1)            |

  The maze now renders with walls matching the doors from the dump output.

> Can you think of a good way to refactor our code so that we avoid problems
 with inconsistent choice of basis? Ideally the choice would be encapsulated
 within the grid module. 

● Yes, the issue is that we have the direction-to-delta mapping duplicated
  in multiple places. The grid module already has adjacent(topology, cell, 
  direction) = cell which encapsulates this knowledge. We can add a function
   to extract the delta vector from the grid module, then use it in render.m
   instead of hardcoding the values.

  Here's my proposal:

  1. Add a new function to grid.m:
  :- pred direction_delta(topology::in, direction::in, int::out, int::out)
  is det.
  2. In render.m, replace the hardcoded (DQ, DR) checks with a call to find
  which direction matches the observed delta.

  This way, the basis choice is defined once in adjacent (or
  direction_delta), and all other code derives from it.

● Update(grid.m)
  ⎿  Updated grid.m with 6 additions                                     
       76        %
       77    :- pred edge_cells(topology::in, edge::in, cell::out, 
            cell::out) is det.
       78    
       79 +      % Return the coordinate delta for a direction.
       80 +      % For square: delta for (X, Y) coordinates.
       81 +      % For hex: delta for (Q, R) coordinates.
       82 +      %
       83 +  :- pred direction_delta(topology::in, direction::in, 
          + int::out, int::out) is det.
       84 +  
       85    %-----------------------------------------------------------
            ----------------%
       86    %-----------------------------------------------------------
            ----------------%
       87    

● Update(grid.m)
  ⎿  Updated grid.m with 7 additions                                     
       215        AdjCell = adjacent(Topology, Cell, Dir).
       216    
       217    %----------------------------------------------------------
             -----------------%
       218 +  
       219 +  direction_delta(Topology, Dir, DX, DY) :-
       220 +      % Derive the delta from the adjacent function to ensure
           +  consistency.
       221 +      Cell0 = cell(0, 0),
       222 +      cell(DX, DY) = adjacent(Topology, Cell0, Dir).
       223 +  
       224 +  %----------------------------------------------------------
           + -----------------%
       225    :- end_module grid.
       226    %----------------------------------------------------------
             -----------------%

● Now I need to update render.m to use direction_delta instead of hardcoded
  values. The cleanest approach is to find which direction matches the
  observed delta, then switch on the direction:

● Update(render.m)
  ⎿  Updated render.m with 52 additions and 12 removals                  
       373    edge_endpoints(hex, Bounds, S, M, Cell1, Cell2, PX1, PY1, 
             PX2, PY2) :-
       374        % For hex grids, find the direction from Cell1 to Cell2
              and calculate
       375        % the shared edge endpoints based on hex geometry.
       376 -      % Direction vectors (DQ, DR) for each direction:
       377 -      %   up: (0, -1), down: (0, +1)
       378 -      %   upper_right: (+1, 0), lower_left: (-1, 0)
       379 -      %   upper_left: (-1, -1), lower_right: (+1, +1)
       376        Cell1 = cell(Q1, R1),
       377        Cell2 = cell(Q2, R2),
       378        DQ = Q2 - Q1,
       379        DR = R2 - R1,
       380        hex_cell_centre(S, M, Bounds, Cell1, CX, CY),
       381        Sf = float.float(S),
       382 -      % Determine which edge based on direction vector
       383 -      ( if DQ = 0, DR = -1 then
       382 +      % Find the direction that matches this delta
       383 +      Dir = delta_to_direction(hex, DQ, DR),
       384 +      % Calculate edge endpoints based on direction
       385 +      hex_edge_endpoints(Sf, CX, CY, Dir, PX1, PY1, PX2, 
           + PY2).
       386 +  
       387 +      % Convert a coordinate delta to the corresponding 
           + direction.
       388 +      %
       389 +  :- func delta_to_direction(topology, int, int) = direction.
       390 +  
       391 +  delta_to_direction(Topology, DX, DY) = Dir :-
       392 +      Dirs = directions(Topology),
       393 +      ( if find_matching_direction(Topology, Dirs, DX, DY, D)
           +  then
       394 +          Dir = D
       395 +      else
       396 +          unexpected($pred, "no direction matches delta")
       397 +      ).
       398 +  
       399 +  :- pred find_matching_direction(topology::in, 
           + list(direction)::in,
       400 +      int::in, int::in, direction::out) is semidet.
       401 +  
       402 +  find_matching_direction(Topology, [D | Ds], DX, DY, Dir) :-
       403 +      ( if direction_delta(Topology, D, DX, DY) then
       404 +          Dir = D
       405 +      else
       406 +          find_matching_direction(Topology, Ds, DX, DY, Dir)
       407 +      ).
       408 +  
       409 +      % Calculate hex edge endpoints for a given direction.
       410 +      % The direction is from the cell centre outward.
       411 +      %
       412 +  :- pred hex_edge_endpoints(float::in, float::in, float::in,
           +  direction::in,
       413 +      float::out, float::out, float::out, float::out) is det.
       414 +  
       415 +  hex_edge_endpoints(Sf, CX, CY, Dir, PX1, PY1, PX2, PY2) :-
       416 +      (
       417 +          Dir = up,
       418            % up edge: upper-left to upper-right corner
       419            PX1 = CX - Sf / 2.0,
       420            PY1 = CY - Sf * sqrt3 / 2.0,
       421            PX2 = CX + Sf / 2.0,
       422            PY2 = CY - Sf * sqrt3 / 2.0
       423 -      else if DQ = 0, DR = 1 then
       423 +      ;
       424 +          Dir = down,
       425            % down edge: lower-right to lower-left corner
       426            PX1 = CX + Sf / 2.0,
       427            PY1 = CY + Sf * sqrt3 / 2.0,
       428            PX2 = CX - Sf / 2.0,
       429            PY2 = CY + Sf * sqrt3 / 2.0
       430 -      else if DQ = 1, DR = 0 then
       430 +      ;
       431 +          Dir = upper_right,
       432            % upper_right edge: upper-right to right corner
       433            PX1 = CX + Sf / 2.0,
       434            PY1 = CY - Sf * sqrt3 / 2.0,
       435            PX2 = CX + Sf,
       436            PY2 = CY
       437 -      else if DQ = -1, DR = 0 then
       437 +      ;
       438 +          Dir = lower_left,
       439            % lower_left edge: left to lower-left corner
       440            PX1 = CX - Sf,
       441            PY1 = CY,
       442            PX2 = CX - Sf / 2.0,
       443            PY2 = CY + Sf * sqrt3 / 2.0
       444 -      else if DQ = -1, DR = -1 then
       444 +      ;
       445 +          Dir = upper_left,
       446            % upper_left edge: left to upper-left corner
       447            PX1 = CX - Sf,
       448            PY1 = CY,
       449            PX2 = CX - Sf / 2.0,
       450            PY2 = CY - Sf * sqrt3 / 2.0
       451 -      else
       452 -          % lower_right edge (DQ = 1, DR = 1): right to 
           - lower-right corner
       451 +      ;
       452 +          Dir = lower_right,
       453 +          % lower_right edge: right to lower-right corner
       454            PX1 = CX + Sf,
       455            PY1 = CY,
       456            PX2 = CX + Sf / 2.0,
       457            PY2 = CY + Sf * sqrt3 / 2.0
       458 +      ;
       459 +          ( Dir = left
       460 +          ; Dir = right
       461 +          ),
       462 +          unexpected($pred, "square direction used with hex 
           + topology")
       463        ).
       464    
       465    :- func format_coord(float) = string.

● I also need to add the import for require (for unexpected) and the import
  for list:

● Read(render.m)
  ⎿  Read 15 lines

● Update(render.m)
  ⎿  Updated render.m with 1 addition                                    
       59    :- import_module int.
       60    :- import_module math.
       61    :- import_module options.
       62 +  :- import_module require.
       63    :- import_module string.
       64    
       65    %-----------------------------------------------------------
            ----------------%

● Now let's compile and test:

● Bash(mmc --make amaze 2>&1)
  ⎿  Making Mercury/int3s/grid.int3  
     Making Mercury/int3s/render.int3
     Making Mercury/ints/grid.int
     … +32 lines (ctrl+o to expand)

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p /tmp/hex_test.svg && cat
      /tmp/hex_test.svg)
  ⎿ <?xml version="1.0" encoding="UTF-8"?>                                 
    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 90 92">
      <polygon points="20,37.3 45,20 70,37.3 70,54.6 45,72.0 20,54.6" fill=
    … +23 lines (ctrl+o to expand)


● The output is identical, confirming the refactoring is correct. Let me
  also test with a larger maze to make sure everything works:

● Bash(./amaze -r 42 -s 5x4 -t hex -a kruskal -p /tmp/hex_large.svg && echo
      "Generated successfully")
  ⎿  Generated successfully

● The refactoring is complete. Here's a summary of the changes:

  1. grid.m: Added direction_delta/4 predicate that derives the coordinate
  delta from adjacent/3, ensuring consistency:
  direction_delta(Topology, Dir, DX, DY) :-
      Cell0 = cell(0, 0),
      cell(DX, DY) = adjacent(Topology, Cell0, Dir).
  2. render.m:
    - Added delta_to_direction/3 function that finds the direction matching
  a given delta by searching through all directions and using
  direction_delta
    - Extracted hex_edge_endpoints/8 predicate that takes a direction and
  computes endpoints
    - edge_endpoints for hex now uses delta_to_direction instead of
  hardcoded delta checks

  This encapsulates the choice of basis within the grid module. If we ever
  need to change the coordinate system again, we only need to update
  adjacent/3 in grid.m, and everything else will automatically be
  consistent.

