IMC 2007 Q24
24. In the diagram on the right, how many squares, of any size, are there whose entries add up to an even total?
- A 12
- B 20
- C 32
- D 36
- E 45
| 1 | 2 | 3 | 4 | 5 |
| 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 |
| 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 |
Intermediate Mathematical Challenge 2007, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 36
Firstly, there are 12 unit squares which contain an even number. Every 2 × 2 square in the diagram has entries which consist of two odd numbers and two even numbers and hence have an even total. There are 16 of these. Each 3 × 3 square in the diagram, however, has entries which consist of five odd numbers and four even numbers (giving an odd total), or four odd numbers and five even numbers (giving an even total). There are 4 of the latter: those with 8, 12, 14 or 18 in the centre. Every 4 × 4 square in the diagram has entries which consist of eight odd numbers and eight even numbers and hence have an even total. There are 4 of these. Finally, the full 5 × 5 square contains 13 odd numbers and 12 even numbers, giving an odd total. So the required number is 12 + 16 + 4 + 4, that is 36.
Solution © UK Mathematics Trust, from the IMC 2007 solutions.