AMC 10 · 2018 · #20

Grade 8 geometry-2d
Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position
symmetry-argumentreflection-symmetryrotation-isometry complementary-countingsystematic-enumeration ↑ Prerequisites: symmetry-argument
📏 Medium solution 💡 2 insights
Problem
A 7× 7 grid of squares is colored black and white. A coloring is called symmetric if it looks the same after any 90^° rotation about the center and after reflection across either diagonal or either midline of the big square. At least one square of each color must appear. How many symmetric colorings are there?

Pick an answer.

(A)
510
(B)
1022
(C)
8190
(D)
8192
(E)
65,534

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

The whole question is about what the eight symmetries (four rotations and four reflections) do to the squares, so Tool #17 (Visualize Spatial Relationships) leads: mentally rotate and fold the grid to see which squares are locked to the same color. Tool #1 (Draw a Diagram) pins the squares to a coordinate grid so the folding is exact. Tool #7 (Identify Subproblems) splits the squares into a few position types (center, midline arm, diagonal arm, interior), and Tool #2 (Make a Systematic List) counts how many independent color choices that leaves. Finally the words 'at least one of each color' are a Tool #16 (Change Focus / Count the Complement) cue: count every coloring, then remove the two all-one-color cases.

1STEP 1

Squares move in linked groups

The eight symmetries — four rotations, four reflections — force any square and its image to share a color, so cells lock into groups.

symmetries={0^°,90^°,180^°,270^°}∪{2 diagonals, 2 midlines}=8
2STEP 2

Fold into one-eighth

The four mirror lines cut the square into 8 wedges that all fold onto one wedge, so each linked group keeps exactly one square there.

fundamental wedge={(a,b):0 ≤ a ≤ b ≤ 3}
3STEP 3

Count the independent squares

Sorting that wedge — center, midline arm, diagonal arm, interior — gives 1+3+3+3=10 independent squares, one per linked group.

1+3+3+3=10 groups
4STEP 4

Two colors for each group

Each of the 10 groups is freely black or white, so the multiplication principle gives 2¹⁰=1024 colorings before the both-colors rule.

2¹⁰=1024
5STEP 5

Remove the two banned grids

The both-colors rule bans exactly two of the 1024 grids — all-black and all-white — so 1024-2=1022, choice (B).

1024-2=1022 → (B)
Answer
1022
The total of 49 squares splits correctly into the 10 groups: the center is a group of 1; each midline-arm square (0,b) has a group of 4, giving 3× 4=12; each diagonal-arm square (a,a) has a group of 4, giving 3× 4=12; each interior square (a,b) with a ＜ b has a group of 8, giving 3× 8=24. Then 1+12+12+24=49, exactly the whole grid, so no square was missed or double-counted. The answer 1022 is just below 1024=2¹⁰, matching choice (B). The large choices 8190 and 8192 would come from 2¹³, which would mean 13 free groups — too many once the symmetry is applied.
💡Key takeaway

Fold the symmetric grid along its mirror lines until only 10 squares are truly free; each is black or white, giving 2¹⁰=1024 pictures, then drop the all-black and all-white ones for 1022, choice (B).

  • Squares move in linked groups
  • Fold into one-eighth
  • Count the independent squares
  • Two colors for each group
  • Remove the two banned grids

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