AMC 10 · 2014 · #18
Grade 7 geometry-2dPick an answer.
"How many arrangements" with a finite pool screams Tool #2 (Make a Systematic List) — but a raw list of 120 orderings is wasteful. First shrink the work two ways. Tool #16 (Count the Complement) notices that the total is 15, so an arc summing to k leaves a complementary arc summing to 15-k; that instantly makes 1--5, 10--15 free and pairs the doubtful targets as 6⇔ 9 and 7⇔ 8, so we only ever test 6 and 7. Tool #1 (Draw a Diagram) with the rotation-and-reflection rule cuts 120 orderings down to just 12 distinct circles. Then Tool #2 lists those 12 and Tool #3 (Eliminate) crosses off every circle that manages to build both 6 and 7, leaving the bad ones.
List the sums you always get
Most totals come out free.
Singles, all-but-one, and the whole circle are automatic, so most targets can never be the problem.
2.NBT.B.5Make A Systematic ListOnly 6 and 7 can go wrong
Only two targets can ever go wrong.
An arc and everything left over always add to 15, so reaching a number automatically reaches its partner.
An arc and everything left over always add to the same total, so reaching one number reaches its partner.
▸ Why?
The circle is exactly the arc plus the rest, so their totals fill the whole.
▸ Why?
Every arc has a leftover and every leftover is an arc, so the two are two readings of one choice.
Only 12 different circles exist
Symmetry leaves only 12 distinct circles.
Spinning or flipping a circle does not change which numbers are neighbors, so those copies count once.
4.G.A.3Draw A DiagramTest each circle for 6 and 7
Each circle is tested on those two targets.
Since 6 and 7 each form only three ways, checking whether any of those pieces sits together is quick for every circle.
7.SP.C.8Eliminate PossibilitiesCount the bad circles
Exactly 2 circles fail, choice (B).
Blocking 6 forces one circle and blocking 7 forces a different one, and there is no third way.
7.SP.C.8Eliminate PossibilitiesThe whole circle is 15, so every arc and its leftover add to 15 — that makes all targets free except 6 and 7, and only two of the twelve different circles fail to build one of them.
- List the sums you always get
- Only 6 and 7 can go wrong
- Only 12 different circles exist
- Test each circle for 6 and 7
- Count the bad circles