IMC 2020 Q17
17. The Knave of Hearts stole some tarts. He ate half of them, and half a tart more. The Knave of Diamonds ate half of what was left, and half a tart more. Then the Knave of Clubs ate half of what remained, and half a tart more. This left just one tart for the Knave of Spades.
How many tarts did the Knave of Hearts steal?
- A 63
- B 31
- C 19
- D 17
- E 15
Intermediate Mathematical Challenge 2020, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 15
Suppose that at a particular stage there are \(m\) tarts available for a Knave to eat and that there are \(n\) left after he has finished eating.
Then \(n = m - (\tfrac{1}{2}m + \tfrac{1}{2}) = \tfrac{1}{2}m - \tfrac{1}{2}\).
Therefore, \(m = 2n + 1\).
As the Knave of Spades received one tart, then the number of tarts which the Knave of Clubs was given was 2 × 1 + 1 = 3.
Similarly, the number of tarts which the Knave of Diamonds was given was 2 × 3 + 1 = 7.
Finally, the number of tarts which the Knave of Hearts stole was 2 × 7 + 1 = 15.
Solution © UK Mathematics Trust, from the IMC 2020 solutions. Longer versions with investigations: Solutions and investigations (PDF).