AMC 10 · 2016 · #20
Grade 7 arithmeticPick an answer.
Counting cycles head-on is hard, but every group of three teams falls into exactly one of two shapes. Either the three results loop (a cycle), or one team in the trio beat the other two (call it the 'top' team). Tool #16 (Count the Complement) turns the hard question into an easy subtraction: cycles equal all trios minus the trios that have a top team. The non-cycle trios are easy to count because each has exactly one top team, so tool #2 (Make a Systematic List) tallies them by walking through each team as a possible top and choosing two of the teams it beat. Tool #1 (Draw a Diagram) with arrows makes the two shapes visible so the split is convincing, and tool #4 (Introduce a Variable) first pins down how many teams there even are.
Find how many teams there are
The record fixes the field at 21.
A team's win and loss totals must add up to everyone else it played, so they hand you the team count for free.
6.EE.B.7Introduce A VariableCount every trio of teams
Every trio counts to 1330.
Count ordered picks then divide out the duplicate orderings, and a 'how many sets of three' question becomes one multiplication and one division.
7.SP.C.8Make A Systematic ListSee the only two shapes a trio can make
A trio has only two possible shapes.
With three one-way arrows among three teams, either they loop or someone wins both their games; there is no third option.
With three one-way arrows among three teams, either they loop or someone wins both games; there is no third option.
▸ Why?
Every trio falls into exactly one of the two shapes, so the counts simply add.
▸ Why?
Once the loop is ruled out, some team must have beaten both others, with no alternative left.
Count the non-cycle trios
The non-cycles are easy to count.
Because each non-cycle trio has exactly one team that beat both others, sorting them by that team counts every one of them once.
7.SP.C.8Make A Systematic ListSubtract to get the cycles
Subtracting gives 385, choice (A).
When two clean categories split the whole, counting one and subtracting is faster than counting the messy one directly.
7.NS.A.3Change Focus Count The ComplementEvery group of three teams either loops or has one team that beat the other two, so count the easy 'one team on top' groups and subtract them from all the groups to find the loops.
- Find how many teams there are
- Count every trio of teams
- See the only two shapes a trio can make
- Count the non-cycle trios
- Subtract to get the cycles