PK 2017 Q15
15. Maja decided to enter numbers into the cells of a 3 × 3 grid. She wanted to do this in such a way that the numbers in each of the four 2 × 2 grids that form part of the 3 × 3 grid have the same totals. She has already written numbers in three of the corner cells, as shown in the diagram. Which number does she need to write in the bottom right corner?
- A 0
- B 1
- C 4
- D 5
- E impossible to determine
| 3 | 1 | |
| 2 | ? |
Pink Kangaroo 2017, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 0
Let the numbers around the top left cell be \(a\), \(b\) and \(c\) as shown. Then the sum of the top left 2 × 2 square (and hence all the 2 × 2 squares) is \(a + b + c + 3\). The top right 2 × 2 square already contains \(a\) and \(b\) and 1, so the middle right cell must contain \(c + 2\). The bottom left 2 × 2 square contains \(b + c + 2\) so the bottom middle cell is \(a + 1\). The bottom right 2 × 2 square already contains \(a + b + c + 3\) so the missing value is zero. There are many ways to complete the grid; one way is shown here.
| 3 | a | 1 |
| c | b | c + 2 |
| 2 | a + 1 | ? |
| 3 | 7 | 1 |
| 4 | 5 | 6 |
| 2 | 8 | 0 |
Solution © UK Mathematics Trust, from the PK 2017 solutions.