### Pyramids

What are the missing numbers in the pyramids?

### 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?

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 $112358$.