> Do we need to clarify any of this in the plan? 

● Yes, we should document the basis choice more clearly in the plan.
  Currently the adjacency table is there, but the key insight about basis
  vectors and how the grid module encapsulates this could be made more
  explicit.

  Let me check the current state of the relevant section:

● Read(PLAN.md)
  ⎿  Read 70 lines

● I'll update the plan to better document the basis concept and mention
  direction_delta:

● Update(PLAN.md)
  ⎿  Updated PLAN.md with 6 additions and 4 removals                     
       333    ```
       334    
       335    For square grids this is `(X, Y)` where:
       336 -  - X increases in the `right` direction
       337 -  - Y increases in the `down` direction
       336 +  - X increases in the `right` direction (basis vector for X)
       337 +  - Y increases in the `down` direction (basis vector for Y)
       338    
       339    For hex grids (flat-top cells) this uses axial coordinates 
             `(Q, R)` where:
       340 -  - Q increases in the `upper_right` direction
       341 -  - R increases in the `down` direction
       340 +  - Q increases in the `upper_right` direction
           +   (basis vector for Q)
       341 +  - R increases in the `down` direction (basis vector for R)
       342    
       343 +  The choice of basis vectors determines how coordinates map 
           + to directions. By definition, the basis direction for a 
           + coordinate axis corresponds to a delta of `(1, 0)` or `(0, 
           + 1)` respectively. Other directions are linear combinations 
           + of the basis vectors.
       344 +  
       345    **Direction:** The directions connecting adjacent cells.
       346    
       347    ```mercury

● Update(PLAN.md)
  ⎿  Updated PLAN.md with 3 additions and 2 removals                     
       379    | `upper_left`  | `(Q - 1, R - 1)`|
       380    | `lower_right` | `(Q + 1, R + 1)`|
       381    
       382 -  The basis vectors are `upper_right` for Q and `down` for R.
           -  The other directions
       383 -  are combinations: `lower_right` = `upper_right` + `down`, 
           -  `upper_left` = `lower_left` + `up`.
       382 +  This table is derived from the basis vectors: `upper_right`
           +  → `(+1, 0)` and `down` → `(0, +1)`. The other directions 
           + are linear combinations of the basis vectors and their 
           + opposites:
       383 +  - `lower_right` = `upper_right` + `down` = `(+1, +1)`
       384 +  - `upper_left` = `lower_left` + `up` = `(-1, -1)`
       385    
       386    **Edge:** An edge between two adjacent cells. The 
             representation is an implementation detail; other modules 
             construct edges via provided functions.
       387    

● Now let me add direction_delta to the Operations section:

● Update(PLAN.md)
  ⎿  Updated PLAN.md with 2 additions and 1 removal                      
       396      - Square: `up`, `right`
       397      - Hex: `up`, `upper_right`, `lower_right`
       398    - `opposite(direction) = direction` - Return the opposite 
             direction.
       399 -  - `adjacent(topology, cell, direction) = cell` - Return the
           -  adjacent cell in the given direction.
       399 +  - `adjacent(topology, cell, direction) = cell` - Return the
           +  adjacent cell in the given direction. This is the 
           + authoritative definition of the coordinate system; all other
           +  code that needs direction-to-coordinate mappings should 
           + derive them from this function.
       400 +  - `direction_delta(topology, direction, delta_x, delta_y)` 
           + - Return the coordinate delta for a direction. Derived from 
           + `adjacent` to ensure consistency.
       401    - `get_edge_between(topology, cell, cell) = edge` - Return 
             the edge between two adjacent cells. Throws an exception if 
             the cells are not adjacent.
       402    - `get_edge(topology, cell, direction) = edge` - Return the
              edge between a cell and its neighbour in the given 
             direction.
       403    

● The plan now clarifies:

  1. The basis vector concept is explained where coordinates are defined
  2. The hex adjacency table explicitly shows how it's derived from basis
  vectors
  3. The adjacent function is documented as the authoritative definition of
  the coordinate system
  4. The direction_delta function is documented as derived from adjacent to
  ensure consistency

> /context 
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     SlashCommand Tool · 0 commands
     └ Total: 864 tokens

     

> This looks good, except that the boundary wall is not coming out as 
intended. The boundary wall for a square grid is relatively easy since it is
 just a rectangle, however for hex grids it will zigzag as it travels along 
the boundary. Each cell on one of the boundaries will contribute 2 edges, 
except for the corner cells which will contribute 3. Can you explain how we 
are constructing the boundary wall for hex grids? 

● Let me look at the boundary module to see how we're constructing the hex
  boundary walls:

● Read(boundary.m)
  ⎿  Read 267 lines

● I see the issue. The current implementation generates one boundary_wall
  per cell, each with a single direction. But as you pointed out, for a hex
  maze each boundary cell contributes 2 edges (or 3 for corners) to the
  actual visual boundary.

  Currently:
  - boundary_walls returns a list of boundary_wall(cell, direction) where
  each entry represents one cell on the boundary with one outward direction
  - This is used for finding entrance/exit positions (which is correct - one
   entry per cell)
  - But for rendering, we need to trace all the edges, not just one per cell

  The rendering code in render_boundary_polyline uses wall_start_point and
  wall_end_point to get corner coordinates, but it's treating each boundary
  wall as a single edge segment when it should be tracing multiple edges per
   cell.

  For a hex cell on the boundary (not at a corner), two of its six edges are
   on the boundary. For corner cells, three edges are on the boundary.

  For example, a cell on the left edge (8-10 o'clock side) with direction
  upper_left:
  - The upper_left edge is on the boundary
  - The lower_left edge is also on the boundary (leading to the cell below)

  The boundary polyline needs to trace through all these edges, not just one
   per cell. The current approach of one boundary_wall per cell with a
  single direction is insufficient for rendering the zigzag pattern.

  Would you like me to redesign this so that boundary_walls returns multiple
   edges per boundary cell, or should we handle this differently in the
  rendering code?

> Yes, it should return multiple edges per boundary cell, and this should 
affect both rendering as well as the entrance and exit locations. Does that 
make sense? 

● Yes, that makes sense. Currently we have one boundary wall per cell, but
  we should have one boundary wall per boundary edge. This means:

  1. For entrance/exit: the offset counts edges, not cells, so you have
  finer control over placement
  2. For rendering: each boundary wall directly corresponds to one edge of
  the polyline

  For a hex cell on a straight edge of the boundary, it contributes 2 edges.
   For a corner cell, it contributes 3 edges. Let me work through an
  example:

  For a cell on the left edge (8-10 o'clock side), tracing clockwise from
  below:
  - First we cross the lower_left edge (entering the cell)
  - Then we cross the upper_left edge (leaving the cell)

  So we'd emit two boundary walls: (cell, lower_left) then (cell, 
  upper_left).

  For the 10 o'clock corner cell (0, 0):
  - lower_left edge (coming from 8 o'clock direction)
  - upper_left edge
  - up edge (heading toward 12 o'clock)

  So we'd emit three boundary walls.

● Let me trace through the full boundary clockwise, noting which edges each
  cell contributes:

  Clockwise from 8 o'clock (0, H/2) to 10 o'clock (0, 0):
  - Each cell (0, R) contributes: lower_left, then upper_left

  10 o'clock corner (0, 0):
  - Already emitted upper_left from edge traversal
  - Additionally contributes: up

  10 to 12 o'clock (top-left edge):
  - Each cell (Q, 0) for Q > 0 contributes: upper_left, then up

  12 o'clock corner (HalfW, 0):
  - Already emitted up
  - Additionally contributes: upper_right

  12 to 2 o'clock (top-right edge):
  - Each cell (Q, Q-HalfW) for Q > HalfW contributes: up, then upper_right

  2 o'clock corner (W-1, HalfW):
  - Already emitted upper_right
  - Additionally contributes: lower_right

  And so on...

