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Pyramids

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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Mindreader

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

Fibonacci Padlock

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2
If the first two digits are $a$ and $b$ (with $a\neq 0$) then the six terms will be $a$, $b$, $a+b$, $a+2b$, $2a+3b$ and $3a+5b$, so we must have $3a+5b \leq 9$.

If $b=0$ then $a$ can be $1$, $2$ or $3$. If $b=1$ then $a$ can only be $1$, and $b$ cannot be greater than $2$.

Hence there are four possibilities, namely $101123$, $202246$, $303369$ and $112358$.

This problem is taken from the UKMT Mathematical Challenges.
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