Pentominoes

Anyone, who like me, spent an inappropriate amount of time in their younger years playing Tetris will be familiar with tetronimoes, as they are the shapes that are used in the game. Each tetronimo is made up of four 1x1 squares joined together to make a shape. Any Tetris player worth their salt will recall that there are seven tetronimoes:

• the 4x1 long thin shape - useful for slotting down the sides to complete a "tetris" (getting four lines to disappear at the same time);
• the 2x2 square;
• the 'T' shape;
• the two 'L' shaped pieces - one that is more of a 'J' than an 'L'; and
• the two (for want of a better description) wavy-line pieces, which also came in two versions like the L and the J.

Now, if you allowed reflective as well as rotational symmetry, then the L and the J would actually be the same piece, as would the two wavy lines, and the total number of tetronimoes would actually be five. But as any Tetris player will recall, it was possible to rotate your pieces to make them fit together, but reflection wasn't an option, so there are seven.

Well, what if we extend by one 1x1 square to make our tetronimoes into pentominoes? Then there are 18 different shapes that can be made. Of these 18 shapes, 6 are like the first three tetronimoes listed above in that they don't come as a pair (as they are their own reflection), whereas the other 12 come as pairs, one the reflection of the other. So if we allowed reflection then (where the number of tetronimoes would reduce from 7 to 5) the number of pentominoes would reduce from 18 to 12.

But, we aren't going to allow reflection, so we will stick with 18.

The total area made up by the 18 pentominoes is (given that each is made up of 5 1x1 squares) 18 x 5 = 90. If we want to make a rectangular shape with area 90, we have the following choices:

• 90 x 1
• 45 x 2
• 30 x 3
• 18 x 5
• 15 x 6
• 10 x 9

Now, if we wanted to cover the rectangle exactly by fitting together our pentominoes (a bit like a game of Tetris) then it is clear that the first two are not possible, as after all, some pentominoes are three units long by three wide (e.g. the cross).

However, I think that the rest of the rectangles can be filled using the pentominoes (although I haven't found solutions for all of them - note - self control required at this stage to avoid asking Google to check! - that might have to be a challenge for another day), and in particular, I have a wooden puzzle which asks you to complete a 10 x 9 rectangle.

And here is a solution.


(Note - solution posted here for posterity, but also so that when I have tipped all the pieces out, and it is really starting to annoy me, I can at least look back at the picture so that I can put the pieces back in again!)