AMC 10 · 2004 · #8
Grade 4 arithmeticPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Simulating 37 rounds one by one would work but is slow and error-prone. Instead, play just the first few rounds carefully as a systematic list (Tool #2) and watch what happens over one full turn of leadership. The three starting totals 15,14,13 take turns being the leader, and after three rounds every player has lost exactly 1 token while the ordering resets. That repeating three-round cycle is the pattern (Tool #5): every 3 rounds the whole board slides down by 1. Once you see it, you jump straight to the near-empty state instead of listing 37 lines, then play out only the final round by hand.
Play the first three rounds
Track (A,B,C): each round the leader loses 3 and the other two gain 1. (15,14,13)→(12,15,14)→(13,12,15)→(14,13,12).
Each of the three starting piles takes one turn as leader, so after three rounds the situation comes back around in the same order.
4.NBT.B.4Make A Systematic ListSpot the three-round cycle
Three rounds on, every pile is 1 smaller and A leads again, so after 3k rounds the totals are (15-k, 14-k, 13-k).
Three rounds later the board is a perfect copy of the start with every number lowered by one, so the pattern just repeats down the ladder.
Three rounds later the board is a perfect copy of the start with every pile lowered by one.
▸ Why?
Each of the three piles takes one turn as leader, so after three rounds the roles come back around.
▸ Why?
Every pile drops by the same single token, so the gaps between the piles are exactly what they were.
Jump to the nearly-empty state
Smallest pile C hits 1 when 13-k=1, so k=12. After 3×12=36 rounds the totals are (3,2,1) — nobody is empty yet.
Extending the pattern by 12 cycles lands you one step from the end without listing every round.
4.OA.A.3Look For A PatternPlay the final round
At (3,2,1) leader A holds exactly 3, gives all of them away and lands on 0, so the game ends at 36+1=37 rounds, choice (B).
A leader holding exactly 3 tokens gives them all away in one round, so that is the round the game must stop.
4.OA.A.3Make A Systematic ListWhen a process repeats in a cycle, find how much changes each cycle and multiply — then you only have to play out the last little bit by hand.
- Play the first three rounds
- Spot the three-round cycle
- Jump to the nearly-empty state
- Play the final round