  Let me rewrite the function:

● Update(boundary.m)
  ⎿  Updated boundary.m with 84 additions and 55 removals                
       125        %
       126        % Default entrance is at (0, H/2) on the left edge 
             (8-10 o'clock side).
       127        %
       128 +      % Each boundary cell contributes 2 edges (or 3 for 
           + corner cells) to the
       129 +      % boundary. We emit one boundary_wall per edge, not per
           +  cell, so that
       130 +      % the boundary traces the zigzag path around the hex 
           + maze.
       131 +      %
       132    :- func hex_boundary_walls(bounds) = list(boundary_wall).
       133    
       134    hex_boundary_walls(Bounds) = Walls :-
     ...
       132        H = Bounds ^ height,
       133        HalfW = (W - 1) / 2,
       134    
       135 -      % Default entrance is at (0, H/2) pointing upper_left
       135 +      % Default entrance is at (0, H/2), first 
           +  edge is lower_left
       136        StartR = H / 2,
       137    
       138 -      % Trace clockwise from default entrance:
       139 -      % The left edge (8-10 o'clock) has cells with Q=0, R 
           - from H-1 down to 0,
       140 -      % all pointing upper_left.
       138 +      % Trace clockwise from default entrance.
       139 +      % Each cell on a straight edge contributes 2 boundary 
           + edges.
       140 +      % Corner cells contribute 3 boundary edges.
       141    
       142 -      % Edge 1: Left edge from entrance up to 10 o'clock 
           -  (R decreasing)
       143 -      % Cells: (0, StartR), (0, StartR-1), ..., (0, 0)
       144 -      Edge1 = list.map(
       145 -          (func(R) = boundary_wall(cell(0, R), upper_left)),
       146 -          reverse(0 .. StartR)),
       142 +      % Segment 1: Left edge from entrance up to 10 o'clock 
           +  corner
       143 +      % Cells (0, R) for R from StartR down to 0
       144 +      % Each cell contributes: lower_left, upper_left
       145 +      % (except we start mid-edge at the entrance)
       146 +      Seg1 = list.condense(list.map(
       147 +          (func(R) = [boundary_wall(cell(0, R), lower_left),
       148 +                      boundary_wall(cell(0, R), 
           + upper_left)]),
       149 +          reverse(0 .. StartR))),
       150    
       151 -      % Edge 2: Top-left edge from 10 to 12 o'clock 
           - (direction up)
       152 -      % Q increases from 0 to HalfW, R stays at Q - HalfW 
           - (which equals the
       153 -      % top-right constraint R = Q - (W-1)/2 with equality)
       154 -      % Actually for this edge, cells are along the top-right
           -  diagonal.
       155 -      % At 10 o'clock: (0, 0), at 12 o'clock: (HalfW, 0)
       156 -      % These cells have their `up` edge on the boundary.
       157 -      Edge2 = list.map(
       158 -          (func(Q) = boundary_wall(cell(Q, 0), up)),
       159 -          0 .. HalfW),
       151 +      % Segment 2: 10 o'clock corner (0, 0) contributes up 
           + edge
       152 +      % (lower_left and upper_left already emitted in Seg1)
       153 +      Seg2 = [boundary_wall(cell(0, 0), up)],
       154    
       155 -      % Edge 3: Top-right edge from 12 to 2 o'clock 
           - (direction upper_right)
       156 -      % Q increases from HalfW to W-1 (inclusive, to include 
           - 12 o'clock corner)
       157 -      % R = Q - HalfW (on the top-right diagonal)
       158 -      % These cells have their `upper_right` edge on the 
           - boundary.
       159 -      Edge3 = list.map(
       160 -          (func(Q) = boundary_wall(cell(Q, Q - HalfW), 
           - upper_right)),
       161 -          HalfW .. (W - 1)),
       155 +      % Segment 3: Top-left edge from 10 to 12 o'clock
       156 +      % Cells (Q, 0) for Q from 1 to HalfW
       157 +      % Each cell contributes: upper_left, up
       158 +      Seg3 = list.condense(list.map(
       159 +          (func(Q) = [boundary_wall(cell(Q, 0), upper_left),
       160 +                      boundary_wall(cell(Q, 0), up)]),
       161 +          1 .. HalfW)),
       162    
       163 -      % Edge 4: Right edge from 2 to 4 o'clock (direction 
           - lower_right)
       164 -      % Q = W-1, R increases from HalfW to HalfW + H - 1
       165 -      % These cells have their `lower_right` edge on the 
           - boundary.
       166 -      Edge4 = list.map(
       167 -          (func(R) = boundary_wall(cell(W - 1, R), 
           - lower_right)),
       168 -          HalfW .. (HalfW + H - 1)),
       163 +      % Segment 4: 12 o'clock corner (HalfW, 0) contributes 
           + upper_right edge
       164 +      % (upper_left and up already emitted in Seg3)
       165 +      Seg4 = [boundary_wall(cell(HalfW, 0), upper_right)],
       166    
       167 -      % Edge 5: Bottom-right edge from 4 to 6 o'clock
           -   (direction down)
       168 -      % Q decreases from W-1 to HalfW (inclusive, to include 
           - 6 o'clock corner)
       169 -      % R = HalfW + H - 1 (constant, the bottom row for Q >= 
           - HalfW)
       170 -      % These cells have their `down` edge on the boundary.
       171 -      Edge5 = list.map(
       172 -          (func(Q) = boundary_wall(cell(Q, HalfW + H - 1), 
           - down)),
       173 -          reverse(HalfW .. (W - 1))),
       167 +      % Segment 5: Top-right edge from 12 to 2 o'clock
       168 +      % Cells (Q, Q - HalfW) for Q from HalfW+1 to W-1
       169 +      % Each cell contributes: up, upper_right
       170 +      Seg5 = list.condense(list.map(
       171 +          (func(Q) = [boundary_wall(cell(Q, Q - HalfW), up),
       172 +                      boundary_wall(cell(Q, Q - HalfW), 
           + upper_right)]),
       173 +          (HalfW + 1) .. (W - 1))),
       174    
       175 -      % Edge 6: Bottom edge from 6 to 8 o'clock (direction 
           - lower_left)
       176 -      % Q decreases from HalfW to 0
       177 -      % R = HalfW + H - 1 (bottom row)
       178 -      % Wait, that's not right. At 6 o'clock: ((W-1)/2, 
           - (W-1)/2 + H - 1)
       179 -      % At 8 o'clock: (0, H - 1)
       180 -      % So R = Q + H - 1 still holds for these cells.
       181 -      % These cells have their `lower_left` edge on the 
           - boundary.
       182 -      Edge6Lower = list.map(
       183 -          (func(Q) = boundary_wall(cell(Q, Q + H - 1), 
           - lower_left)),
       184 -          reverse(0 .. HalfW)),
       175 +      % Segment 6: 2 o'clock corner (W-1, HalfW) contributes 
           + lower_right edge
       176 +      % (up and upper_right already emitted in Seg5)
       177 +      Seg6 = [boundary_wall(cell(W - 1, HalfW), 
           + lower_right)],
       178    
       179 -      % Edge 7: Left edge from 8 o'clock back up to entrance 
           - (R decreasing)
       180 -      % Cells: (0, H-1), (0, H-2), ..., (0, StartR+1)
       181 -      % These cells have their `upper_left` edge on the 
           - boundary.
       182 -      Edge7 = list.map(
       183 -          (func(R) = boundary_wall(cell(0, R), upper_left)),
       184 -          reverse((StartR + 1) .. (H - 1))),
       179 +      % Segment 7: Right edge from 2 to 4 o'clock
       180 +      % Cells (W-1, R) for R from HalfW+1 to HalfW+H-1
       181 +      % Each cell contributes: upper_right, lower_right
       182 +      Seg7 = list.condense(list.map(
       183 +          (func(R) = [boundary_wall(cell(W - 1, R), 
           +  upper_right),
       184 +                      boundary_wall(cell(W - 1, R), 
           + lower_right)]),
       185 +          (HalfW + 1) .. (HalfW + H - 1))),
       186    
       187 -      Walls = Edge1 ++ Edge2 ++ Edge3 ++ Edge4 ++ Edge5 ++ 
           - Edge6Lower ++ Edge7.
       187 +      % Segment 8: 4 o'clock corner (W-1, HalfW+H-1) 
           + contributes down edge
       188 +      % (upper_right and lower_right already emitted in Seg7)
       189 +      Seg8 = [boundary_wall(cell(W - 1, HalfW + H - 1), 
           + down)],
       190    
       191 +      % Segment 9: Bottom-right edge from 4 to 6 o'clock
       192 +      % Cells (Q, HalfW + H - 1) for Q from W-2 down to HalfW
       193 +      % Each cell contributes: lower_right, down
       194 +      Seg9 = list.condense(list.map(
       195 +          (func(Q) = [boundary_wall(cell(Q, HalfW + H - 1), 
           + lower_right),
       196 +                      boundary_wall(cell(Q, HalfW + H - 1), 
           + down)]),
       197 +          reverse(HalfW .. (W - 2)))),
       198 +  
       199 +      % Segment 10: 6 o'clock corner (HalfW, HalfW+H-1) 
           + contributes lower_left edge
       200 +      % (lower_right and down already emitted in Seg9)
       201 +      Seg10 = [boundary_wall(cell(HalfW, HalfW + H - 1), 
           + lower_left)],
       202 +  
       203 +      % Segment 11: Bottom-left edge from 6 to 8 o'clock
       204 +      % Cells (Q, Q + H - 1) for Q from HalfW-1 down to 0
       205 +      % Each cell contributes: down, lower_left
       206 +      Seg11 = list.condense(list.map(
       207 +          (func(Q) = [boundary_wall(cell(Q, Q + H - 1), 
           + down),
       208 +                      boundary_wall(cell(Q, Q + H - 1), 
           + lower_left)]),
       209 +          reverse(0 .. (HalfW - 1)))),
       210 +  
       211 +      % Segment 12: 8 o'clock corner (0, H-1) contributes 
           + upper_left edge
       212 +      % (down and lower_left already emitted in Seg11)
       213 +      Seg12 = [boundary_wall(cell(0, H - 1), upper_left)],
       214 +  
       215 +      % Segment 13: Left edge from 8 o'clock back to entrance
       216 +      % Cells (0, R) for R from H-2 down to StartR+1
       217 +      % Each cell contributes: lower_left, upper_left
       218 +      Seg13 = list.condense(list.map(
       219 +          (func(R) = [boundary_wall(cell(0, R), lower_left),
       220 +                      boundary_wall(cell(0, R), 
           + upper_left)]),
       221 +          reverse((StartR + 1) .. (H - 2)))),
       222 +  
       223 +      Walls = Seg1 ++ Seg2 ++ Seg3 ++ Seg4 ++ Seg5 ++ Seg6 ++
           +  Seg7 ++ Seg8 ++
       224 +              Seg9 ++ Seg10 ++ Seg11 ++ Seg12 ++ Seg13.
       225 +  
       226    %----------------------------------------------------------
             -----------------%
       227    
       228    split_boundary(Walls, Entrance, Exit, Segment1, Segment2) 
             :-

