IMC 2006 Q12
12. When a solid cube is held up to the light, how many of the following shapes could its shadow have?
- A 0
- B 1
- C 2
- D 3
- E 4
Intermediate Mathematical Challenge 2006, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 3
Three of the shadows are possible, the first, third and fourth. The diagrams below show four positions obtained by rotating a cube about \(XY\), a line through the midpoints of a pair of opposite edges. Three of the shadows in the question correspond to three of these positions. Though the second shadow matches the angles and four of the lengths given by the second position, it does not match the other two lengths and so is not possible.
In fact, the second shadow corresponds to a cuboid which is half a cube, as shown alongside.
Solution © UK Mathematics Trust, from the IMC 2006 solutions.