Pentomino puzzle page

                        Pentominoes are geometrical figures made by joining 5 squares side by side. There are 12 possible figures, as shown below.

                        The total area covered by these 12 figures equals 60 (5 ´ 12) unity squares. It is, therefore, possible that the 12 figures could be arranged in such a way to fill exactly a rectangle measuring 6 ´ 10, or 5 ´ 12, or 4 ´ 15 or 3 ´ 20 (the unity is the side of the square), because all these rectangles have the same total area.

                        What do you think? Is it possible?

                        By the way, it is permitted to rotate the figures to any position, including out-of-plane rotations. However, to fill a rectangle, the sides of the squares must be maintained parallel to the sides of the rectangle, a limitation that drastically reduces the number of possible positions for the figures. The maximum numbers of positions are:

                        It is not only possible to fill the previously mentioned rectangles, but there are many different solutions. The numbers are in the table below.

                        Can you find all the 3719 solutions?

                        Of course, this does not seem feasible for human beings: if you could find one solution every day, it would take 10 years to accomplish the task.

                        The fun of the thing is not to pursue the all-solutions stage, but to find as many solutions as possible.

                        The program “Pentomino2”, which is free and you can easily download and use without any restrictions, can be used both for playing and for finding the number of solutions. You can add some figures or start with an empty board, and the computer will be able to find the number of solutions for you. You can see only the number of solutions, but it is possible to see the solutions themselves, also.

                        Before using the program, you may find it useful to know that this program does not check for symmetry: if you start with an empty board, the number of solutions found by the computer must be divided by 4, because there are always 4 solutions that are identical:

                        However, if any number of pieces is added by hand before asking the computer to find the solutions, this problem does not exist anymore. The computer does not move the pieces added by hand, so if you had added the yellow piece below, only one of the (above) four possible solutions could be found.

                        By the way: use the left mouse button (press and drag) to move the pieces, and use the right mouse button to rotate the individual pieces (by just clicking).

Downloading

                        To download, just choose one of the possibilities below. The much smaller (exe) file works on most computers (Windows). It downloads in a shorter time and does not require installation, but contains the whole program. Unzip it to any desired address of your computer; if you like, a shortcut can be added to your desktop, for instance. Then just click on the file and the program will start.

                        If the (exe) file does not run on your computer (it will probably say that there is a file missing, a *.dll, a *.ocx, etc.), then you have to download the more extensive file and install the program.

Short executable file: Pentomino2(exe) [1.55 MB]

Complete program for installation: Pentomino2(full) [10.8 MB]