AMC 10 · 2015 · #25
Grade 11 geometry-2dpatternPick an answer.
Adding 2015 hops one at a time is hopeless, and the hops never repeat because the lengths keep growing. But the directions do repeat: 12 × 30^°=360^°, so only twelve headings ever occur. Tool #1 (Draw a Diagram) puts those twelve headings on one picture and shows the deeper fact — headings six apart point exactly opposite, so all twelve add to zero. Tool #4 (Introduce a Variable) names the twelve unit headings u₀,…,u₁₁ so a hop becomes (length) × u_j and the whole trip becomes one sum. Tool #5 (Look for a Pattern) is the crux: chop the hops into blocks of twelve. Inside a block the lengths look like (block start)+j, and the (block start) part is multiplied by the sum of all twelve headings — which is zero. So every block of twelve hops moves the bee by exactly the same vector, however far out the spiral has grown. That converts 2015 hops into 168 copies of one short push. Tool #11 (Work Backwards) handles the awkward fact that 2015 is not a multiple of 12: overshoot to P₂₀₁₆ (which is 168 clean blocks) and then undo the single hop that went too far. Tool #15 (Organize Information in More Ways) does the bookkeeping that turns the twelve-hop push into exact coordinates by re-pairing the terms.
Number the hops
Each hop is a length times a heading.
Hop number, hop length, and hop heading all tick up together, so one index k controls the entire flight plan.
9.F-IF.A.3Introduce A VariableTwelve headings, in opposite pairs
There are only twelve headings, in opposite pairs.
Turning half a circle reverses a direction, so the twelve spokes of the wheel cancel in six opposite pairs.
Turning half a circle reverses a direction, so the twelve spokes cancel in six opposite pairs.
▸ Why?
The full turn is shared equally among the twelve headings, so half a turn is exactly six of them.
▸ Why?
A step added to its own opposite leaves nothing behind, so each such pair contributes no movement.
Every twelve hops gives the same push
Every block of twelve gives the same push.
Splitting each length into "block start" plus "offset" throws away everything that depends on which block you are in.
9.A-SSE.A.2Look For A PatternMeasure that push exactly
That push is measured exactly.
A hop and the hop six later point opposite ways and differ by exactly 6 in length, so each such pair is just a 6-inch step backwards.
10.G-SRT.C.6Organize Information In More WaysRide 168 blocks to P₂₀₁₆
Riding whole blocks reaches a round landing.
2016 is a whole number of twelve-hop blocks, so getting there is one identical push repeated 168 times.
7.NS.A.2Look For A PatternUndo the extra hop
Undoing one extra hop corrects it.
Reaching P₂₀₁₆ is easy, so go one hop too far on purpose and then take that hop back.
10.G-SRT.C.6Work BackwardsDistance back to the start
The two coordinates come out equal.
Equal legs mean the distance is just one leg times √2, the diagonal of a square.
8.G.B.8Draw A DiagramMatch the required form
Matching the form gives 2024, choice (A).
√2·√3=√6 splits the single product into exactly the two square-free pieces the problem asks for.
11.N-RN.A.2Introduce A VariableTwelve turns of 30^° bring the heading right back to the start, so cut the 2015 hops into blocks of twelve — the growing lengths cancel, every block pushes the bee the same short distance, and 168 identical pushes are easy to add.
- Number the hops
- Twelve headings, in opposite pairs
- Every twelve hops gives the same push
- Measure that push exactly
- Ride 168 blocks to P₂₀₁₆
- Undo the extra hop
- Distance back to the start
- Match the required form