+---+ +---+---+ +---+ +---+---+ +---+ +---+---+---+
| O | | P P | | Q | | R R | | S | | T T T |
+ + + + + + +---+ +---+ + +---+ +---+ +---+
| O | | P P | | Q | | R R | | S S | | T |
+ + + +---+ + + +---+ + +---+ + + +
| O | | P | | Q | | R | | S | | T |
+ + +---+ + +---+ +---+ + + +---+
| O | | Q Q | | S |
+ + +---+---+ +---+
| O |
+---+
+---+ +---+ +---+ +---+ +---+ +---+ +---+---+
| U | | U | | V | | W | | X | | Y | | Z Z |
+ +---+ + + + + +---+ +---+ +---+ +---+ + +---+ +
| U U U | | V | | W W | | X X X | | Y Y | | Z |
+---+---+---+ + +---+---+ +---+ +---+ +---+ +---+ +---+ + + +---+
| V V V | | W W | | X | | Y | | Z Z |
+---+---+---+ +---+---+ +---+ + + +---+---+
| Y |
+---+
I've labelled each of them with a letter from O to Z to allow them to be quickly referred to as e.g. "the R pentomino". I got this particular labelling from Conway, and its main advantage is that by remembering that it uses the last twelve letters of the alphabet and that the shapes of the letters give hints about the shapes of the pentominos, it allows you to remember the list of the twelve pentominos reasonably easily. Its disadvantage is that some of the hints aren't very good - in particular, O (think of a very tall and thin letter O) and Q (think of an O with a tail), and it isn't exactly obvious which of S and Z is which. (So some people prefer to think of the O and Q pentominoes as instead being more obviously named as the I and L pentominoes, for example: the set of letters I,L,P,R-Z is less memorable than O-Z, but the hints about the pentomino shapes are better.)
Anyway, the puzzles of tiling rectangles with the twelve pentominoes are fairly well-known. As the twelve pentominoes are made up of five squares each, they cover a total of sixty squares, and so can potentially tile 3x20, 4x15, 5x12 and 6x10 rectangles. And in fact, all four of those rectangles can be tiled, most of them in quite a lot of ways - the 6x10 rectangle can be tiled in 2339 different ways, the 5x12 rectangle in 1010 different ways and the 4x15 rectangle in 368 different ways (not counting rotations and reflections of solutions as different). That basically puts the puzzle of finding all the tilings of those rectangles beyond the reach of sensible human endeavours - those numbers have all been found by computer searches.
But there are just two possible tilings of the 3x20 rectangle by the twelve pentominoes, and the puzzle of finding both of them can be solved by humans without the aid of computers. What I'm wondering is what the best line of reasoning is that leads to the two solutions - "best" essentially meaning eliminating as many cases as possible in a 'wholesale' way, to avoid doing a massive search of lots of different possibilities. As an example of a step in such a line of reasoning, consider how much of the middle row of the 3x20 rectangle each of the P-Z pentominoes must cover: the X pentomino must cover three of its squares, the P, R, S and W pentominoes must each cover at least two of its squares and the Q, T, U, V, Y and Z pentominoes must each cover at least one of its squares. So between them, they must cover at least 17 of the 20 squares of the middle row, and so the O pentomino cannot run along the middle row, but must run along either the top row or the bottom row (and we might as well assume that it's the top row, since solutions with it on the top row and solutions with it on the bottom row are mirror images of each other).
Note that what I am after is the reasoning. Just posting the solutions without reasoning gets no credit! Posting interesting lines of reasoning that pin down some aspects of the solutions gets rather more, even if they don't fully solve the problem.
Gengulphus