What are some interesting math problems to explore and solve?

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Please look for 1) golden rectangle 2) Maguval squares
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If m and n are odd positive integers, then m2 + n2 is even, but not divisible by 4. Justify.
Since m is odd positive integer then we can write m=2a+1 and n=2b+1, such that a,b >0 m2+n2=(2a+1)^2+(2b+1)^2 = 4a^2+1+4a+4b^2+1+4b = 4a^2+4b^2+4a+4b+2 = 2(2a^2+2b^2+2a+2b+1) Above eqn is divisible by 2 but not divisible by 4.
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