GK 2017 Q16
16. Ella wants to write a number into each cell of a 3 × 3 grid so that the sum of the numbers in any two cells that share an edge is the same. She has already written two numbers, as shown in the diagram.
When Ella has completed the grid, what will be the sum of all the numbers in the grid?
- A 18
- B 20
- C 21
- D 22
- E 23
| 2 | ||
| 3 | ||
Grey Kangaroo 2017, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 22
Label the numbers Ella writes down as shown in the diagram.
| 2 | \(a\) | \(b\) |
| \(c\) | \(d\) | 3 |
| \(e\) | \(f\) | \(g\) |
Since the sum of the numbers in any two adjacent cells is the same, \(2 + a = a + b\) and hence \(b = 2\). Therefore \(b + 3 = 2 + 3 = 5\). Hence the sum of the numbers in any two adjacent cells is 5. It is now straightforward to see that \(a = c = f = 3\) and that \(b = d = e = g = 2\). Therefore the sum of all the numbers in the grid is 5 × 2 + 4 × 3 = 22.
Solution © UK Mathematics Trust, from the GK 2017 solutions.