14. Jenny decided to enter numbers into the cells of a $3 \times 3$ table so that the sum of the numbers in all four possible $2 \times 2$ cells will be the same. The numbers in three of the corner cells have already been written, as shown.
Which number should she write in the fourth corner cell?
A 0
B 1
C 4
D 5
E 6
Solution
B

Let the numbers in the centre left cell and the centre right cell be $x$ and $y$ and let the number in the lower left corner be $z$, as shown in the diagram. Since the sum of the numbers in all four possible $2 \times 2$ cells should be the same, by considering the top left $2 \times 2$ cell and the top right $2 \times 2$ cell, since the top two cells in the central column are common, we have $2+x=4+y$ and hence $x=y+2$.
Similarly by considering the bottom left $2 \times 2$ cell and the bottom right $2 \times 2$ cell where the lower two cells in the central column are common, we have $z+x=y+3$ and hence $z+y+2=y+3$, which has solution $z=1$. Therefore the value of the number in the fourth corner cell is 1 .