● Now let me compile and test:

● Bash(mmc --make amaze 2>&1)
  ⎿  Making Mercury/int3s/boundary.int3
     Making Mercury/ints/boundary.int
     Making Mercury/cs/boundary.c
     … +22 lines (ctrl+o to expand)

● It compiled with warnings. Let me test it:

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p /tmp/hex_test.svg && cat
      /tmp/hex_test.svg)
  ⎿ <?xml version="1.0" encoding="UTF-8"?>                                 
    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 90 92">
      <polygon points="20,37.3 45,20 70,37.3 70,54.6 45,72.0 20,54.6" fill=
    … +23 lines (ctrl+o to expand)


● The boundary is now zigzagging, but there are issues - the polyline has
  some incorrect points. The problem is likely in the wall_start_point and
  wall_end_point functions in render.m which were designed for the old
  one-edge-per-cell model.

  Let me look at how these are used and think about what they should do now.
   With one boundary_wall per edge:
  - wall_start_point(wall) should return where this edge starts (clockwise)
  - wall_end_point(wall) should return where this edge ends (clockwise)

  These should be the endpoints of the edge itself. Let me check the current
   implementation:

● Read(render.m)
  ⎿  Read 130 lines

● I see the issue. The wall_start_point and wall_end_point functions were
  written with the assumption that we're tracing the boundary in a
  particular order and the "start" and "end" relate to where we're arriving
  from. But now that each boundary wall is just an edge, these should simply
   return the two endpoints of that edge.

  Let me think about each hex edge and its endpoints (tracing clockwise
  around the cell from the perspective of being outside the maze):

  - up edge: from upper-left corner (-S/2, -S*sqrt3/2) to upper-right corner
   (+S/2, -S*sqrt3/2)
  - upper_right edge: from upper-right corner (+S/2, -S*sqrt3/2) to right
  corner (+S, 0)
  - lower_right edge: from right corner (+S, 0) to lower-right corner (+S/2,
   +S*sqrt3/2)
  - down edge: from lower-right corner (+S/2, +S*sqrt3/2) to lower-left
  corner (-S/2, +S*sqrt3/2)
  - lower_left edge: from lower-left corner (-S/2, +S*sqrt3/2) to left
  corner (-S, 0)
  - upper_left edge: from left corner (-S, 0) to upper-left corner (-S/2, 
  -S*sqrt3/2)

  Wait, that's tracing clockwise around the cell itself. For the boundary,
  we're tracing clockwise around the maze, which means we're on the outside
  looking in. So the direction is reversed - we traverse each edge in the
  opposite order.

  Actually, looking at the polyline construction in
  render_boundary_polyline, it builds points like:
  StartPoint = wall_end_point(StartWall)  -- end of entrance
  CornerPoints = list.map(wall_start_point, Segment)  -- start of each wall
  in segment
  EndPoint = wall_start_point(EndWall)  -- start of exit

  This is getting confusing. Let me simplify by making wall_start_point and
  wall_end_point just return the edge endpoints based on clockwise traversal
   of the maze boundary:

● Update(render.m)
  ⎿  Updated render.m with 31 additions and 23 removals                  
       575    wall_start_point(hex, Bounds, S, M, boundary_wall(Cell, 
             Dir)) = Str :-
       576        hex_cell_centre(S, M, Bounds, Cell, CX, CY),
       577        Sf = float.float(S),
       578 -      % For clockwise traversal around the maze boundary, the
           -  "start" of each
       579 -      % boundary wall edge is where we arrive when tracing 
           - clockwise.
       580 -      % This depends on which edge of the cell is the 
           - boundary edge.
       578 +      % Return the starting point of this edge when tracing 
           + the boundary clockwise.
       579 +      % The edge is identified by the cell and the outward 
           + direction.
       580 +      % Clockwise around the maze = counter-clockwise around 
           + each cell.
       581 +      hex_edge_start_point(Sf, CX, CY, Dir, PX, PY),
       582 +      Str = string.format("%s %s", [s(format_coord(PX)), 
           + s(format_coord(PY))]).
       583 +  
       584 +      % Get the starting point of a hex edge when tracing 
           + clockwise around the maze.
       585 +      % This is counter-clockwise around the cell itself.
       586 +      %
       587 +  :- pred hex_edge_start_point(float::in, float::in, 
           + float::in, direction::in,
       588 +      float::out, float::out) is det.
       589 +  
       590 +  hex_edge_start_point(Sf, CX, CY, Dir, PX, PY) :-
       591        (
       592 -          Dir = upper_left,
       593 -          % upper_left edge on boundary: arriving from below 
           - (lower-left corner)
       594 -          PX = CX - Sf / 2.0,
       595 -          PY = CY + Sf * sqrt3 / 2.0
       596 -      ;
       592            Dir = up,
       593 -          % up edge on boundary: arriving from left 
           - (upper-left corner)
       594 -          PX = CX - Sf / 2.0,
       593 +          % up edge: start at upper-right corner (going left)
       594 +          PX = CX + Sf / 2.0,
       595            PY = CY - Sf * sqrt3 / 2.0
       596        ;
       597 -          Dir = upper_right,
       598 -          % upper_right edge on boundary: arriving from above
           -  (upper-right corner)
       599 -          PX = CX + Sf / 2.0,
       597 +          Dir = upper_left,
       598 +          % upper_left edge: start at upper-left corner 
           + (going down-left)
       599 +          PX = CX - Sf / 2.0,
       600            PY = CY - Sf * sqrt3 / 2.0
       601        ;
       602 -          Dir = lower_right,
       603 -          % lower_right edge on boundary: arriving from above
           -  (right corner)
       604 -          PX = CX + Sf,
       602 +          Dir = lower_left,
       603 +          % lower_left edge: start at left corner (going 
           + down-right)
       604 +          PX = CX - Sf,
       605            PY = CY
       606        ;
       607            Dir = down,
       608 -          % down edge on boundary: arriving from right 
           - (lower-right corner)
       609 -          PX = CX + Sf / 2.0,
       608 +          % down edge: start at lower-left corner (going 
           + right)
       609 +          PX = CX - Sf / 2.0,
       610            PY = CY + Sf * sqrt3 / 2.0
       611        ;
       612 -          Dir = lower_left,
       613 -          % lower_left edge on boundary: arriving from 
           - below-right (lower-right corner)
       612 +          Dir = lower_right,
       613 +          % lower_right edge: start at lower-right corner 
           + (going up-right)
       614            PX = CX + Sf / 2.0,
       615            PY = CY + Sf * sqrt3 / 2.0
       616        ;
       617 +          Dir = upper_right,
       618 +          % upper_right edge: start at right corner (going 
           + up-left)
       619 +          PX = CX + Sf,
       620 +          PY = CY
       621 +      ;
       622            ( Dir = left
       623            ; Dir = right
       624            ),
       625 -          % Square directions should not occur with hex 
           - topology
       625            PX = 0.0,
       626            PY = 0.0
       627 -      ),
       628 -      Str = string.format("%s %s", [s(format_coord(PX)), 
           - s(format_coord(PY))]).
       627 +      ).
       628    
       629        % Get the end point of a wall (clockwise direction).
       630        % This is the point we reach last when tracing the 
             boundary clockwise.

