Questions 1-5

1: What is a "Pentomino"?

A pentomino is a mathematical shape, formed by taking five squares and connecting them together along their edges. Here are some pentominoes:

2: How many pentominoes are there?

That depends on how you count them. Here's an example:

Is that two different pentominoes? Or is it the same pentomino, but just flipped over? Or looked at in a mirror? There's not a right or wrong answer to those questions - sometimes, it makes sense to think of them as different, sometimes the same.

Now, take a look at these:

Maybe that's four different shapes, maybe that's just one. In this case, they're just turned 90 degrees from one to the next. I'd usually say that that's one pentomino.

If you imagine a pentomino as a physical object, maybe something you cut out of cardboard and put on the table, these rotations could be done by leaving the cardboard on the table and sliding it around. If we can also pick up the cardboard off the table to do the flips like we did earlier, as well as rotating it on the table, then we get 8 total possibilities for this one pentomino:

I'll call that 8 different orientations for a single pentomino. There's something "the same" about all of these shapes, and that's what I'm going to mostly want to talk about.

Ok, so we've looked at only a few pentomino shapes so far, how many different pentominoes are there?

There are twelve:

TODO: revise to only have the one set.

3: Who came up with the name "Pentomino"?

Solomon W. Golomb in his 1965 book Polyominoes: Puzzles, Patterns, Problems, and Packings.

Martin Gardner mentioned them in his Mathematical Games column in the Scientific American magazine.

4: What if there were fewer than five squares?

Pentominoes are an extension of "dominoes", which have two squares joined together.

TODO: picture

You could put three squares together to make a "triomino" - there are two different triominoes, one with a bend, and one that's straight.

TODO: picture

Four squares make a tetromino, which you might have seen in the game of Tetris. There are 5 different tetrominoes, but in Tetris, they color the flipped versions differently, which helps because the game is all about rotating the pieces to fit into the grid.

TODO: picture square straight T L S

5: What about more than five squares?

If you put six squares together, that's a hexomino, and there are different hexominoes.

TODO: picture

You can keep putting squares together to form sept-, oct-, non-, dec-, ominoes and more.