The twelve pentominoes
Tonight's idea
A domino is two squares joined along an edge. Join five squares the same way and you have a pentomino. Turning a shape round or flipping it over does not make a new one. With that rule there are exactly twelve, each named after a letter it resembles.
How can anyone be sure none is missing? Search in an orderly way. Sort the shapes by their longest straight row of squares.
- Five in a row: only I.
- Four in a row: the fifth square goes beside an end square (L) or beside one of the middle two (Y).
- Three in a row: two squares are left to place. That gives eight shapes: F, N, P, T, U, V, X, Z.
- Never more than two in a row: only the staircase W.
That is \(1 + 2 + 8 + 1 = 12\). Twelve shapes of five squares cover \(12 \times 5 = 60\) squares, so the full set might fill a rectangle of area 60.
Watch one: filling a 6 by 10 rectangle
The rectangle has \(6 \times 10 = 60\) squares. Check the letters: all twelve pieces are there, once each, with no gaps and no overlaps.
Finding an arrangement like this by hand can take a long time. A good plan is to place the awkward pieces first. X is the worst. It can never cover a corner square, and its four arms leave notches that other pieces must fill. Save easy pieces such as L and P for the end.
Place, turn and flip the pieces yourself: mathdetours.com/demos/pentomino.html.
Your turn
- Every pentomino has area 5. The perimeter is the distance round the outside, counted in sides of a small square. Find the perimeter of I, of X and of P. Eleven of the twelve share one perimeter. Which is the odd one out?
- Two copies of the same pentomino can sometimes make a 2 by 5 rectangle. Three of the twelve can do it. Which three? Sketch each rectangle.
- (a) List every rectangle with whole-number sides and area 60. (b) Two of them can never be filled by the twelve pentominoes. Which two, and why?
- A tetromino is four squares joined edge to edge. Use the longest-row method to find them all. Draw them and say how many there are.
Stretch: no flipping
Suppose the pieces are painted on one side, so you may turn them but not flip them over. Some pentominoes now come in two versions, a left-handed one and a right-handed one. Which ones? How many different pieces are there now?
Answers
- I: \(5 + 1 + 5 + 1 = 12\). X: 12. P: 10. P is the odd one out; the other eleven all have perimeter 12. P is the only one containing a 2 by 2 block, so more of its sides are hidden inside the shape.
- I, L and P. Two I's lie side by side. Two L's lock together, one turned half way round. Two P's do the same.
- (a) 1 by 60, 2 by 30, 3 by 20, 4 by 15, 5 by 12, 6 by 10. (b) 1 by 60 and 2 by 30. The X is three squares wide whichever way you turn it (so are several others), so it cannot fit in a strip only 1 or 2 squares wide. The other four rectangles can all be filled. The 3 by 20 is the hardest: it has only two solutions.
- Five. Four in a row: the straight one. Three in a row: the fourth square goes beside an end (an L) or beside the middle (a T). Never more than two in a row: the 2 by 2 square, and the zigzag made of two dominoes, one shifted along by a square.
Stretch. A shape with a mirror line looks the same flipped, so it gains nothing: that is I, T, U, V, W and X. The other six, F, L, N, P, Y and Z, each come in two versions. That makes \(6 + 2 \times 6 = 18\) pieces.