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Computationally solving "Easy Cube" and "Soma Cube" games / puzzles

Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

Easy Cube is a reflexion game from Japan. It consists of a set of 7 pieces which are "tetris"-like (each piece is a simple geometrical shape made from 3,4 or 5 cubes), and a collection of 48 problems, where the goal is to use the pieces to fill a given shape (in 2d, or in 3d for the hardest problems). The game box The 7 pieces 2 examples of problems It is pleasant to play, with the difficulty slowly increasing whit the hardest problems. I decided to try to solve it using Mathematica. I did not use any advanced mathematics for this (which anyway I would not know :-)), and I wanted a method general enough so it could solve all the problems without any specific adpatation. I went for a purely random method, where the program places a first piece at random, then try to put a second piece, etc. When it fails to add a piece after a fixed number of attemps, it simply starts the whole process again. So this program does not learn anything... It just tries many many different possibilities, until it finds a solution ! Using Wolfram language, the code to make it and to vizualize to result was quite simple and effective (I show the code details at the end of this post for people interested). In particular, going from the 2d case to the 3d case needed very small changes only. The two followings animations show the program in action for a 2d problem and a 3d problem The number of attempts needed to reach a solution is of course random, and depends on the difficulty of the problem. Typically, it takes between a few hundreds and a few thousands iterations, which takes between a few seconds to 1-2 minutes. Here a sample of 4 solutions found for a 2d problem : And 2 solutions from a 3d problem : Finally, I considered it if was possible to use this method to find all solutions for a given problem (the number of different solutions is an info given on the description of each problem). In principle, because the method is random, it is not well suited for this (one can never be sure that all solutions have been found). However, since the method is quite fast, I decided to try anyway. For a given problem, I computed 10000 solutions. Here is a plot of the number of attempts needed for each solution, and a histogram of this: On average, finding a solution to this problem requested 2882 attempts... From the 10000 solutions, one wants only different solutions. This is easily done with the DeleteDuplicates command: SolgridQ26ListMReduced = DeleteDuplicates@SolgridQ26ListM; Length@SolgridQ26ListMReduced 362 which gives 362 different solutions. However, since this problem geometry has a simple central symmetry, one also want to get rid of "symmetric copies" of the solutions. Defining the symmetry operation as the GsymGrid function, this is again done with a DeleteDuplicates command : SolgridQ26ListMReducedSym = DeleteDuplicates[ SolgridQ26ListMReduced, #2 == GSymGrid[#1] &]; Length@SolgridQ26ListMReducedSym 188 which gives 188 unique solutions. This is precisely the number which is given in the problem description ! So the method was able to find all solutions. Here is plot with the 188 solutions: Details of the code Definition of the pieces (2d case). Each piece is a simple list of the points make the piece. When then construct, for each piece, a list of copies of it with all possible orientations. Pieces Definitions PTriangle = {{0, 0}, {0, 1}, {1, 0}}; PSquare = {{0, 0}, {0, 1}, {1, 0}, {1, 1}}; PLine = {{0, 0}, {0, 1}, {0, 2}}; PL = {{0, 0}, {0, 1}, {1, 0}, {2, 0}}; PPodium = {{0, 0}, {0, 1}, {-1, 0}, {1, 0}}; PSnake = {{0, 0}, {0, 1}, {1, 0}, {-1, 1}}; PBulky = {{0, 0}, {1, 0}, {0, 1}, {1, 1}, {-1, 1}}; (* Rotation by angle \[Pi]/2 matrix) Mrot = {{0, 1}, {-1, 0}}; ( Reflection by y-axis matrix ) Mref = {{-1, 0}, {0, 1}}; MUnit = {{1, 0}, {0, 1}}; (Operators which perform a given number of rotation, and of \ reflection) RotatePiece[piece_, n_] := Nest[Mrot.# &, #, n] & /@ piece ReflectPiece[piece_, n_] := If[n == 0, (MUnit.#) & /@ piece, (Mref.