C2. In a $6 \times 6$ grid of numbers:
(i) all the numbers in the top row and the leftmost column are the same;
(ii) each other number is the sum of the number above it and the number to the left of it;
(iii) the number in the bottom right corner is 2016.
What are the possible numbers in the top left corner?
Solution
Suppose we start by letting the number in the top left corner be $t$. Since, by rule (i), the numbers in the top row and left-hand column are all the same, we can fill them all in as $t$. Then we can use rule (ii) to fill in the rest of the grid, as shown.
From rule (iii), we now obtain
$$
252 t=2016 .
$$

Dividing each side by 252 , we get
$$
t=8 \text {. }
$$

So the only possible number in the top left corner is 8 .