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What are the missing numbers in the pyramids?

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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Below 400

Stage: 3 and 4 Short Challenge Level: Challenge Level:1


Note that the number at the end of the $n$th row is $n^2$, so 400 will lie at the end of the 20th row. The row below will end in 21$^2$, i.e., 441, so the number directly below 400 will be 440.

This problem is taken from the UKMT Mathematical Challenges.

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