● Now let me update the wall_end_point for hex:

● Update(render.m)
  ⎿  Updated render.m with 30 additions and 22 removals                  
       667    wall_end_point(hex, Bounds, S, M, boundary_wall(Cell, Dir))
              = Str :-
       668        hex_cell_centre(S, M, Bounds, Cell, CX, CY),
       669        Sf = float.float(S),
       670 -      % For clockwise traversal, the "end" of each boundary 
           - wall edge
       670 +      % Return the ending point of this edge when tracing the
           +  boundary clockwise.
       671 +      hex_edge_end_point(Sf, CX, CY, Dir, PX, PY),
       672 +      Str = string.format("%s %s", [s(format_coord(PX)), 
           + s(format_coord(PY))]).
       673 +  
       674 +      % Get the ending point of a hex edge when tracing 
           + clockwise around the maze.
       675 +      % This is counter-clockwise around the cell itself.
       676 +      %
       677 +  :- pred hex_edge_end_point(float::in, float::in, float::in,
           +  direction::in,
       678 +      float::out, float::out) is det.
       679 +  
       680 +  hex_edge_end_point(Sf, CX, CY, Dir, PX, PY) :-
       681        (
       682 -          Dir = upper_left,
       683 -          % upper_left edge: left corner
       684 -          PX = CX - Sf,
       685 -          PY = CY
       686 -      ;
       682            Dir = up,
       683 -          % up edge: upper-right corner
       684 -          PX = CX + Sf / 2.0,
       683 +          % up edge: end at upper-left corner
       684 +          PX = CX - Sf / 2.0,
       685            PY = CY - Sf * sqrt3 / 2.0
       686        ;
       687 -          Dir = upper_right,
       688 -          % upper_right edge: right corner
       689 -          PX = CX + Sf,
       687 +          Dir = upper_left,
       688 +          % upper_left edge: end at left corner
       689 +          PX = CX - Sf,
       690            PY = CY
       691        ;
       692 -          Dir = lower_right,
       693 -          % lower_right edge: lower-right corner
       694 -          PX = CX + Sf / 2.0,
       692 +          Dir = lower_left,
       693 +          % lower_left edge: end at lower-left corner
       694 +          PX = CX - Sf / 2.0,
       695            PY = CY + Sf * sqrt3 / 2.0
       696        ;
       697            Dir = down,
       698 -          % down edge: lower-left corner
       699 -          PX = CX - Sf / 2.0,
       698 +          % down edge: end at lower-right corner
       699 +          PX = CX + Sf / 2.0,
       700            PY = CY + Sf * sqrt3 / 2.0
       701        ;
       702 -          Dir = lower_left,
       703 -          % lower_left edge: left corner
       704 -          PX = CX - Sf,
       702 +          Dir = lower_right,
       703 +          % lower_right edge: end at right corner
       704 +          PX = CX + Sf,
       705            PY = CY
       706        ;
       707 +          Dir = upper_right,
       708 +          % upper_right edge: end at upper-right corner
       709 +          PX = CX + Sf / 2.0,
       710 +          PY = CY - Sf * sqrt3 / 2.0
       711 +      ;
       712            ( Dir = left
       713            ; Dir = right
       714            ),
       715 -          % Square directions should not occur with hex 
           - topology
       715            PX = 0.0,
       716            PY = 0.0
       717 -      ),
       718 -      Str = string.format("%s %s", [s(format_coord(PX)), 
           - s(format_coord(PY))]).
       717 +      ).
       718    
       719    %----------------------------------------------------------
             -----------------%
       720    

● Now let me compile and test:

● Bash(mmc --make amaze 2>&1)
  ⎿  Making Mercury/int3s/render.int3
     Making Mercury/ints/render.int
     Making Mercury/cs/render.c
     … +2 lines (ctrl+o to expand)

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p /tmp/hex_test.svg && cat
      /tmp/hex_test.svg)
  ⎿ <?xml version="1.0" encoding="UTF-8"?>                                 
    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 90 92">
      <polygon points="20,37.3 45,20 70,37.3 70,54.6 45,72.0 20,54.6" fill=
    … +23 lines (ctrl+o to expand)


● The boundary is now zigzagging properly! Looking at the first polyline, it
   traces the boundary with the characteristic hex zigzag pattern. But
  there's an issue - the second polyline has 20 54.6 and 25 46.0,20 54.6
  which doesn't look right (there's a weird backtrack at the end).

  Let me look at a simpler test to verify - let me dump the boundary walls
  to see if they're correct:

● Bash(./amaze -r 3 -s 3x2 -t hex -a dfs -p -d 2>&1 | head -40)
  ⎿  Maze:                         
       Topology: hex
       Size: 3x2
     … +14 lines (ctrl+o to expand)

● Good - the entrance is now (0,1) lower_left which is the first edge of
  cell (0,1) when tracing clockwise. The exit is (2,1) lower_right.

  Looking at the SVG output again, I see the issue with the second polyline
  ending with 25 46.0,20 54.6 - there's a backtrack. This suggests something
   is wrong with how we're constructing the polyline points or with the
  boundary wall order.

  Let me check the render_boundary_polyline logic:

● Read(render.m)
  ⎿  Read 45 lines

● I see - the polyline construction uses:
  1. StartPoint = wall_end_point(StartWall) - where the gap ends
  2. CornerPoints = list.map(wall_start_point, Segment) - start of each edge
   in segment
  3. EndPoint = wall_start_point(EndWall) - where the next gap begins

  But with the new edge-per-wall model, consecutive edges share a vertex. So
   we shouldn't need to add both start and end points for the segment - just
   the start of each edge should be enough because the end of each edge is
  the start of the next edge.

  Actually, looking more carefully, the issue is that for a continuous path
  we need:
  - Start at the end of the entrance edge (where the gap ends)
  - Then for each edge in the segment, we trace from start to end
  - End at the start of the exit edge (where the next gap begins)

  But since consecutive edges share vertices, we only need to output one
  point per edge - the start point. The last edge's end point is the start
  of the exit edge.

  Wait, actually the current approach is almost right. Let me trace through
  what should happen:

  For the first polyline (from entrance to exit):
  - StartPoint = wall_end_point(Entrance) - where the entrance gap ends
  - For each wall in Segment: wall_start_point(wall) - the start of each
  edge
  - EndPoint = wall_start_point(Exit) - where the exit gap begins

  This should work because:
  - We start at the end of the entrance edge
  - Each segment edge's start point is where we arrive
  - We end at the start of the exit edge

  The problem might be that consecutive edges in the segment share the
  endpoint/startpoint, so we're duplicating points. Let me check by
  examining the actual coordinates more carefully.

  Actually, I realize there may be an issue in the boundary generation
  itself. Let me check if consecutive walls properly share vertices. For
  example, after (0, 1) lower_left comes (0, 1) upper_left - the end of
  lower_left should equal the start of upper_left.

  For cell (0, 1):
  - lower_left edge end: left corner = (CX - S, CY)
  - upper_left edge start: upper-left corner = (CX - S/2, CY - S*sqrt3/2)

  These are different points - the end of lower_left is NOT the same as the
  start of upper_left. This is wrong!

  The issue is my understanding of the edge traversal order. When tracing
  clockwise around the maze (counter-clockwise around each cell), after the
  lower_left edge we should be at the left corner, and the next edge should
  start there. But upper_left starts at the upper-left corner, not the left
  corner.

  Let me reconsider the hex cell edge order when going counter-clockwise
  around the cell:
  - Starting from lower-left corner, going counter-clockwise:
    - down edge: lower-left to lower-right
    - lower_right edge: lower-right to right
    - upper_right edge: right to upper-right
    - up edge: upper-right to upper-left
    - upper_left edge: upper-left to left
    - lower_left edge: left to lower-left

  So after lower_left (ends at lower-left corner), the next edge
  counter-clockwise is down (starts at lower-left corner). But in the
  boundary, we're emitting lower_left then upper_left for the same cell,
  which isn't continuous!

  The problem is that we're emitting edges in the wrong order within each
  cell. For a cell on the left edge of the maze (pointing upper_left and
  lower_left), when tracing clockwise around the maze:
  - We approach from below (from the cell at lower R)
  - First boundary edge is lower_left (the edge below the cell)
  - Second boundary edge is upper_left (the edge above the cell)

  Wait, but these edges aren't adjacent on the cell! They're on opposite
  sides of the left corner.

  I think the issue is that lower_left and upper_left for the same cell are
  on opposite sides of the cell's left corner, not adjacent edges. So for a
  cell on the left boundary, both these edges are on the boundary, but
  there's a corner (the left corner) between them, not a direct connection.

  Hmm, but looking at a flat-topped hexagon, the lower_left edge goes from
  left corner to lower-left corner, and upper_left goes from upper-left
  corner to left corner. So they DO share the left corner! But my
  hex_edge_start_point and hex_edge_end_point are wrong.

