AMC 10 · 2017 · #22
Grade 11 probabilityPick an answer.
Chasing four coin piles through four rounds buries the problem in bookkeeping. Tool #15 (Organize Information in More Ways) re-records each round as a single arrow from giver to receiver, so the entire game becomes one picture of four arrows on four labeled dots, and the goal turns into a clean structural statement: every dot must send as many arrows as it receives. Tool #1 (Draw a Diagram) is what makes that reframing visible on paper. Tool #7 (Identify Subproblems) then splits the count into two independent halves — which four arrows appear, and in which order the four rounds produce them. Tool #2 (Make a Systematic List) finishes it by listing the only arrow patterns that can balance, with no overlaps and nothing missed.
Turn each round into one arrow
One round is one arrow.
The white balls carry no consequence, so all a round really tells you is who paid whom.
10.S-CP.A.1Organize Information In More WaysEveryone even means out equals in
Back to even means each player receives as often as they give.
Coins only travel along arrows, so getting back to the start means each player's outflow is exactly cancelled by their inflow.
Coins only travel along arrows, so returning to the start means each player's outflow exactly cancels their inflow.
▸ Why?
What one player hands over another receives, so the total never moves and only the split does.
▸ Why?
Each arrow is counted once as an outflow and once as an inflow, so the two tallies must match everywhere.
Balanced arrows break into loops
Balanced arrows break into loops.
If every dot's outflow matches its inflow, a walk along arrows can never dead-end, so it always circles back.
7.SP.C.8Identify SubproblemsCount the two-loop patterns
The two-loop shapes include using one pair twice.
Choose the picture first and the round order second — the two choices do not interfere, so multiply them.
11.S-CP.B.9Make A Systematic ListCount the four-loop patterns
The single loop through all four is counted separately.
A loop through all four players is just a circular seating chart, and four people seat around a circle in (4-1)! = 6 ways.
11.S-CP.B.9Make A Systematic ListAdd the cases and divide
Adding and dividing gives five one-hundred-ninety-seconds.
When every outcome is equally likely, the probability is simply the good count over the total count.
7.SP.C.7Make A Systematic ListDraw every payment as an arrow: everyone lands back where they started exactly when the arrows close up into loops, and then all that is left to do is count the loops.
- Turn each round into one arrow
- Everyone even means out equals in
- Balanced arrows break into loops
- Count the two-loop patterns
- Count the four-loop patterns
- Add the cases and divide