GK 2017 Q22
22. Stan had 125 small cubes. He glued some of them together to form a large cube with nine tunnels, each perpendicular to two opposite faces and passing through the cube, as shown in the diagram.
How many of the small cubes did he not use?
- A 52
- B 45
- C 42
- D 39
- E 36
Grey Kangaroo 2017, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 39
Each of the nine tunnels in Stan's cube is five cubes long. However, the three tunnels starting nearest to the top front vertex of the cube all intersect one cube in. Similarly, the three tunnels starting at the centres of the faces all intersect at the centre of the large cube and the final three tunnels all intersect one cube in from the other end to that shown. Hence the number of small cubes not used is 9 × 5 − 3 × 2 = 45 − 6 = 39.
Solution © UK Mathematics Trust, from the GK 2017 solutions.