5. All the digits $2,3,4,5$ and 6 are placed in the grid, one in each cell, to form two three-digit numbers that are squares.
Which digit is placed in the centre of the grid?
A 2
B 3
C 4
D 5
E 6
SolUTION
A

The three-digit squares use just the digits $2,3,4,5$ and 6 . The smallest square which uses these digits is $15^{2}=225$ and the largest is $25^{2}=625$. However, the squares that go in the grid cannot repeat digits. Therefore the only squares that we need consider are the three-digit squares that use three of digits $2,3,4,5$ and 6 , once each. It may be checked (see Exercise 5.1, below) that there are just three squares which meet these conditions. They are $16^{2}=256,18^{2}=324$ and $25^{2}=625$.

The digit 2 occurs twice as the tens digit of these three squares. Therefore, the digit that goes in the central square of the grid is 2 .
In the context of the SMC we could stop here. However, for a complete solution you would need to add that, as the squares 324 and 625 use each of the digits 3 , 4,5 and 6 just once, it is possible to place the digits in the grid so as to make these squares. The diagram shows one way to do this.