#) & /@ piece] Each piece with all the possible orientations PTriangleAll = {PTriangle, RotatePiece[PTriangle, 1], RotatePiece[PTriangle, 2], RotatePiece[PTriangle, 3]}; PSquareAll = {PSquare}; PLineAll = {PLine, RotatePiece[PLine, 1]}; PLAll = Module[{PLr = ReflectPiece[PL, 1]}, {PL, RotatePiece[PL, 1], RotatePiece[PL, 2], RotatePiece[PL, 3], PLr, RotatePiece[PLr, 1], RotatePiece[PLr, 2], RotatePiece[PLr, 3]}]; PPodiumAll = {PPodium, RotatePiece[PPodium, 1], RotatePiece[PPodium, 2], RotatePiece[PPodium, 3]}; PSnakeAll = {PSnake, RotatePiece[PSnake, 1], ReflectPiece[PSnake, 1], RotatePiece[ReflectPiece[PSnake, 1], 1]}; PBulkyAll = Module[{PBulkyr = ReflectPiece[PBulky, 1]}, {PBulky, RotatePiece[PBulky, 1], RotatePiece[PBulky, 2], RotatePiece[PBulky, 3], PBulkyr, RotatePiece[PBulkyr, 1], RotatePiece[PBulkyr, 2], RotatePiece[PBulkyr, 3]}]; tPieces = {PBulkyAll, PLAll, PPodiumAll, PSnakeAll, PSquareAll, PLineAll, PTriangleAll}; tPiecesNames = {"Bu", "Ll", "Po", "Sn", "Sq", "Li", "Tr"}; Functions which place the pieces* Piece placement functions PutPiece[piece_, piecename_, grid_] := Module[{IniPt, IniPtxy, Pts, Vcheck, PtsinGrid, Pos, i, gridx}, gridx = grid; IniPt = RandomChoice[Position[gridx, 0]]; Pts = Map[IniPt + # &, RandomChoice[piece]]; PtsinGrid = gridx[[Sequence @@ #]] & /@ Pts; Vcheck = If[PtsinGrid == Table[0, Length[PtsinGrid]], "success", "failure"]; If[Vcheck == "success", Do[gridx[[Sequence @@ Pts[[i]]]] = piecename, {i, Length[Pts]}]; {"success", gridx}, {"failure", gridx}] ] AddOnePiece[gridin_, piece_, piecename_, nmax_] := Module[{cc = "failure", nn = 0, aa}, While[(cc != "success") && (nn < nmax), aa = PutPiece[piece, piecename, gridin]; cc = aa[[1]]; nn = nn + 1]; aa] Grid definition, and plotting functions The initial grid is a ensemble of 0, embedded in a series of 1, forming a matrix. When a piece is placed the 0's it occupies are replaced by the name of the piece General Grid Definitions TheGrid0 = Table[1, {i, -6, 6, 1}, {j, -6, 6, 1}]; GridDefine[coord_] := Module[{gridout}, gridout = TheGrid0; Do[gridout[[Sequence @@ (coord[[i]] + {7, 7}) ]] = 0, {i, Length[coord]}]; gridout] DrawPieceRectangle[piecename_, grid_, color_] := {color, Map[Rectangle[{#[[1]] - 0.5 - 7, #[[2]] - 0.5 - 7}, {#[[1]] + 0.5 - 7, #[[2]] + 0.5 - 7}, RoundingRadius -> 0.] &, Position[grid, piecename]]} mycolors = ColorData[24]; PlotGrid[grid_, plotrange_] := Graphics[{{White, Rectangle[{-6, -6}, {6, 6}]}, DrawPieceRectangle[0, grid, White], DrawPieceRectangle[1, grid, Black], DrawPieceRectangle[tPiecesNames[[#]], grid, mycolors[If[# != 6, #, # + 4]]] & /@ Range[7]}, PlotRange -> plotrange] The final solving functions Full solving function Get the solution, and at each step provides a plot named p, which can be dynamically visualized using a cell with Dynamic[p] SolveGrid[InitGrid_, plotrange_] := Module[{imax, nattempt, grid, i, vc}, imax = 0; nattempt = 0; While[imax < 8, grid[1] = InitGrid; i = 1; vc = "success"; While[(i < 8) && (vc == \!\(\* TagBox[ StyleBox["\"\<success\>\"", ShowSpecialCharacters->False, ShowStringCharacters->True, NumberMarks->True], FullForm]\)), {vc, grid[i + 1]} = AddOnePiece[grid[i], tPieces[[i]], tPiecesNames[[i]], 30]; p = PlotGrid[grid[i + 1], plotrange]; If[vc == "success" \|\| i < 7, i++; imax = i, i++]]; nattempt++]; Print[nattempt]; PlotGrid[grid[8], plotrange]; grid[8]] Get ony the solution, without final plot, nor dynamical vizualisation SolveGridBare[InitGrid_] := Module[{imax, nattempt, grid, i, vc}, imax = 0; nattempt = 0; While[imax < 8, grid[1] = InitGrid; i = 1; vc = "success"; While[(i < 8) && (vc == \!\(\* TagBox[ StyleBox["\"\<success\>\"", ShowSpecialCharacters->False, ShowStringCharacters->True, NumberMarks->True], FullForm]\)), {vc, grid[i + 1]} = AddOnePiece[grid[i], tPieces[[i]], tPiecesNames[[i]], 30]; If[vc == "success" \|\| i < 7, i++; imax = i, i++]]; nattempt++]; Print[nattempt]; grid[8]] SolveGridBare2[InitGrid_] := Module[{imax, nattempt, grid, i, vc}, imax = 0; nattempt = 0; While[imax < 8, grid[1] = InitGrid; i = 1; vc = "success"; While[(i < 8) && (vc == \!