Even squares
Can you find squares within a number grid whose entries add up to an even total?
Age
11 to 14
Challenge level
Problem
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 |
Student Solutions
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
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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
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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
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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