Beginning Pentomino Problems Nearly all of the problems in this section require less than the full set of twelve pentominos, which make them a little easier. A-1 3 x 5 Rectangles. The example below shows how three pentominos can be put together to make a 3x5 rectangle. Find another rectangle of the same size using the N, P, and U pentominos. See how many of the other five ways of building this rectangle you can find using different combinations of pentominos. A-2 4 x 5 Rectangles. This rectangle is formed using four pentominos. (a) Find at least two other ways of filling a rectangle of the same size using four pentominos. (b) Find a solution where a pentomino piece does not touch the outer edge of the rectangle. (c) Find a solution so that the four pentominos used touch at the same point. (This is called a crossroads solution.) (d) Find a solution so that the rectangle can be divided into two identical shapes. A-3 5 x 5 Squares. Now we have a square that is built from five pentominos. (a) Find at least two other ways of filling a square of the same size using five pentominos. (b) For each pentomino, try to find a solution where the given piece does not touch an edge. Does this answer change if the I pentomino were not used in the solution? A-4 Many Rectangles. Including the three problems, there are 14 rectangles that can be constructed that do not use the full set of pentominos. One of the rectangles is very simple to solve, even easier than the first problem.
80 - 99 D under 80 Horrible!! Partial Solutions for some of the previous questions A-1 3x5 Rectangles. The other five solutions use the following groups of pieces: {P, U, V}, {N, P, U}, {P, U, Y}, {F, P, U}, and {L, P, V}. A-2 4x5 Rectangles. (b) There are 2 solutions, one for the P and the U. (c) One solution uses {L, P, V, Y}. (d) One solution uses {F, L, U, V}. A-3 5 x 5 Squares. (b) Solutions for the F, P, T, U, V, W, and Z. If the rectangle does not contain the I, then solutions only exist for the P, U, W, and Z. A-4 Many Rectangles. Size of Rectangle Number of Solutions 1 x 5 1 3 x 5 7 2 x 10 2 4 x 5 50 5 x 5 107 3 x 10 145 5 x 6 541 5 x 7 1396 4 x 10 2085 5 x 8 3408 3 x 15 201 5 x 9 5902 5 x 10 6951 5 x 11 4103 A-5 Congruent Groups. {F, T}
Section B - Intermediate Pentomino Problems This section contains problems that require the use of all twelve pentominoes. You will discover that even the first ten problems are very difficult to solve. Perseverance and patience is essential to solve any of the problems in this section. Problems B-1 through B-6: 8x8 Square with Four Unit Holes. B-1 B-2 B-3 B-4 B-5 B-6 Problems B-11 through B-13: Three Hearts. B-11 B-12 B-13 Problems B-14 through B-16: Three Crosses. B-14 B-15 B-16 Problems B-17 through B-22: Variations of the 88 Square with Four Unit Holes. More interesting solutions can be sought for problems B-1, B-4 and Each pattern can be divided into congruent parts. The division is indicated by the heavy line. B-17 B-18 B-19 B-20 B-21 B-22 Problems B-23 through B-28: Variations of the 610 Rectangle. are over two thousand solutions to the basic 6  10 rectangular pattern. These six problems impose additional constraints to make the pattern more challenging. B-23 Obtain a solution to the 610 rectangle by constructing two simultaneous 65 rectangles. (Note that this solution can also form a 512 rectangle.) B-24 Build the 610 rectangle such that it contains (a) a 35 sub-rectangle. (b) a 45 sub-rectangle. Can the sub-rectangles be constructed so that it is completely contained within the larger rectangle?