  Let me reconsider. For clockwise around the maze (counter-clockwise around
   each cell):
  - lower_left edge: from lower-left corner → left corner (going
  counter-clockwise)
  - upper_left edge: from left corner → upper-left corner (going
  counter-clockwise)

  So lower_left END = left corner, and upper_left START = left corner. Let
  me fix the functions:

● Update(render.m)
  ⎿  Updated render.m with 43 additions and 18 removals                  
       588        float::out, float::out) is det.
       589    
       590    hex_edge_start_point(Sf, CX, CY, Dir, PX, PY) :-
       591 +      % When tracing clockwise around the maze, we traverse 
           + each cell's boundary
       592 +      % edges counter-clockwise. The start point is where we 
           + enter the edge.
       593 +      %
       594 +      % Hex corners (relative to centre, with right = 0 
           + degrees):
       595 +      %   right:       (+S, 0)
       596 +      %   upper-right: (+S/2, -S*sqrt3/2)
       597 +      %   upper-left:  (-S/2, -S*sqrt3/2)
       598 +      %   left:        (-S, 0)
       599 +      %   lower-left:  (-S/2, +S*sqrt3/2)
       600 +      %   lower-right: (+S/2, +S*sqrt3/2)
       601 +      %
       602 +      % Edge traversal (counter-clockwise around cell = 
           + clockwise around maze):
       603 +      %   down:        lower-left → lower-right
       604 +      %   lower_right: lower-right → right
       605 +      %   upper_right: right → upper-right
       606 +      %   up:          upper-right → upper-left
       607 +      %   upper_left:  upper-left → left  [wait, should be: 
           + left → upper-left]
       608 +      %   lower_left:  left → lower-left  [wait, should be: 
           + lower-left → left]
       609 +      %
       610 +      % Actually, going counter-clockwise around the cell:
       611 +      %   From lower-left corner, the edges in order are:
       612 +      %   lower_left (lower-left → left), upper_left (left → 
           + upper-left),
       613 +      %   up (upper-left → upper-right), upper_right 
           + (upper-right → right),
       614 +      %   lower_right (right → lower-right), down 
           + (lower-right → lower-left)
       615 +      %
       616        (
       617 -          Dir = up,
       618 -          % up edge: start at upper-right corner (going left)
       619 -          PX = CX + Sf / 2.0,
       620 -          PY = CY - Sf * sqrt3 / 2.0
       621 -      ;
       622 -          Dir = upper_left,
       623 -          % upper_left edge: start at upper-left corner 
           - (going down-left)
       617 +          Dir = lower_left,
       618 +          % lower_left edge: lower-left corner → left corner
       619            PX = CX - Sf / 2.0,
       620 -          PY = CY - Sf * sqrt3 / 2.0
       620 +          PY = CY + Sf * sqrt3 / 2.0
       621        ;
       622 -          Dir = lower_left,
       623 -          % lower_left edge: start at left corner (going 
           - down-right)
       622 +          Dir = upper_left,
       623 +          % upper_left edge: left corner → upper-left corner
       624            PX = CX - Sf,
       625            PY = CY
       626        ;
       627 -          Dir = down,
       628 -          % down edge: start at lower-left corner (going 
           -  right)
       627 +          Dir = up,
       628 +          % up edge: upper-left corner → upper-right corner
       629            PX = CX - Sf / 2.0,
       630 -          PY = CY + Sf * sqrt3 / 2.0
       630 +          PY = CY - Sf * sqrt3 / 2.0
       631        ;
       632 -          Dir = lower_right,
       633 -          % lower_right edge: start at lower-right corner 
           - (going up-right)
       632 +          Dir = upper_right,
       633 +          % upper_right edge: upper-right corner → right 
           + corner
       634            PX = CX + Sf / 2.0,
       635 -          PY = CY + Sf * sqrt3 / 2.0
       635 +          PY = CY - Sf * sqrt3 / 2.0
       636        ;
       637 -          Dir = upper_right,
       638 -          % upper_right edge: start at right corner (going 
           - up-left)
       637 +          Dir = lower_right,
       638 +          % lower_right edge: right corner → lower-right 
           + corner
       639            PX = CX + Sf,
       640            PY = CY
       641        ;
       642 +          Dir = down,
       643 +          % down edge: lower-right corner → lower-left corner
       644 +          PX = CX + Sf / 2.0,
       645 +          PY = CY + Sf * sqrt3 / 2.0
       646 +      ;
       647            ( Dir = left
       648            ; Dir = right
       649            ),

● Update(render.m)
  ⎿  Updated render.m with 25 additions and 18 removals                
       703        float::out, float::out) is det.
       704    
       705    hex_edge_end_point(Sf, CX, CY, Dir, PX, PY) :-
       706 +      % End points matching the start points 
           + (counter-clockwise around cell):
       707 +      %   lower_left: lower-left → left (end = left)
       708 +      %   upper_left: left → upper-left (end = upper-left)
       709 +      %   up: upper-left → upper-right (end = upper-right)
       710 +      %   upper_right: upper-right → right (end = right)
       711 +      %   lower_right: right → lower-right (end = 
           + lower-right)
       712 +      %   down: lower-right → lower-left (end = lower-left)
       713        (
       714 -          Dir = up,
       715 -          % up edge: end at upper-left corner
       716 -          PX = CX - Sf / 2.0,
       717 -          PY = CY - Sf * sqrt3 / 2.0
       718 -      ;
       719 -          Dir = upper_left,
       720 -          % upper_left edge: end at left corner
       714 +          Dir = lower_left,
       715 +          % lower_left edge: end at left corner
       716            PX = CX - Sf,
       717            PY = CY
       718        ;
       719 -          Dir = lower_left,
       720 -          % lower_left edge: end at lower-left corner
       719 +          Dir = upper_left,
       720 +          % upper_left edge: end at upper-left corner
       721            PX = CX - Sf / 2.0,
       722 -          PY = CY + Sf * sqrt3 / 2.0
       722 +          PY = CY - Sf * sqrt3 / 2.0
       723        ;
       724 -          Dir = down,
       725 -          % down edge: end at lower-right corner
       724 +          Dir = up,
       725 +          % up edge: end at upper-right corner
       726            PX = CX + Sf / 2.0,
       727 -          PY = CY + Sf * sqrt3 / 2.0
       727 +          PY = CY - Sf * sqrt3 / 2.0
       728        ;
       729 -          Dir = lower_right,
       730 -          % lower_right edge: end at right corner
       729 +          Dir = upper_right,
       730 +          % upper_right edge: end at right corner
       731            PX = CX + Sf,
       732            PY = CY
       733        ;
       734 -          Dir = upper_right,
       735 -          % upper_right edge: end at upper-right corner
       734 +          Dir = lower_right,
       735 +          % lower_right edge: end at lower-right corner
       736            PX = CX + Sf / 2.0,
       737 -          PY = CY - Sf * sqrt3 / 2.0
       737 +          PY = CY + Sf * sqrt3 / 2.0
       738        ;
       739 +          Dir = down,
       740 +          % down edge: end at lower-left corner
       741 +          PX = CX - Sf / 2.0,
       742 +          PY = CY + Sf * sqrt3 / 2.0
       743 +      ;
       744            ( Dir = left
       745            ; Dir = right
       746            ),

● Bash(mmc --make amaze 2>&1 && ./amaze -r 3 -s 3x2 -t hex -a dfs -p
      /tmp/hex_test.svg && cat /tmp/hex_test.svg)
  ⎿  Making Mercury/int3s/render.int3
     Making Mercury/ints/render.int
     Making Mercury/cs/render.c
     … +28 lines (ctrl+o to expand)

● Still some issues - there's 20 54.6,20 54.6 (duplicate point) and the
  second polyline ends with 20 54.6,25 63.3 which looks wrong.

  The issue might be in the boundary wall generation order. Let me check
  what walls we're producing for this maze by adding some debug output.
  Actually, let me think more carefully about the edge order within each
  cell.

  Looking at the first polyline, the boundary should go:
  - From entrance (0,1) lower_left edge end → around → to exit (2,1)
  lower_right edge start

  Wait, I think there's confusion between which edge is "lower_left"
  geometrically versus how we're interpreting it.

  Let me think about this differently. A flat-topped hexagon has 6 edges:
  - Top edge (horizontal, at top)
  - Upper-right edge (diagonal, going down-right from top-right corner)
  - Lower-right edge (diagonal, going down-left from right corner)
  - Bottom edge (horizontal, at bottom)
  - Lower-left edge (diagonal, going up-left from bottom-left corner)
  - Upper-left edge (diagonal, going up-right from left corner)

  Each edge is identified by the direction to the adjacent cell. The
  upper_left direction points to the cell that's up and to the left, so the
  edge in that direction is the upper-left edge of the hexagon.

