IMC 2005 Q16
16. A 1 × 2 × 3 block is placed on an 8 × 8 board, as shown with the 1 × 2 face ‘X’ at the bottom. It is rolled over an edge without slipping onto the 1 × 3 face Y, then onto the 2 × 3 face Z, then onto X, Y, Z again in that order. How many different squares on the board has the block occupied altogether, including the starting and ending positions?
- A 18
- B 19
- C 20
- D 21
- E 22
Intermediate Mathematical Challenge 2005, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 19
The two dark squares are those on which the block stands originally. After the first move, it occupies the squares labelled ‘1’ or ‘1/5’. After the second move it occupies the squares labelled ‘2’ and so on. After the fifth and final move, it occupies the squares labelled ‘5’ or ‘1/5’.
As can be seen, 19 squares are occupied altogether.
Solution © UK Mathematics Trust, from the IMC 2005 solutions.