\(\* TagBox[ StyleBox["\"\<success\>\"", ShowSpecialCharacters->False, ShowStringCharacters->True, NumberMarks->True], FullForm]\)), {vc, grid[i + 1]} = AddOnePiece[grid[i], tPieces[[i]], tPiecesNames[[i]], 30]; If[vc == "success" \|\| i < 7, i++; imax = i, i++]]; nattempt++]; {grid[8], nattempt}] Example of use coordQ35 = Join[Table[{i, 2}, {i, -2, 3}], Table[{i, -2}, {i, -2, 3}], Table[{i, 1}, {i, 0, 3}], Table[{i, 0}, {i, 0, 3}], Table[{i, -1}, {i, 0, 3}], Table[{-2, j}, {j, -1, 1}]]; TheGridQ35 = GridDefine[coordQ35]; SolgridQ35ex1 = SolveGridBare[TheGridQ35]; PlotGrid[SolgridQ35ex1, {{-3.5, 4.5}, {-3.5, 3.5}}] A dynamic view of the process (similar to the animations shown above) is obtained by running a cell "Dynamic[p]" before calling SolveGrid[TheGridQ35,{{-3.5, 4.5}, {-3.5, 3.5}}] Pieces definition for the 3 case (here all the possible orientations are obtained by applying randomly rotations and reflection on the initial piece) Pieces Definitions PTriangle = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}}; PSquare = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}, {1, 1, 0}}; PLine = {{0, -1, 0}, {0, 0, 0}, {0, 1, 0}}; PL = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}, {0}}; PPodium = {{0, 0, 0}, {0, 1, 0}, {-1, 0, 0}, {1, 0, 0}}; PSnake = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}, {-1, 1, 0}}; PBulky = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {1, 1, 0}, {-1, 1, 0}}; (* Rotation by angle \[Pi]/2 matrix) Mrotz = {{0, 1, 0}, {-1, 0, 0}, {0, 0, 1}}; Mrotx = {{1, 0, 0}, {0, 0, 1}, {0, -1, 0}}; Mrotx = {{0, 0, -1}, {0, 1, 0}, {1, 0, 0}}; ( Reflection matrix ) Mrefx = {{-1, 0, 0}, {0, 1, 0}, {0, 0, 1}}; Mrefy = {{1, 0, 0}, {0, -1, 0}, {0, 0, 1}}; Mrefz = {{1, 0, 0}, {0, 1, 0}, {0, 0, -1}}; MUnit = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}; (Operators which perform a given number of rotation, and of \ reflection) RotatePiecex[piece_, n_] := Nest[Mrotx.# &, #, n] & /@ piece RotatePiecey[piece_, n_] := Nest[Mrotx.# &, #, n] & /@ piece RotatePiecez[piece_, n_] := Nest[Mrotz.# &, #, n] & /@ piece ReflectPiecex[piece_, n_] := If[n == 0, (MUnit.#) & /@ piece, (Mrefx.#) & /@ piece] ReflectPiecey[piece_, n_] := If[n == 0, (MUnit.#) & /@ piece, (Mrefy.#) & /@ piece] ReflectPiecez[piece_, n_] := If[n == 0, (MUnit.#) & /@ piece, (Mrefz.#) & /@ piece] Each piece with all the possible orientations MakeAllPieces[piece_] := DeleteDuplicates@ Table[Module[{Rota, Rotb, Rotc, Refa, Refb, Refc, na, nb, nc, ma, mb, mc}, Rota = RandomChoice[{RotatePiecex, RotatePiecey, RotatePiecez}]; Rotb = RandomChoice[{RotatePiecex, RotatePiecey, RotatePiecez}]; Rotc = RandomChoice[{RotatePiecex, RotatePiecey, RotatePiecez}]; Refa = RandomChoice[{ReflectPiecex, ReflectPiecey, ReflectPiecez}]; Refb = RandomChoice[{ReflectPiecex, ReflectPiecey, ReflectPiecez}]; Refc = RandomChoice[{ReflectPiecex, ReflectPiecey, ReflectPiecez}]; na = RandomInteger[{0, 3}]; nb = RandomInteger[{0, 3}]; nc = RandomInteger[{0, 3}]; ma = RandomInteger[{0, 1}]; mb = RandomInteger[{0, 1}]; mc = RandomInteger[{0, 1}]; Refc[Rotc[Refb[Rotb[ Refa[Rota[piece, na], ma], nb], mb], nc], mc]], {i, 1, 2000}] PTriangleAll = MakeAllPieces[PTriangle]; Length@PTriangleAll 24 PSquareAll = MakeAllPieces[PSquare]; Length@PSquareAll 24 PLineAll = MakeAllPieces[PLine]; Length@PLineAll 6 PLAll = MakeAllPieces[PL]; Length@PLAll 24 PPodiumAll = MakeAllPieces[PPodium]; Length@PPodiumAll 24 PSnakeAll = MakeAllPieces[PSnake]; Length@PSnakeAll 24 PBulkyAll = MakeAllPieces[PBulky]; Length@PBulkyAll 24 tPieces = {PBulkyAll, PLAll, PPodiumAll, PSnakeAll, PSquareAll, PLineAll, PTriangleAll}; tPiecesNames = {"Bu", "Ll", "Po", "Sn", "Sq", "Li", "Tr"}; Grid definition and visualisation for the 3d case* General Grid Definitions TheGrid0 = Table[1, {i, -6, 6, 1}, {j, -6, 6, 1}, {k, -6, 6, 1}]; GridDefine[coord_] := Module[{gridout}, gridout = TheGrid0; Do[gridout[[Sequence @@ (coord[[i]] + {7, 7, 7}) ]] = 0, {i, Length[coord]}]; gridout] DrawPieceCube[piecename_, grid_, color_] := {Glow[color], Opacity[0.75], Specularity[0], EdgeForm[Thick], Map[Cuboid[{#[[1]] - 0.5 - 7, #[[2]] - 0.5 - 7, #[[3]] - 0.5 - 7}] &, Position[grid, piecename]]} mycolors = ColorData[24]; PlotGrid3D[grid_, plotrange_] := Graphics3D[{({White,Cuboid[{-6,-6,-6},{6,6,6}]},) DrawPieceCube[0, grid, White],(DrawPieceCube[1,grid,Black],) DrawPieceCube[tPiecesNames[[#]], grid, mycolors[If[# != 6, #, # + 4]]] & /@ Range[7]}, PlotRange -> plotrange, Lighting -> None, Boxed -> False] PlotGrid3D[grid_, plotrange_, pov_] := Graphics3D[{({White,Cuboid[{-6,-6,-6},{6,6,6}]},) DrawPieceCube[0, grid, White],(DrawPieceCube[1,grid,Black],) DrawPieceCube[tPiecesNames[[#]], grid, mycolors[If[# != 6, #, # + 4]]] & /@ Range[7]}, PlotRange -> plotrange, Lighting -> None, Boxed -> False, ViewPoint -> pov]

