There is a grid of 20 squares by 10 squares. How many different rectangles are possible?
(Note that square is a rectangle).

Okay, here goes round 2:

Same brute force of an idea as my previous post, but I started at the bottom left of the 10 x 20 grid (10 being the base and 20 being the height) this time and went straight through from level to level always using the bottom left most square as my anchor.

Using this method, the first level has 10 unique rectangles:
1x1, 1x2, 1x3 ... 1x10

The second level has 9 unique rectangles:
2x2, 2x3, 2x4 ... 2x10

This continues eliminating one from each subsequent level all the way until you reach level 11. (Total number of unique rectangles thus far equals 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1). After this point, every single rectangle you draw will not have another single corresponding rectangle at further levels because the base length has already been exceeded (note: the sum of the last 10 levels will therefore be 10 x 10).

Therefore the final number for all the unique rectangles will be 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + (10 x 10) = 155.

If this has been a confusing solution, I can try to clarify it better. If this helps, here they are all named:

1x1, 1x2, 1x3, 1x4, 1x5, 1x6, 1x7, 1x8, 1x9, 1x10
2x2, 2x3, 2x4, 2x5, 2x6, 2x7, 2x8, 2x9, 2x10
3x3, 3x4, 3x5, 3x6, 3x7, 3x8, 3x9, 3x10
4x4, 4x5, 4x6, 4x7, 4x8, 4x9, 2x10
5x5, 5x6, 5x7, 5x8, 5x9, 5x10
6x6, 6x7, 6x8, 6x9, 6x10
7x7, 7x8, 7x9, 7x10
8x8, 8x9, 8x10
9x9, 9x10
10x10
11x1, 11x2, 11x3, 11x4, 11x5, 11x6, 11x7, 11x8, 11x9, 11x10
12x1, 12x2, 12x3, 12x4, 12x5, 12x6, 12x7, 12x8, 12x9, 12x10
13x1, 13x2, 13x3, 13x4, 13x5, 13x6, 13x7, 13x8, 13x9, 13x10
14x1, 14x2, 14x3, 14x4, 14x5, 14x6, 14x7, 14x8, 14x9, 14x10
15x1, 15x2, 15x3, 15x4, 15x5, 15x6, 15x7, 15x8, 15x9, 15x10
16x1, 16x2, 16x3, 16x4, 16x5, 16x6, 16x7, 16x8, 16x9, 16x10
17x1, 17x2, 17x3, 17x4, 17x5, 17x6, 17x7, 17x8, 17x9, 17x10
18x1, 18x2, 18x3, 18x4, 18x5, 18x6, 18x7, 18x8, 18x9, 18x10
19x1, 19x2, 19x3, 19x4, 19x5, 19x6, 19x7, 19x8, 19x9, 19x10
20x1, 20x2, 20x3, 20x4, 20x5, 20x6, 20x7, 20x8, 20x9, 20x10
Edited on January 21, 2004, 6:17 pm

Posted by Benjamin J. Ladd on 2004-01-21 18:03:47