PK 2020 Q23
23. Alice, Belle and Cathy had an arm-wrestling contest. In each game two girls wrestled, while the third rested. After each game, the winner played the next game against the girl who had rested. In total, Alice played 10 times, Belle played 15 times and Cathy played 17 times. Who lost the second game?
- A Alice
- B Belle
- C Cathy
- D Either Alice or Belle could have lost the second game.
- E Either Belle or Cathy could have lost the second game.
Pink Kangaroo 2020, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A Alice
Since each game involved two girls, the number of games played is (10 + 15 + 17) ÷ 2 = 21. Alice played 10 of these, and rested for 11 of them. The maximum amount of resting possible is obtained by alternately losing and resting. To rest 11 times, Alice must have rested the odd-numbered games (1st, 3rd, etc) and lost the even numbered games (2nd, 4th, etc). Hence Alice lost the second game.
Solution © UK Mathematics Trust, from the PK 2020 solutions.