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Answer: 36
1 by 1 squares:
Must contain an even number
Up to 25 there are 12 even numbers
2 by 2 squares:
examples:
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
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
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
Even Squares
Age 11 to 14
ShortChallenge Level 
Answer: 36
1 by 1 squares:
Must contain an even number
Up to 25 there are 12 even numbers
2 by 2 squares:
examples:
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
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
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
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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