Section A - Beginning Pentomino Problems

Nearly all of the problems in this section require less than the full set of twelve pentominoes, which make them much easier than the problems in later sections.

A-1 3 x 5 Rectangles. The example below shows how three pentominoes can be put together to make a 3x5 rectangle. Find another rectangle of the same size using the N, P, and U pentominoes. See how many of the other five ways of building this rectangle you can find using different combinations of pentominoes.

A-2 4 x 5 Rectangles. This rectangle is formed using four pentominoes.

(a) Find at least two other ways of filling a rectangle of the same size using four pentominoes.
(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 pentominoes 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 pentominoes.

(a) Find at least two other ways of filling a square of the same size using five pentominoes.
(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 first three problems, there are 14 rectangles that can be constructed that do not use the full set of pentominoes. One of the rectangles is very simple to solve, even easier than the first problem. See if you can discover all the other different rectangles that can be constructed and a solution for each one.

Problems A-5 and A-6: Congruent Groups.

These problems require two (or more) pentominoes to be put together to make one total shape, and then to find the same number of pentominoes that make the same total shape as the first one. Two groups of pentominoes that can form the same shape are called congruent groups. In the example below, the I and L pentominoes are congruent to the N and W pentominoes because they can form the same shape.

A-5 Find two pentominoes which will make the same total shape as the one given below for the I and U.

A-6 Put the U and Y pentominoes together to make the same total shape as the

(a) N and P,
(b) N and Z,
(c) V and X,
(d) F and N,
(e) P and T,
(f) L and T, and
(g) L and Z.

Note: The total shape for each of these will be different.

A-7 The Duplication Problem. Four pentominoes can be put together to make a copy of the P pentomino which is two times as wide and two times as high as the original piece.

Make a duplication of any pentomino other than the P. Two of them cannot be done. Which ones?

Notice that the solution of the P duplication above in fact uses its smaller counterpart in the solution. Find a solution of the P duplication that does not use the P pentomino. For every other duplication, determine if there is a solution that does use its smaller counterpart and then determine if there is a solution that does not use its smaller counterpart.

Problems A-8 through A-11: Simultaneous Solutions.

A pattern is given and you need to use some of the pentominoes to cover the pattern. You then need to use some of the remaining pentominoes to cover the same pattern. This process is called finding simultaneous solutions of the given pattern.

A-8 Find three simultaneous solutions to each of the 10-square patterns below.

 (a) (b) (c) (d)

A-9 Find three simultaneous solutions to each of the 15-square patterns below.

 (a) (b)

A-10 Simultaneous Rectangles. Determine how many pairs of rectangles can be constructed simultaneously using a single set of pentominoes. As an example, problem A-1 demonstrates that two 3x5 rectangles can be constructed simultaneously.

A-11 Simultaneous Duplications. Find all pairs of pentominoes such that their duplications can be constructed simultaneously.

A-12 A Pentomino Farm. The image below shows the full set of twelve pentominoes arranged as a fence to enclose a field. The rule used to join them is that they must touch along the full edge of a square and not just at the corners. The enclosed field has an area of 43 unit squares, but the pieces have not been used very efficiently.

The problem is to find a pentomino fence enclosing the greatest possible area. You can grade your attempts by the following table:

 Area Grade 120 or above A 110 - 119 B 100 - 109 C 80 - 99 D under 80 Horrible!!