AMC 8 · 2017 · #15
Grade 4 counting
Pick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem is purely spatial — we are walking on a grid — so Tool #1 (Draw a Diagram) is the natural starting point. With the picture in hand we use Tool #7 (Identify Subproblems) to split the 3-move walk into three independent counting questions: A → M, then M → C, then C → 8. Tool #5 (Look for a Pattern) helps once we notice that every M has the same number of valid C-neighbors and every C has the same number of valid 8-neighbors, so we can multiply the three small counts instead of enumerating all paths.
The central A's four orthogonal neighbors are all M's (up, down, left, right), so the first move has 4 choices.
Just describing positions like "above, below, left, right" of A is a Kindergarten geometry idea.
K.G.A.1Draw A DiagramFrom any M the neighbors are the already-used A plus three C's, so by symmetry every M offers 3 C-choices.
Spotting that the same count of 3 repeats at every M is a Grade 4 "find the repeating rule" pattern observation.
4.OA.C.5Look For A PatternBoth kinds of C — the inner corner and the outer tip — touch exactly two 8's, so each C gives 2 choices for the last move.
Confirming that the count of 2 holds for both types of C is another Grade 4 pattern-rule check.
4.OA.C.5Look For A PatternMultiply the three independent counts by the multiplication principle: 4 × 3 × 2 = 24 paths — answer choice (D).
Multiplying the choices at each step is exactly the Grade 3 "number of groups times number per group" multiplication idea.
3.OA.A.3Identify SubproblemsThis AMC 8 problem only needs Grade 4 pattern-spotting plus the simple Grade 3 idea that the total number of paths is the product of the choices at each step!
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