Easy Cube is a reflexion game from Japan. It consists of a set of 7 pieces which are "tetris"-like (each piece is a simple geometrical shape made from 3,4 or 5 cubes), and a collection of 48 problems, where the goal is to use the pieces to fill a given shape (in 2d, or in 3d for the hardest problems).

The game box enter image description here

The 7 pieces enter image description here

2 examples of problems enter image description here

It is pleasant to play, with the difficulty slowly increasing whit the hardest problems. I decided to try to solve it using Mathematica. I did not use any advanced mathematics for this (which anyway I would not know :-)), and I wanted a method general enough so it could solve all the problems without any specific adpatation. I went for a purely random method, where the program places a first piece at random, then try to put a second piece, etc. When it fails to add a piece after a fixed number of attemps, it simply starts the whole process again. So this program does not learn anything... It just tries many many different possibilities, until it finds a solution ! Using Wolfram language, the code to make it and to vizualize to result was quite simple and effective (I show the code details at the end of this post for people interested). In particular, going from the 2d case to the 3d case needed very small changes only.

The two followings animations show the program in action for a 2d problem and a 3d problem enter image description here

enter image description here

The number of attempts needed to reach a solution is of course random, and depends on the difficulty of the problem. Typically, it takes between a few hundreds and a few thousands iterations, which takes between a few seconds to 1-2 minutes. Here a sample of 4 solutions found for a 2d problem : Sample solutions from a 2d problem

And 2 solutions from a 3d problem : Sample solutions from a 2d problem

Finally, I considered it if was possible to use this method to find all solutions for a given problem (the number of different solutions is an info given on the description of each problem). In principle, because the method is random, it is not well suited for this (one can never be sure that all solutions have been found). However, since the method is quite fast, I decided to try anyway. For a given problem, I computed 10000 solutions. Here is a plot of the number of attempts needed for each solution, and a histogram of this: Number of attempts for 10000 solutions

On average, finding a solution to this problem requested 2882 attempts...

From the 10000 solutions, one wants only different solutions. This is easily done with the DeleteDuplicates command:

SolgridQ26ListMReduced = 
 DeleteDuplicates@SolgridQ26ListM; Length@SolgridQ26ListMReduced

362

which gives 362 different solutions. However, since this problem geometry has a simple central symmetry, one also want to get rid of "symmetric copies" of the solutions. Defining the symmetry operation as the GsymGrid function, this is again done with a DeleteDuplicates command :

SolgridQ26ListMReducedSym = 
 DeleteDuplicates[
  SolgridQ26ListMReduced, #2 == 
    GSymGrid[#1] &]; Length@SolgridQ26ListMReducedSym

188

which gives 188 unique solutions. This is precisely the number which is given in the problem description ! So the method was able to find all solutions. Here is plot with the 188 solutions:

Details of the code

Definition of the pieces (2d case). Each piece is a simple list of the points make the piece. When then construct, for each piece, a list of copies of it with all possible orientations.

Pieces Definitions
PTriangle = {{0, 0}, {0, 1}, {1, 0}};
PSquare = {{0, 0}, {0, 1}, {1, 0}, {1, 1}};
PLine = {{0, 0}, {0, 1}, {0, 2}};
PL = {{0, 0}, {0, 1}, {1, 0}, {2, 0}};
PPodium = {{0, 0}, {0, 1}, {-1, 0}, {1, 0}};
PSnake = {{0, 0}, {0, 1}, {1, 0}, {-1, 1}};
PBulky = {{0, 0}, {1, 0}, {0, 1}, {1, 1}, {-1, 1}};
(* Rotation by angle \[Pi]/2 matrix*)
Mrot = {{0, 1}, {-1, 0}};
(* Reflection by y-axis matrix *)
Mref = {{-1, 0}, {0, 1}};
MUnit = {{1, 0}, {0, 1}};
(*Operators which perform a given number of rotation, and of \
reflection*)
RotatePiece[piece_, n_] := Nest[Mrot.# &, #, n] & /@ piece
ReflectPiece[piece_, n_] := 
 If[n == 0, (MUnit.#) & /@ piece, (Mref.#) & /@ piece]
Each piece with all the possible orientations
PTriangleAll = {PTriangle, RotatePiece[PTriangle, 1], 
   RotatePiece[PTriangle, 2], RotatePiece[PTriangle, 3]};
PSquareAll = {PSquare};
PLineAll = {PLine, RotatePiece[PLine, 1]};
PLAll = Module[{PLr = ReflectPiece[PL, 1]}, {PL, RotatePiece[PL, 1], 
    RotatePiece[PL, 2], RotatePiece[PL, 3], PLr, RotatePiece[PLr, 1], 
    RotatePiece[PLr, 2], RotatePiece[PLr, 3]}];
PPodiumAll = {PPodium, RotatePiece[PPodium, 1], 
   RotatePiece[PPodium, 2], RotatePiece[PPodium, 3]};
PSnakeAll = {PSnake, RotatePiece[PSnake, 1], ReflectPiece[PSnake, 1], 
   RotatePiece[ReflectPiece[PSnake, 1], 1]};
PBulkyAll = 
  Module[{PBulkyr = ReflectPiece[PBulky, 1]}, {PBulky, 
    RotatePiece[PBulky, 1], RotatePiece[PBulky, 2], 
    RotatePiece[PBulky, 3], PBulkyr, RotatePiece[PBulkyr, 1], 
    RotatePiece[PBulkyr, 2], RotatePiece[PBulkyr, 3]}];
tPieces = {PBulkyAll, PLAll, PPodiumAll, PSnakeAll, PSquareAll, 
   PLineAll, PTriangleAll};
tPiecesNames = {"Bu", "Ll", "Po", "Sn", "Sq", "Li", "Tr"};

