Procedural Puzzle Challenge Generation in Fujisan 富士山 Dr. Mark Goadrich – Hendrix College James Droscha – Glastyn Games

FujisaN: A Solitaire Puzzle Game

Fujisan Elements 2 Rows x 12 Columns of spaces with values between 0 and 5 inclusive 4 Shinto Priests start off the board on either side GOAL: Move Priests one at a time until all four are at the summit. 24 Values 1 2×2 Summit 10 Steps 24 Spaces 4 Priests

Fujisan Movement Rules A Priest may move onto a space if that space’s value matches the number of unoccupied spaces the Priest must move in a straight line, left or right, to get there (including the destination space itself, but not including the Priest’s starting space). Occupied spaces (containing intervening Priests) are not counted when determining the distance from a Priest to a given space. A Priest may pass over the summit as part of a move. A Priest that lands on the mountain’s summit can no longer move left or right, but may still move freely up or down within the column. A Priest may move freely up and down between the two spaces of any given step of the mountain.

How To make Fujisan Challenges? The solver must use the pieces of the game to setup their own challenges. Is the setup process easy for the solver? Can all such challenges be solved? How difficult are the created challenges?

Solitaire Puzzles and Procedural Content Generation

Solitaire Puzzle Games Fifteen Puzzle, Tipover, Peg Solitaire, Sokoban, Monument Valley

Procedural Content Generation Offline Constructed Deterministic Necessary Online Generate-and-Test Randomized Optional Making Challenges for Solitare Puzzle Games is like making Levels for Video Game Puzzle Games

RUSH HOUR Offline Constructed Deterministic Necessary Randomized

Klondike Solitaire Online Generate-and-Test Randomized Optional Necessary?

5 setup Algorithms

Piecepack Game System 24 square tiles, indexed on the obverse in four suits (suns, moons, crowns, and arms) of six values each (null, ace, 2, 3, 4, and 5) and divided on the reverse into a 2×2 space grid. 24 round coins, each sized to fit comfortably into one space on a tile, marked on the obverse with one of the six values and on the reverse with one of the four suits. 4 cubic dice, one per suit, each side marked with one of the six values. 4 pawns, one per suit, each sized to fit comfortably into one space on a tile.

1. Pure Random TEDIOUS ≈ 1018 Challenges Take one die from the piecepack. For each space, roll the die and place a coin that matches the number rolled on the space. ≈ 1018 Challenges 0 – 24 of each value

2. Any Coin BETTER ≈ 1015 Challenges Shuffle the 24 coins face-down. For each space on the board, randomly select one coin and place it face-up on this space. ≈ 1015 Challenges 4 of each value

3. Piecepack rules OK ≈ 1010 Challenges Shuffle the 24 coins face-down, and separate into four groups based on their suit. Then repeatedly place two coins on the two right-most available spaces, choosing from each of the suits in turn (sun, moon, crown, arms). ≈ 1010 Challenges 4 of each value Step values cannot be identical

Alternate Pieces? 20 Engraved Tiles 15 Dominoes double-five mirrored configuration minus all-0 tile 15 Dominoes double-five configuration minus tiles with identical value

4. Engraved Tiles EASY ≈ 107 Challenges Shuffle the tiles face-down. Then assemble the mountain using face- up tiles, using six for the bottom row, five for the next row, then four, then three, and finally two. ≈ 107 Challenges 0-7 of each value Step values cannot be identical Step values only repeated at summit

5. Dominoes EASY ≈ 105 Challenges Shuffle the dominoes face-down. Place 12 of these dominoes face-up in a row to create the mountain. ≈ 105 Challenges 2-5 of each value Step values cannot be identical Step values cannot be repeated

Results

Fujisan Simulator C# implementation 1000 Challenges per Setup .210133325040. .245231154405. (1,13) -> (1,8) .2452311.4405  (1,8) -> (1,3) .24.231154405  (1,3) -> (0,3) .21.133325040. .245231154405  (0,3) -> (0,1) ..10133325040. (0,0) -> (0,2)  ..0133325040. (0,1) -> (1,1)  2.0133325040. C# implementation 5 PCG Setups A* Solver 1000 Challenges per Setup 10 Trials of 100 Challenges https://github.com/mgoadric/fujisan

Remaining Questions Can all such challenges be solved? How difficult are the created challenges?

Solvability Recorded percent of solvable challenges for each strategy t-test results Random < Any Coin Any Coin < piecepack piecepack < Domino

Difficulty Random Any Coin Recorded minimum length solution for each solvable challenge Follows Poisson distribution Kruskal-Wallis H-test p-value = 1.93 x 10−8 Engraved Tiles is the culprit! Missing 0 values? piecepack Engraved Dominoes

Understanding The Results Good Fujisan challenges have Non-obvious moves (need 0 values) Well-connected steps (non-identical step values) Well-distributed values (non-repeated step values)

Future Work Formalizing Easy of Physical Setup Shuffling and Sorting different for computers and humans Enriching Difficulty Metric Are all moves equal? What about branching factor? Can PCG for Solitaire Puzzles be Online and Constructive? Must avoid being tedious Must avoid giving away the solution Dominoes lack mountain aesthetic?