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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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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?

Square Ratio

Stage: 3 Short Challenge Level: Challenge Level:1


Let the length of a short side of a rectangle be $x$ and the length of a long side be $y$. Then the whole square has side of length $(y+x)$, whilst the small square has side of length $y-x$. As the area of the whole square is four times the area of the small square, the length of the side of the whole square is twice the length of the side of the small square. Therefore $y+x=2(y-x)$, i.e., $y=3x$ so $x:y=$1:3.

This problem is taken from the UKMT Mathematical Challenges.