Functions which place the pieces

Piece placement functions
PutPiece[piece_, piecename_, grid_] := 
 Module[{IniPt, IniPtxy, Pts, Vcheck, PtsinGrid, Pos, i, gridx}, 
  gridx = grid;
  IniPt = RandomChoice[Position[gridx, 0]];
  Pts = Map[IniPt + # &, RandomChoice[piece]];
  PtsinGrid = gridx[[Sequence @@ #]] & /@ Pts;
  Vcheck = 
   If[PtsinGrid ==  Table[0, Length[PtsinGrid]], "success", "failure"];
  If[Vcheck == "success", 
   Do[gridx[[Sequence @@ Pts[[i]]]] = piecename, {i, 
     Length[Pts]}]; {"success", gridx}, {"failure", gridx}]
  ]
AddOnePiece[gridin_, piece_, piecename_, nmax_] := 
 Module[{cc = "failure", nn = 0, aa}, 
  While[(cc != "success") && (nn < nmax), 
   aa = PutPiece[piece, piecename, gridin]; cc = aa[[1]]; 
   nn = nn + 1]; aa]

Grid definition, and plotting functions The initial grid is a ensemble of 0, embedded in a series of 1, forming a matrix. When a piece is placed the 0's it occupies are replaced by the name of the piece

General Grid Definitions
TheGrid0 = Table[1, {i, -6, 6, 1}, {j, -6, 6, 1}];
GridDefine[coord_] :=  
 Module[{gridout}, gridout = TheGrid0; 
  Do[gridout[[Sequence @@ (coord[[i]] + {7, 7}) ]] = 0, {i, 
    Length[coord]}]; gridout]
DrawPieceRectangle[piecename_, grid_, color_] := {color, 
  Map[Rectangle[{#[[1]] - 0.5 - 7, #[[2]] - 0.5 - 7}, {#[[1]] + 0.5 - 
       7, #[[2]] + 0.5 - 7}, RoundingRadius -> 0.] &, 
   Position[grid, piecename]]}
mycolors = ColorData[24];
PlotGrid[grid_, plotrange_] := 
 Graphics[{{White, Rectangle[{-6, -6}, {6, 6}]},
   DrawPieceRectangle[0, grid, White], 
   DrawPieceRectangle[1, grid, Black], 
   DrawPieceRectangle[tPiecesNames[[#]], grid, 
      mycolors[If[# != 6, #, # + 4]]] & /@ Range[7]}, 
  PlotRange -> plotrange]

The final solving functions

Full solving function
Get the solution, and at each step provides a plot named p, which can be dynamically visualized using a cell with Dynamic[p]
SolveGrid[InitGrid_, plotrange_] := 
 Module[{imax, nattempt, grid, i, vc}, imax = 0; nattempt = 0; 
  While[imax < 8, grid[1] = InitGrid; i = 1; vc = "success"; 
   While[(i < 8) && (vc == \!\(\*
TagBox[
StyleBox["\"\<success\>\"",
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)),
     {vc, grid[i + 1]} =  
     AddOnePiece[grid[i], tPieces[[i]], tPiecesNames[[i]], 30]; 
    p = PlotGrid[grid[i + 1], plotrange]; 
    If[vc == "success" || i < 7, i++; imax = i, i++]]; nattempt++]; 
  Print[nattempt]; PlotGrid[grid[8], plotrange]; grid[8]]
Get ony the solution, without final plot, nor dynamical vizualisation
SolveGridBare[InitGrid_] := 
 Module[{imax, nattempt, grid, i, vc}, imax = 0; nattempt = 0; 
  While[imax < 8, grid[1] = InitGrid; i = 1; vc = "success"; 
   While[(i < 8) && (vc == \!\(\*
TagBox[
StyleBox["\"\<success\>\"",
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)),
     {vc, grid[i + 1]} =  
     AddOnePiece[grid[i], tPieces[[i]], tPiecesNames[[i]], 30]; 
    If[vc == "success" || i < 7, i++; imax = i, i++]]; nattempt++]; 
  Print[nattempt]; grid[8]]
SolveGridBare2[InitGrid_] := 
 Module[{imax, nattempt, grid, i, vc}, imax = 0; nattempt = 0; 
  While[imax < 8, grid[1] = InitGrid; i = 1; vc = "success"; 
   While[(i < 8) && (vc == \!\(\*
TagBox[
StyleBox["\"\<success\>\"",
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)),
     {vc, grid[i + 1]} =  
     AddOnePiece[grid[i], tPieces[[i]], tPiecesNames[[i]], 30]; 
    If[vc == "success" || i < 7, i++; imax = i, i++]]; 
   nattempt++]; {grid[8], nattempt}]

