Even Squares
Can you find squares within a number grid whose entries add up to an even total?
In the diagram below, how many squares (of any size) are there whose entries add up to an even total?
1 | 2 | 3 | 4 | 5 |
6 | 7 | 8 | 9 | 10 |
11 | 12 | 13 | 14 | 15 |
16 | 17 | 18 | 19 | 20 |
21 | 22 | 23 | 24 | 25 |
Answer: 36
1 by 1 squares:
Must contain an even number
Up to 25 there are 12 even numbers
2 by 2 squares:
examples:
Image
![Even Squares Even Squares](/sites/default/files/styles/large/public/thumbnails/content-id-5772-2%252520by%2525202%2525201.png?itok=_ntl2ZWi)
Image
![Even Squares Even Squares](/sites/default/files/styles/large/public/thumbnails/content-id-5772-2%252520by%2525202%2525202.png?itok=OJxKTvMi)
2 by 2 squares always contain 2 even and 2 odd numbers so all have an even total
2 by 2 squares can't have their first number in the last row or column, so there are 16 2 by 2 squares in the grid
3 by 3 squares:
They can have 4 odd numbers or 5 odd numbers
Image
![Even Squares Even Squares](/sites/default/files/styles/large/public/thumbnails/content-id-5772-3%252520by%2525203%2525201.png?itok=TVdvUDJx)
Image
![Even Squares Even Squares](/sites/default/files/styles/large/public/thumbnails/content-id-5772-3%252520by%2525203%2525202.png?itok=cg-EOCm2)
5 odd numbers means the sum is odd, 4 odd numbers means the sum is even
3 by 3 squares containing 4 odd numbers begin on even numbers
How many are there in the grid? The square starting at 6 is good but the one starting at 14 is not
Image
![Even Squares Even Squares](/sites/default/files/styles/large/public/thumbnails/content-id-5772-starting%252520at%2525206%252520full.png?itok=2rErBSib)
Image
![Even Squares Even Squares](/sites/default/files/styles/large/public/thumbnails/content-id-5772-starting%252520at%25252014.png?itok=qE97_ZHK)
The 3 by 3 square can start at 2, 6, 8 or 12 (4 possible)
4 by 4 squares:
These contain 8 even numbers and 8 odd numbers (they are 4 2 by 2 squares stuck together)
There is one in each corner of the whole square (4 possible)
5 by 5 squares:
Not allowed - the grid contains 12 even and 13 odd numbers
Total: 12 + 16 + 4 + 4 = 36