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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?

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The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Even So

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Find some triples of whole numbers $ a $, $ b $ and $ c $ such that $ a^2 + b^2 + c^2 $ is a multiple of 4. Is it necessarily the case that $ a $, $ b $ and $ c $ must all be even? If so, can you explain why?