# 10x10 divided into the most number of rectangles of different area

How can a 10x10 be divided into rectangles such that there are as many as possible and they all have different area? Can you find multiple solutions that are not mirror/rotation of each other?

Good luck!

• so it must be divided into only rectangles, all be used, and all rectangles must have integer sizes? – Jasen Oct 13 '19 at 6:21
• @Jasen that is correct. – Dmitry Kamenetsky Oct 13 '19 at 7:57

A solution:

 2 I I I I I I I I I
2 I I I I I I I I I
3 C C C C E E E E E
3 C C C C E E E E E
3 C C C C E E E E E
A A A A A A A A A A
1 9 9 9 9 9 9 9 9 9
4 4 8 8 8 8 8 8 8 8
4 4 5 5 5 5 5 6 6 6
7 7 7 7 7 7 7 6 6 6


This uses 13 rectangles, of sizes 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18.

Proof that this is the maximum:

Note that we can't use sizes 11 and 13: that would require 1x11 and 1x13 rectangles which are too large to fit in a 10x10. So we take the least 14 rectangles that are possible: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16. But those give a sum of 112 so we can conclude that 14 rectangles couldn't fit in a grid with 100 squares.

A solution with different sizes of rectangles.

 _ _ _ _ _ _ _ _ _ _
|1|2 _|7 _ _ _ _ _ _|
|4 _ _ _|6 _ _ _ _ _|
|3 _ _|5 _ _ _ _|10 |
|14 |16     |8  |   |
|   |       |   |   |
|   |       |   |   |
|   |_ _ _ _|_ _|_ _|
|   |15       |9    |
|   |         |     |
|_ _|_ _ _ _ _|_ _ _|
13 rectangles with sizes 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16