Answer:
The rectangles that have exactly 100 squares are 1 by 100 Â 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 Â
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, Â Â 1, Â Â Â 2, Â Â 3, Â Â Â 4, Â Â Â 5, Â Â 6, Â Â Â Â 7, Â Â Â 8, Â Â Â Â 9, Â Â Â Â Â 10, Â Â Â Â 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 Â 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.
