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Problem of the Week

Problem B

Plus Primes

Problem

The diagram below shows a set of `plus signs' of di�erent sizes, each formed from a set of unitsquares. These unit squares can be rearranged to form rectangles of various sizes; for example,the unit squares in the �rst plus sign can form one rectangle with dimensions 1 unit by 5 units.

a) Rearrange the unit squares of each of the other plus signs to form as many rectangles aspossible. Then construct the next three plus signs in this sequence and do the same forthem.

b) Construct a table (or label your diagrams) showing, for each of the six `plus signs', thenumber of unit squares and the dimensions of the di�erent rectangles which can beformed.

c) What is special about the number of unit squares in those `plus signs' which can only berearranged to form ONE rectangle?

d) Some of the `plus signs' permit rearrangements that are square, a special type ofrectangle. What is special about the number of unit squares in those `plus signs'?

e) What kind of number of unit squares are in the `plus signs' which can be rearranged intomore than one rectangle?

f) After the sixth `plus sign', what will be the next one with a perfect square number ofunit squares?

Solution

a),b) In the diagram below, the six `plus signs' and their associated rectangles (includingsquares) are shown, numbered 1. to 6. The number of unit squares is shown at thebottom right of each `plus sign', and the associated rectangles are labelled with theirdimensions (e.g., for 5., there are two rectangles, one with dimensions 1 unit by 21 units,and one 3 units by 7 units.)

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c) The `plus signs' 1., 3., and 4. have only one possible rectangle. In this case, the numberof unit squares is a prime number, 5, 13, and 17, respectively, each number having onlytwo factors, 1 and itself.

d) The `plus signs' 2. and 6., which permit squares (3 by 3 units and 5 by 5 units) have aperfect square, or square number of unit squares, 9 and 25, respectively.

e) The `plus signs' 2., 5., and 6., which have two rectangles have 9, 21, and 25 unit squaresrespectively. These are composite numbers having more than two factors; 9 has factors 1,3, 9, while 21 has factors 1, 3, 7, and 21, and 25 has factors 1, 5, and 25.

f) The next `plus signs' with a perfect square number of unit squares will be the one with 49(7× 7) unit squares. Since it will have 4 `arms' of length 12, it will be the 12th `plus sign'.

Extension:

What is the �rst `plus sign' which can be rearranged to form three di�erent rectangles?