Example of use

coordQ35 = 
  Join[Table[{i, 2}, {i, -2, 3}], Table[{i, -2}, {i, -2, 3}], 
   Table[{i, 1}, {i, 0, 3}], Table[{i, 0}, {i, 0, 3}], 
   Table[{i, -1}, {i, 0, 3}], Table[{-2, j}, {j, -1, 1}]];

TheGridQ35 = GridDefine[coordQ35];
SolgridQ35ex1 = SolveGridBare[TheGridQ35];
PlotGrid[SolgridQ35ex1, {{-3.5, 4.5}, {-3.5, 3.5}}]

A dynamic view of the process (similar to the animations shown above) is obtained by running a cell "Dynamic[p]" before calling SolveGrid[TheGridQ35,{{-3.5, 4.5}, {-3.5, 3.5}}]

Pieces definition for the 3 case (here all the possible orientations are obtained by applying randomly rotations and reflection on the initial piece)

Pieces Definitions
PTriangle = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}};
PSquare = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}, {1, 1, 0}};
PLine = {{0, -1, 0}, {0, 0, 0}, {0, 1, 0}};
PL = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}, {0}};
PPodium = {{0, 0, 0}, {0, 1, 0}, {-1, 0, 0}, {1, 0, 0}};
PSnake = {{0, 0, 0}, {0, 1, 0}, {1, 0, 0}, {-1, 1, 0}};
PBulky = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {1, 1, 0}, {-1, 1, 0}};
(* Rotation by angle \[Pi]/2 matrix*)
Mrotz = {{0, 1, 0}, {-1, 0, 0}, {0, 0, 1}};
Mrotx = {{1, 0, 0}, {0, 0, 1}, {0, -1, 0}};
Mrotx = {{0, 0, -1}, {0, 1, 0}, {1, 0, 0}};
(* Reflection  matrix *)
Mrefx = {{-1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
Mrefy = {{1, 0, 0}, {0, -1, 0}, {0, 0, 1}};
Mrefz = {{1, 0, 0}, {0, 1, 0}, {0, 0, -1}};
MUnit = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
(*Operators which perform a given number of rotation, and of \
reflection*)
RotatePiecex[piece_, n_] := Nest[Mrotx.# &, #, n] & /@ piece
RotatePiecey[piece_, n_] := Nest[Mrotx.# &, #, n] & /@ piece
RotatePiecez[piece_, n_] := Nest[Mrotz.# &, #, n] & /@ piece
ReflectPiecex[piece_, n_] := 
 If[n == 0, (MUnit.#) & /@ piece, (Mrefx.#) & /@ piece]
ReflectPiecey[piece_, n_] := 
 If[n == 0, (MUnit.#) & /@ piece, (Mrefy.#) & /@ piece]
ReflectPiecez[piece_, n_] := 
 If[n == 0, (MUnit.#) & /@ piece, (Mrefz.#) & /@ piece]
Each piece with all the possible orientations
MakeAllPieces[piece_] := 
 DeleteDuplicates@
  Table[Module[{Rota, Rotb, Rotc, Refa, Refb, Refc, na, nb, nc, ma, 
     mb, mc},
    Rota = RandomChoice[{RotatePiecex, RotatePiecey, RotatePiecez}];
    Rotb = RandomChoice[{RotatePiecex, RotatePiecey, RotatePiecez}]; 
    Rotc = RandomChoice[{RotatePiecex, RotatePiecey, RotatePiecez}];
    Refa = RandomChoice[{ReflectPiecex, ReflectPiecey, ReflectPiecez}];
    Refb = RandomChoice[{ReflectPiecex, ReflectPiecey, ReflectPiecez}];
    Refc = RandomChoice[{ReflectPiecex, ReflectPiecey, ReflectPiecez}];
    na = RandomInteger[{0, 3}]; nb = RandomInteger[{0, 3}]; 
    nc = RandomInteger[{0, 3}];
    ma = RandomInteger[{0, 1}]; mb = RandomInteger[{0, 1}]; 
    mc = RandomInteger[{0, 1}];
    Refc[Rotc[Refb[Rotb[ Refa[Rota[piece, na], ma], nb], mb], nc], 
     mc]], {i, 1, 2000}]
PTriangleAll = MakeAllPieces[PTriangle]; Length@PTriangleAll
24
PSquareAll = MakeAllPieces[PSquare]; Length@PSquareAll
24
PLineAll = MakeAllPieces[PLine]; Length@PLineAll
6
PLAll = MakeAllPieces[PL]; Length@PLAll
24
PPodiumAll = MakeAllPieces[PPodium]; Length@PPodiumAll
24
PSnakeAll = MakeAllPieces[PSnake]; Length@PSnakeAll
24
PBulkyAll = MakeAllPieces[PBulky]; Length@PBulkyAll
24
tPieces = {PBulkyAll, PLAll, PPodiumAll, PSnakeAll, PSquareAll, 
   PLineAll, PTriangleAll};
tPiecesNames = {"Bu", "Ll", "Po", "Sn", "Sq", "Li", "Tr"};