  Now, when tracing the boundary clockwise (counter-clockwise around each
  cell), starting from the lower-left corner of a cell:
  1. Lower-left edge: from lower-left corner → left corner
  2. Upper-left edge: from left corner → upper-left corner
  3. Top edge (up): from upper-left corner → upper-right corner
  4. Upper-right edge: from upper-right corner → right corner
  5. Lower-right edge: from right corner → lower-right corner
  6. Bottom edge (down): from lower-right corner → lower-left corner

  So for cell (0, 1) on the left edge of the maze, we emit lower_left then
  upper_left. The end of lower_left (left corner) should equal the start of
  upper_left (left corner). Let me verify my formulas are consistent:

  Cell (0, 1) has centre at some (CX, CY).
  - lower_left end: left corner = (CX - S, CY) ✓
  - upper_left start: left corner = (CX - S, CY) ✓

  Good, those match now! Let me trace through more carefully what the issue
  might be. The duplicate point 20 54.6,20 54.6 suggests that the end of one
   edge equals the start of the next, which is what we want, but we're
  outputting both.

  Looking at render_boundary_polyline:
  StartPoint = wall_end_point(StartWall)  // entrance edge end
  CornerPoints = list.map(wall_start_point, Segment)  // start of each
  segment edge
  EndPoint = wall_start_point(EndWall)  // exit edge start
  AllPoints = [StartPoint | CornerPoints] ++ [EndPoint]

  If StartPoint equals the first CornerPoint, we'd get a duplicate. Let me
  think about this...

  The entrance is (0,1) lower_left. The wall_end_point of this is the left
  corner of cell (0,1).

  What's the first edge in Segment1? It should be (0,1) upper_left - and its
   start point is also the left corner of cell (0,1)!

  So yes, we're duplicating the vertex. The fix is that we shouldn't include
   both - the end of the entrance edge and the start of the first segment
  edge are the same point.

  Let me simplify the polyline construction:

● Update(render.m)
  ⎿  Updated render.m with 16 additions and 6 removals                   
       513            StartWall, Segment, EndWall, !IO) :-
       514        S = RenderOpts ^ ro_cell_spacing,
       515        M = RenderOpts ^ ro_margin,
       516 -      % Start point: the "end" corner of the start wall 
           - (clockwise direction)
       516 +      % The polyline traces from the end of the StartWall 
           + (where its gap ends),
       517 +      % through all edges in Segment, to the start of EndWall
           +  (where its gap begins).
       518 +      %
       519 +      % Since consecutive edges share vertices, we only need:
       520 +      % - Start point: end of StartWall
       521 +      % - For each edge in Segment: its end point (which is 
           + start of next edge)
       522 +      % - Note: start of first segment edge = end of 
           + StartWall (already included)
       523 +      % - Note: end of last segment edge = start of EndWall
       524 +      %
       525 +      % So we output: end(StartWall), then end of each 
           + segment edge.
       526 +      % The last segment edge's end = start(EndWall), so we 
           + don't need EndPoint separately.
       527 +      %
       528        StartPoint = wall_end_point(Topology, Bounds, S, M, 
             StartWall),
       529 -      % End point: the "start" corner of the end wall 
           - (clockwise direction)
       530 -      EndPoint = wall_start_point(Topology, Bounds, S, M, 
           - EndWall),
       531 -      % Corner points from the segment walls
       532 -      CornerPoints = list.map(wall_corner_point(Topology, 
           - Bounds, S, M), Segment),
       533 -      AllPoints = [StartPoint | CornerPoints] ++ [EndPoint],
       529 +      SegmentEndPoints = list.map(
       530 +          (func(W) = wall_end_point(Topology, Bounds, S, M, 
           + W)),
       531 +          Segment),
       532 +      AllPoints = [StartPoint | SegmentEndPoints],
       533        PointsStr = string.join_list(",", AllPoints),
       534        io.format(Stream,
       535            "  <polyline points=\"%s\" fill=\"none\" 
             stroke=\"%s\" stroke-width=\"%d\" stroke-linecap=\"round\" 
             stroke-linejoin=\"round\"/>\n",

● Bash(mmc --make amaze 2>&1 && ./amaze -r 3 -s 3x2 -t hex -a dfs -p
      /tmp/hex_test.svg && cat /tmp/hex_test.svg)
  ⎿  Making Mercury/int3s/render.int3
     Making Mercury/ints/render.int
     Making Mercury/cs/render.c
     … +31 lines (ctrl+o to expand)

● No more duplicate points. Let me copy this SVG to see it visually and
  verify it looks correct. But first, let me rename the unused parameter to
  suppress the warning:

● Update(render.m)
  ⎿  Updated render.m with 1 addition and 1 removal                   
       510        io::di, io::uo) is det.
       511    
       512    render_boundary_polyline(Stream, RenderOpts, Topology, 
             Bounds,
       513 -          StartWall, Segment, EndWall, !IO) :-
       513 +          StartWall, Segment, _EndWall, !IO) :-
       514        S = RenderOpts ^ ro_cell_spacing,
       515        M = RenderOpts ^ ro_margin,
       516        % The polyline traces from the end of the StartWall 
             (where its gap ends),

● Let me test with a larger hex maze to make sure it looks correct, and also
   test a square maze to make sure we didn't break that:

● Bash(mmc --make amaze 2>&1 | grep -v "^Making")
  ⎿  No matches found

● Bash(./amaze -r 42 -s 5 -t square -a dfs -p /tmp/sq_test.svg && echo "Square
      OK")
  ⎿  Square OK

● Bash(./amaze -r 42 -s 5x4 -t hex -a kruskal -p /tmp/hex_large.svg && echo "Hex
       large OK")
  ⎿  Hex large OK

● The boundary wall generation has been redesigned to emit one entry per
  edge. Here's a summary of the changes:

  1. boundary.m - hex_boundary_walls now emits one boundary_wall per
  boundary edge, not per cell:
    - Cells on straight edges contribute 2 boundary walls (e.g., lower_left
  then upper_left for left-edge cells)
    - Corner cells contribute 3 boundary walls (the third being where the
  corner turns)
    - The order is carefully constructed so consecutive edges share vertices
  2. render.m - Updated hex edge start/end points to properly trace
  counter-clockwise around each cell:
    - lower_left: lower-left corner → left corner
    - upper_left: left corner → upper-left corner
    - up: upper-left corner → upper-right corner
    - upper_right: upper-right corner → right corner
    - lower_right: right corner → lower-right corner
    - down: lower-right corner → lower-left corner
  3. render.m - Simplified render_boundary_polyline to avoid duplicate
  points by outputting end(StartWall) followed by end of each segment edge.

> Looks fantastic! There is just one more change I would like to make, which
 is in how we interpret a single value for the size option. For square grids
 it makes sense to use NxN, but for hex grids it might be better to have H =
 N and W = (H * 2) - 1. Can you update the plan and code to do this instead 
of what we currently do? 

● That makes sense - for hex grids, a single size value N would give a more
  regular hexagon shape with H = N and W = 2*N - 1, rather than the NxN
  interpretation which works for squares.

● Read(PLAN.md)
  ⎿  Read 25 lines

● Update(PLAN.md)
  ⎿  Updated PLAN.md with 2 additions                                   
       142    
       143    - `-t, --topology <square|hex>` - Grid topology (default: 
             square)
       144    - `-s, --size <N|WxH>` - Maze size (default: 10)
       145 +    - For square grids: `N` means `NxN`
       146 +    - For hex grids: `N` means `W = 2*N - 1`, `H = N` 
           + (regular hexagon shape)
       147    - `-r, --random-seed <number>` - Random number seed 
             (default: derived from system time)
       148    - `-e, --entrance-offset <number>` - Entrance offset from 
             centre (default: 0)
       149    - `-x, --exit-offset <number>` - Exit offset from centre 
             (default: same as entrance offset)

● Now let me update the options.m. The issue is that at the point where we
  parse the size, we might not yet know the topology. Let me check the order
   of processing:

● Read(options.m)
  ⎿  Read 50 lines

● Good - the topology is parsed before the size, so we have access to it. I
  need to modify the size parsing to take the topology into account. Let me
  also look at parse_size:

● Read(options.m)
  ⎿  Read 25 lines

● I'll modify parse_size to take the topology as a parameter:

● Update(options.m)
  ⎿  Updated options.m with 14 additions and 5 removals               
       314    algorithm_name(kruskal) = "kruskal".
       315    algorithm_name(prim) = "prim".
       316    
       317 -  :- pred parse_size(
           -  string::in, int::out, int::out) is semidet.
       317 +  :- pred parse_size(topology::in, 
           +  string::in, int::out, int::out) is semidet.
       318    
       319 -  parse_size(Str, Width, Height) :-
       319 +  parse_size(Topology, Str, Width, Height) :-
       320        ( if string.to_int(Str, N) then
       321 -          Width = N,
       322 -          Height = N,
       323 -          N > 0
       321 +          N > 0,
       322 +          (
       323 +              Topology = square,
       324 +              % For square grids: N means NxN
       325 +              Width = N,
       326 +              Height = N
       327 +          ;
       328 +              Topology = hex,
       329 +              % For hex grids: N means W = 2*N - 1, H = N 
           + (regular hexagon shape)
       330 +              Width = 2 * N - 1,
       331 +              Height = N
       332 +          )
       333        else if
       334            [WStr, HStr] = string.split_at_char('x', Str),
       335            string.to_int(WStr, W),

