Question

A 2 by 3 rectangle contains 8 squares. A 3 by 4 rectangle contains 20 squares. A 4 by 6 rectangle contains 50 squares. What size rectangle contains exactly 100 squares? Is there more than one? Can you find them all? Can you prove there are no more?

Answer 1

Answer:

The rectangles that have exactly 100 squares are 1 by 100  4 by 11 and 5 by 8

Step-by-step explanation:

The given parameters are;

A 2 by 3 rectangle contains 8 squares

A 3 by 4 rectangle contains 20 squares

A 4 by 6 rectangle contains 50 squares  

We note that the number of squares varies proportionately to the increase in the dimension of the rectangle, that is we have;

2 by  3 rectangle contains 8 squares

3 by 4 rectangle contains 20 squares increase by 12 = 3 × 4

4 by 5 rectangle contains 40 squares increase by 20 = 4 × 5

The number of squares formed can therefore be arranged in a matrix as follows(please see attached matrix);

Column,     1,       2,     3,      4,       5,     6,        7,       8,         9,           10,         11

Row

1

2

3

4

5

6

7

8

9

10

11

It will be observed that after reaching a point where the row and column are equal, (x, x) the rate of increase in the number of squares become constant and equal to x × ((x - 1) × 0.5) + 1) = x × (x + 1)/2

From the matrix, the rectangles that have exactly 100 squares are 1 by 100  4 by 11 and 5 by 8 and there are no more as presented in the attached matrix, the number of squares formed keeps increasing for each column and row added.