Grid definition and visualisation for the 3d case

General Grid Definitions
TheGrid0 = Table[1, {i, -6, 6, 1}, {j, -6, 6, 1}, {k, -6, 6, 1}];
GridDefine[coord_] :=  
 Module[{gridout}, gridout = TheGrid0; 
  Do[gridout[[Sequence @@ (coord[[i]] + {7, 7, 7}) ]] = 0, {i, 
    Length[coord]}]; gridout]
DrawPieceCube[piecename_, grid_, color_] := {Glow[color], 
  Opacity[0.75], Specularity[0], EdgeForm[Thick], 
  Map[Cuboid[{#[[1]] - 0.5 - 7, #[[2]] - 0.5 - 7, #[[3]] - 0.5 - 
       7}] &, Position[grid, piecename]]}
mycolors = ColorData[24];
PlotGrid3D[grid_, plotrange_] := 
 Graphics3D[{(*{White,Cuboid[{-6,-6,-6},{6,6,6}]},*)

   DrawPieceCube[0, grid, White],(*DrawPieceCube[1,grid,Black],*)
   DrawPieceCube[tPiecesNames[[#]], grid, 
      mycolors[If[# != 6, #, # + 4]]] & /@ Range[7]}, 
  PlotRange -> plotrange, Lighting -> None, Boxed -> False]
PlotGrid3D[grid_, plotrange_, pov_] := 
 Graphics3D[{(*{White,Cuboid[{-6,-6,-6},{6,6,6}]},*)

   DrawPieceCube[0, grid, White],(*DrawPieceCube[1,grid,Black],*)
   DrawPieceCube[tPiecesNames[[#]], grid, 
      mycolors[If[# != 6, #, # + 4]]] & /@ Range[7]}, 
  PlotRange -> plotrange, Lighting -> None, Boxed -> False, 
  ViewPoint -> pov]

POSTED BY: Thibaut Jonckheere

15 Replies

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Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

POSTED BY: Thibaut Jonckheere

Szabolcs Horvát

Posted 9 years ago

It's possible to look at this problem from a slightly different angle and reduce it to a graph-theoretical [clique][1] finding problem. This is useful because we already have a fast (maximal) clique finder built into Mathematica. I will only give an implementation for the most basic case, i.e. filling out the 3 by 3 by 3 cube, as in illustration. In graph theory a clique is a complete subgraph, i.e. a subgraph in which all pairs of nodes are connected. The idea Let us construct a graph, where the nodes are all possible placements of each individual piece in the fit space. That is, we must generate all rotations of each piece (`rotatedPieces` from Sjoerd) and all possible positions of each of these within the fit space. Then we connect two nodes (i.e. placements) if they are not overlapping. If we are not allowed to re-use the same piece twice, then we we only connect non-overlapping placements of different pieces. Now any valid arrangement of pieces is a clique. That's because an arrangement is valid no two pieces overlap, i.e. they are connected in the graph. Finding a maximal clique means cramming enough pieces in the fit space so that no more can be added. We have 7 pieces and want to use them all. So we need to find cliques of size 7. Implementation Let us implement this for a 3x3x3 fit space starting form Sjoerd's `piecesAsObjects` list. The following function generates all translations of a rotated piece within the fit space: allTranslations[p_] := Module[{pad}, pad = (3 - Dimensions[p]); pad // Map[Range[0, #] &] // Tuples // Map[Transpose[{#, pad - #}] &] // Map[ArrayPad[p, #] &] ] We collect these in a list of lists: allTrans = Map[allTranslations /* Apply[Sequence], piecesAsObjects, {2}]; Each sublist contains all possible placements (including rotations) of a piece. The following function tests for overlap between two placements: overlapping[{a_, b_}] := Total[a b, Infinity] > 0 We only want to add connections between different pieces, so we start with `Subsets[allTrans, {2}]`. Then for each piece-pair we add connections for non-overlapping placements. edges = (UndirectedEdge @@@ Select[Tuples[#], Not@overlapping] &) /@ Subsets[allTrans, {2}]; This was the most time consuming part. It took over a second on my machine. This can probably be done much faster with ListConvolve/ListCorrelate, like Sjoerd did. Now we have our graph: g = Graph[Join @@ edges]; The vertices are actual 3D placement arrays, but if you prefer to work with indices instead, use `IndexGraph`. We are now ready to use `FindClique`, but we must do so carefully. There are a huge number of cliques in this graph, and trying to find all of them would take impractically long (and would likely cause the machine to run out of memory). A simple `FindClique[g]` "finds a largest clique in the graph", according to the documentation. This is a bit misleading. You might think that finding only one* largest clique shouldn't be slow. The problem is that the algorithm must iterate through all maximal cliques to decide which is the largest, so it will in fact enumerate all of them. But luckily we already know the size of the largest clique: it is 7, i.e. containing all pieces. So we can stop the search as soon as we find a clique of size 7. This is done by FindClique[g, {7}] We can ask Mathematica to find more, say 10 of them. I'll do this in a way so only placement indexes are shown and the output is clearer: FindClique[IndexGraph[g], {7}, 10] {{12, 132, 301, 428, 473, 578, 625}, {12, 231, 301, 403, 522, 573, 625}, {12, 231, 301, 403, 521, 604, 625}, {91, 251, 301, 389, 518, 611, 625}, {91, 251, 301, 389, 502, 620, 625}, {91, 271, 301, 414, 518, 575, 625}, {91, 201, 301, 413, 527, 544, 625}, {91, 164, 301, 380, 480, 620, 625}, {91, 225, 301, 428, 480, 617, 625}, {91, 225, 301, 428, 520, 568, 625}} Let's write a function for plotting placements: plotClique[cl_] := Graphics3D[ Raster3D[Total[cl Range@Length[cl]], ColorFunction -> ColorData[97]], Boxed -> False, SphericalRegion -> True] plotClique /@ FindClique[g, {Length[pieces]}, 10] We can ask for even more arrangements. Finding 200 takes 2 seconds on my machine. FindClique[IndexGraph[g], {7}, 200] // Length // AbsoluteTiming {1.87257, 200} This is just meant to show the idea. It would be nice to implement this in a general way, for all kinds of fill spaces, and making use of `ListCorrelate`. Update Finding all cliques of size 7 took quite a bit longer. It took ~1000 seconds on my machine and came up with 11520 cliques. Many of these arrangements are equivalent. We have 24 rotations and 2 reflections of a single arrangement. In reality there are only 11520/(224) = 240 distinct ones. This result, together with Wikipedia's claim that there are only 240 truly distinct arrangements, implies that none of the arrangements have any rotational of reflection symmetries. Otherwise we would find fewer than 48240 cliques. Update 2: If I use `IGMaximalCliques[g, {7}]` from the IGraph/M package (an igraph interface I made for Mathematica), I get the result in merely 60 seconds instead of 1000 seconds with the builtin `FindClique`. IGraph/M does not yet support finding only a limited number of cliques. It returns all cliques that satisfy the size criteria. I plan to add this feature in the next IGraph/M release, which will also bring additional clique finding performance improvements. Update 3: Somewhat surprisingly, `IGCliques[g, {7}]` returns the result in only 0.8 seconds. `IGCliques` finds all cliques, not just maximal ones. In this case, all size-7 cliques are clearly also maximal, so we might as well use `IGCliques`.

