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Summing Consecutive Numbers

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

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?

Fibs

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?

Mindreader

Age 11 to 14
Challenge Level

This is what happens, only twisted so that you know what's going on.

The friend picks three consecutive numbers, $x, x+1,$ and $x+2$ where $ x \leq 58$.

He/she then picks a number $3n$, where $n \leq 33$ and tells you what it is.

Adding the 4 numbers gives $3x + 3 + 3n = 3(x + 1 + n)$.

Multiplying by 67 gives the number $201(x + 1 + n)$.

We know $x + 1 + n \leq 58 + 1 + 33 = 92$.

So the 2 digit number is 1 times $(x + 1 + n)$ and the rest is $200(x + 1 + n)$ which is 200 times the 2 digit number, so you double the 2 digit number to get the remaining digits.

You know what $n$ is, so subtract $n$ and 1 from the 2 digit number to get $x$.

From Ian Green, Age 13 Coopers Company and Coburn School, Upminster, Essex.