AMC 8 · 2022 · #23
Grade 7 counting
Pick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The headline question ("how many grids?") is too big to attack directly, but it cracks open the moment we use Tool #7 (Identify Subproblems) to ask a geometry sub-question first: which pairs of lines can be "all △" and "all bigcirc" at the same time? Because the two lines cannot share a cell, only disjoint line pairs survive — and the only disjoint pairs in a 3× 3 grid are two-different-rows or two-different-columns. Diagonals are eliminated immediately. That collapses the problem into a much smaller counting task. Tool #9 (Easier Related Problem) handles the remaining piece: count valid fillings under the column case alone, then double by symmetry. Tool #2 (Systematic List) supplies the casework on "how many monochrome columns appear" so nothing is missed or double-counted.
The △-line and bigcirc-line can't share a cell, so the only disjoint options are two different rows or two different columns.
Sorting the 8 lines into "can" vs "cannot" coexist is a Grade 5 sorting-by-attribute move that shrinks the problem hugely.
5.G.B.3Identify SubproblemsBy row–column symmetry the row count equals the column count, and the two cases never overlap, so count the column case and multiply by 2.
Symmetry between rows and columns is a Grade 5 "properties shared across a category" idea — solve the easier half, then mirror it.
5.G.B.3Solve An Easier Related ProblemCase 1 — one all-△ column and one all-bigcirc column: 3 × 2 ways to place them, times 6 mixed fillings of the last column, gives 36.
Multiplying independent choices for each column is exactly the Grade 7 "compound events via organized lists" counting principle.
7.SP.C.8Make A Systematic ListCase 2 — all three columns monochrome with both shapes: 3 + 3 = 6. Adding Case 1 gives the column total 42.
Splitting the remaining situations into a complete, non-overlapping list and adding is Grade 7 systematic counting.
7.SP.C.8Make A Systematic ListSymmetry gives the row case another 42, and it's disjoint from the column case, so the total is 42 + 42 = 84 — choice (D).
Adding disjoint cases is the addition principle of counting — Grade 7 compound-event reasoning.
7.SP.C.8Solve An Easier Related ProblemThis AMC 8 problem only needs Grade 7 organized-list counting — multiplication and addition of cases — that you already know!