POSTED BY: Szabolcs Horvát

Sjoerd de Vries

Sjoerd de Vries, Rabobank

Posted 9 years ago

POSTED BY: Sjoerd de Vries

Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

Wow, this is great. Your code succeeds at being very efficient while being reasonnably simple and short. I did not know the ListCorrelate[] command, but it is very effective here. Also, randomly trying the pieces is not really efficient. Yes, I agree with this, it ts the most basic way, and the least efficient For instance, randomly combining rotation matrices and hoping you'll eventually get all necessary rotations just doesn't feel good I disagree with this. While it effectively uses RandomChoice[], in practice there is no randomness in the results for all pieces orientations (thanks to the large number of tries), and it is fast enough (and it needs to be done only once for a given set of pieces). I think it is a matter of taste, but for me it was a lazy way to get all possible orientations without the risk of forgetting some combination of transformations. :-) Thank you for the notebook - many things to learn !

POSTED BY: Thibaut Jonckheere

Sjoerd de Vries

Sjoerd de Vries, Rabobank

Posted 9 years ago

Attachments:

POSTED BY: Sjoerd de Vries

Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

Well, I don't know. It would be nice to mix this basic solving with machine learning, to improve it by eliminating some tries which are obviously wrong. For example, one could imagine that the machine could learn that it is bad to leave an empty space between two neighboring pieces that cannot be filled by any piece (this is something a human does naturally when trying to solve the puzzle). On a more elaborate level the machine could learn some typical combinations of 2 or 3 pieces which gives useful shapes. But as I have no experience with machine learning, I don't see what would be the way to train the machine. If any expert has some idea about it...

POSTED BY: Thibaut Jonckheere

Vijay Sharma

Vijay Sharma, Riyaaz.org

Posted 9 years ago

@Thibaut Jonckheere Is it also possible to make the program learn about the game? Say, rather trying brute force all the solutions it can intelligently decide some better moves?

POSTED BY: Vijay Sharma

Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

I have attached the notebook version of the code (SOMA set of pieces) for convenience. It contains all the code (the pieces names are still "incorrect", but the pieces themselves are ok), and a few examples. Attachments:

POSTED BY: Thibaut Jonckheere

Anton Antonov

Anton Antonov, Accendo Data LLC

Posted 9 years ago

Thanks, I will play with this!

POSTED BY: Anton Antonov

Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

POSTED BY: Thibaut Jonckheere

Anton Antonov

Anton Antonov, Accendo Data LLC

Posted 9 years ago

The "Easy cube" seems to be a simplified version of the "Soma cube". So, I wonder how easy would be to adapt the code given in this discussion for "Soma cube" constructs. E.g.

POSTED BY: Anton Antonov

EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 9 years ago

- another post of yours has been selected for the Staff Picks group, congratulations ! We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

POSTED BY: EDITORIAL BOARD

Sander Huisman

Sander Huisman, University of Twente

Posted 9 years ago

I've seen a c++ code dancing links code for sudoku, it looked really complex. I'm quite sure it is not easy ;)

POSTED BY: Sander Huisman

Thibaut Jonckheere

Thibaut Jonckheere, CNRS

Posted 9 years ago

Thank you for the info ! I did not know the dancing links algorithm. Having a look at the wikipedia page about it, it seems very interesting (and way more powerful than my basic algorithm). I have matter to study here.

POSTED BY: Thibaut Jonckheere

Sander Huisman

Sander Huisman, University of Twente

Posted 9 years ago

Nicely done! I always wanted to program the dancing links algorithm that is used for 'exact cover' problems like this one! But your algorithm seems nice too. I'm gonna study it a bit!

POSTED BY: Sander Huisman

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Computationally solving "Easy Cube" and "Soma Cube" games / puzzles

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The idea

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