● Now I need to update the call site to pass the topology:

● Update(options.m)
  ⎿  Updated options.m with 10 additions and 4 removals                
       184                )
       185            ),
       186    
       187 -          % Process size option (default 10x10)
       187 +          % Process size option (default 
           +  10, interpreted based on topology)
       188            (
       189                MaybeSizeStr = no,
       190 -              Width = 10,
       191 -              Height = 10,
       190 +              ( if parse_size(Topology, "10", W0, H0) then
       191 +                  Width = W0,
       192 +                  Height = H0
       193 +              else
       194 +                  % Should never fail for "10"
       195 +                  Width = 10,
       196 +                  Height = 10
       197 +              ),
       198                SizeOk = yes
       199            ;
       200                MaybeSizeStr = yes(SizeStr),
       201 -              ( if parse_size(SizeStr, W, H) then
       201 +              ( if parse_size(Topology, SizeStr, W, H) then
       202                    Width = W,
       203                    Height = H,
       204                    SizeOk = yes

● Now let me compile and test:

● Bash(mmc --make amaze 2>&1)
  ⎿  Making Mercury/int3s/options.int3
     Making Mercury/ints/options.int
     Making Mercury/cs/options.c
     … +2 lines (ctrl+o to expand)

● Bash(echo "=== Hex size 3 (should be 5x3) ===" && ./amaze -t hex -s 3 -d 2>&1
      | head -5
      echo ""…)
  ⎿  Error: Exit code 1
     head: cannot open 'echo' for reading: No such file or directory
     head: cannot open '' for reading: No such file or directory
     head: cannot open 'echo' for reading: No such file or directory
     head: cannot open '=== Hex size 5 (should be 9x5) ===' for reading: No such 
     file or directory

     === Hex size 3 (should be 5x3) ===


● Bash(./amaze -t hex -s 3 -d 2>&1 | head -5)
  ⎿  Maze:                        
       Topology: hex
       Size: 5x3
     … +2 lines (ctrl+o to expand)

● Bash(./amaze -t hex -s 5 -d 2>&1 | head -5)
  ⎿  Maze:                        
       Topology: hex
       Size: 9x5
     … +2 lines (ctrl+o to expand)

● Bash(./amaze -t square -s 5 -d 2>&1 | head -5)
  ⎿  Maze:                        
       Topology: square
       Size: 5x5
     … +2 lines (ctrl+o to expand)

● Bash(./amaze -t hex -s 3x2 -d 2>&1 | head -5)
  ⎿  Maze:                        
       Topology: hex
       Size: 3x2
     … +2 lines (ctrl+o to expand)

● All tests pass:
  - Hex size 3 → 5x3 (W = 2*3-1 = 5, H = 3) ✓
  - Hex size 5 → 9x5 (W = 2*5-1 = 9, H = 5) ✓
  - Square size 5 → 5x5 ✓
  - Explicit 3x2 still works ✓

● Done! The changes are:

  1. PLAN.md - Updated the size option documentation to explain the
  different interpretations:
    - For square grids: N means NxN
    - For hex grids: N means W = 2*N - 1, H = N (regular hexagon shape)
  2. options.m - Modified parse_size to take the topology as a parameter and
   interpret single-value sizes accordingly:
    - Square: N → NxN
    - Hex: N → (2*N-1) x N
    - Explicit WxH still works for both topologies

> Let's update the plan to check off the remaining boxes in iteration 6 and 
mark it as complete. 

● Read(PLAN.md)
  ⎿  Read 41 lines

● Update(PLAN.md)
  ⎿  Updated PLAN.md with 16 additions and 14 removals                   
       1017    - [x] Binary Tree Algorithm
       1018    - [x] Sidewinder Algorithm
       1019    
       1020 -  ### Iteration 6
       1020 +  ### Iteration 6 (Complete)
       1021    
       1022    **Goals:**
       1023    
     ...
       1034    
       1035    **Implementation Tasks:**
       1036    
       1037 -  - [ ] Update grid.m: change hex adjacency so R increases 
            - `down` (currently increases `up`)
       1038 -  - [ ] Update maze.m: implement `in_maze` for hex bounds
       1039 -  - [ 
            -  ] Update maze.m: implement `cells` for hex (enumerate all 
            -  cells in bounds)
       1040 -  - [ ] Update maze.m: implement `internal_edges` for hex
       1041 -  - [ ] Implement hex_boundary_walls in boundary.m
       1042 -  - [ 
            -  ] Implement hex coordinate geometry helpers in render.m
       1043 -  - [ 
            -  ] Update render_cell_background for hex (draw boundary 
            -  polygon)
       1044 -  - [ ] Update edge_endpoints for hex
       1045 -  - [ ] Update wall_start_point / wall_end_point for hex
       1046 -  - [ ] Update cell_centre for hex
       1047 -  - [ ] Update viewbox calculation for hex
       1048 -  - [ 
            -  ] Test with topology-agnostic algorithms (DFS, Kruskal, 
            -  Prim, Aldous-Broder, Wilson, Hunt-and-Kill)
       1049 -  - [ 
            -  ] Add error handling for Binary Tree and Sidewinder with 
            -  hex topology
       1037 +  - [x] Update grid.m: hex adjacency with basis vectors 
            + (upper_right, down)
       1038 +  - [x] Update maze.m: implement `in_maze` for hex bounds
       1039 +  - [x
            +  ] Update maze.m: implement `cells` for hex (enumerate all 
            +  cells in bounds)
       1040 +  - [x] Update maze.m: implement `internal_edges` for hex
       1041 +  - [x] Implement hex_boundary_walls in boundary.m
            +   (one entry per edge, not per cell)
       1042 +  - [x
            +  ] Implement hex coordinate geometry helpers in render.m
       1043 +  - [x
            +  ] Update render_cell_background for hex (draw boundary 
            +  polygon)
       1044 +  - [x] Update edge_endpoints for hex
       1045 +  - [x] Update wall_start_point / wall_end_point for hex
       1046 +  - [x] Update cell_centre for hex
       1047 +  - [x] Update viewbox calculation for hex
       1048 +  - [x
            +  ] Test with topology-agnostic algorithms (DFS, Kruskal, 
            +  Prim, Aldous-Broder, Wilson, Hunt-and-Kill)
       1049 +  - [x
            +  ] Add error handling for Binary Tree and Sidewinder with 
            +  hex topology
       1050 +  - [x] Add direction_delta to grid.m for consistent basis 
            + handling
       1051 +  - [x] Update size option interpretation for hex (N means 
            + W=2N-1, H=N)

● Iteration 6 is now marked as complete. All the hex grid support has been
  implemented:

  - Hex coordinate system with (upper_right, down) basis vectors
  - Hex bounds checking and cell enumeration
  - Hex boundary wall generation (one edge per entry for proper zigzag
  rendering)
  - Hex rendering: cell backgrounds, internal walls, boundary walls,
  solution paths
  - Algorithm compatibility (6 algorithms work with hex, 2 rejected with
  error)
  - direction_delta for consistent basis handling across modules
  - Size option interpretation (-s N gives regular hexagon shape for hex
  grids)

> /context 
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     ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁   ⛁ System tools: 14.6k tokens (7.3%)
     ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁   ⛁ Messages: 119.3k tokens (59.6%)
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     ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛁ ⛀ ⛶ ⛶   ⛝ Autocompact buffer: 45.0k tokens (22.5%)
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     ⛝ ⛝ ⛝ ⛝ ⛝ ⛝ ⛝ ⛝ ⛝ ⛝ 

     SlashCommand Tool · 0 commands
     └ Total: 864 tokens

     

> I have staged our changes to implement iteration 6. Please go ahead and 
commit with a brief message. 

● Bash(git status)
  ⎿  On branch master                                   
     Changes to be committed:
       (use "git restore --staged <file>..." to unstage)
     … +13 lines (ctrl+o to expand)

● Bash(git diff --cached --stat)
  ⎿   PLAN.md                  |  57 +++--      
      amaze.m                  |   3 +-
      boundary.m               | 118 ++++++++++-
     … +8 lines (ctrl+o to expand)

● Bash(git log --oneline -3)
  ⎿  a8fbcea Transcript of the sixth conversation
     a5c9901 Transcript of the fifth conversation
     3e698fe Plan iteration 6: hex grid support

● Bash(git commit -m "Add hexagonal grid topology support (iteration 6)…)
  ⎿  [master cb64457] Add hexagonal grid topology support (iteration 6)
      9 files changed, 754 insertions(+), 114 deletions(-)

● Bash(git status)
  ⎿  On branch master                                                
     Untracked files:
       (use "git add <file>..." to include in what will be committed)
     … +4 lines (ctrl+o to expand)

● Done! Committed iteration 6 with 754 lines added across 9 files.



