AMC 10 · 2004 · #7
Grade 4 arithmeticPick an answer.
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
Playing three rounds by hand returns to the same shape with every pile one smaller.
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
So each three-round cycle subtracts one from all three, giving a clean formula for 3k rounds.
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 copy of the start with every pile one token lower.
▸ Why?
Each of the three piles takes one turn as leader, so after three rounds the roles come back around in the same order.
▸ 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
Running the smallest pile down to one lands after 36 rounds at (3,2,1).
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
One more round empties the leader, so the game lasts 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