GK 2018 Q23
23. Wendy wants to write a number in every cell on the border of a table.
In each cell, the number she writes is equal to the sum of the two numbers in the cells with which this cell shares an edge. Two of the numbers are given in the diagram.
What number should she write in the cell marked \(x\)?
- A 10
- B 7
- C 13
- D −13
- E −3
| 3 | ||||
| \(x\) | ||||
| 10 | ||||
Grey Kangaroo 2018, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 7
Let the values she writes in some of the cells be as shown in the diagram. Since the number in any cell is equal to the sum of the numbers in the two cells that border it, we have \(a = 10 + b\) and hence \(b = a - 10\). Also we have \(b = c + a\) giving \(c = -10\), \(c = b + d\) giving \(d = -a\) and \(d = c + 3\) giving \(d = -7\). Therefore \(a = 7\) and \(b = -3\) and it is now possible to work out the remaining unknown values. Since \(10 = 7 + e\), we obtain \(e = 3\).
Similarly, \(e = 10 + f\) giving \(f = -7\) and then \(f = e + g\) giving \(g = -10\). Also \(g = f + h\) giving \(h = -3\) and \(h = g + x\) giving \(x = 7\).
| 3 | ||||
| \(d\) | ||||
| \(c\) | ||||
| \(b\) | ||||
| \(a\) | \(x\) | |||
| 10 | \(e\) | \(f\) | \(g\) | \(h\) |
Solution © UK Mathematics Trust, from the GK 2018 solutions.