On this page you can find solutions to a wooden Pentomino puzzle, also called Yasumi puzzle. It is a game in which different kinds of blocks have to be placed on a grid. Each block is made up of 5 cells each and there are 12 blocks in total. This is exactly the number of unique blocks that can be made up of 5 cells. In the original game these blocks have to be placed on a grid of 6 by 10. Blocks may be rotated if necessary.
In the picture below you can see all the blocks:
Solutions to the original game
One of the solutions to the original game is shown in the picture on the left. This is not the only solution to the game; there are actually a lot. I wanted to have all solutions to this game and wrote a computer program to find them. Using brute force alone would not have gotten me very far, because the problem is very complex.
For example, each block (except the red one) can be rotated. The red block cannot be rotated, so there is only one way to place it. The green block can be rotated once, so there are two ways to place it. Five other blocks can be rotated twice, so there are four ways to place them. The last five blocks can be rotated three times, so there are eight ways to place them. Therefore, counting rotations only, there are 11 * 21 * 45 * 85 equals 67,108,864 ways to place the blocks.
I used backtracking to crack the problem. The function that checks the feasability of the partial solution is crucial to the success of the program. The function I came up with checks whether the size of the open areas in the grid are divisable by 5. If not, then blocks made up of 5 cells cannot be used to fill the open areas. At that point the partial solution can be ignored and another one can be constructed. The program took about 80 minutes on a Pentium II 366 MHz to compile a list of all solutions. If you would like to have the source code, contact me at the e-mail address on this page.
Solutions to rectangle-shaped grids
The grid used in the original game is not the only one in which the blocks can be placed. Other rectangles have sizes of 5 by 12, 4 by 15 and 3 by 20.
Solutions to stair-shaped grids
Besides rectangle-shaped grids, there are stair-shaped grids that are worth taking a look at.
Solutions to other-shaped grids
Finally, here are the solutions to a shape that does not fit into the other categories.