AMC 10 · 2020 · #20

Grade 8 probability

Archetype Casework by Position / Vertex Type └ Sub-archetype Count of Type

probability-basicrotation-isometrycombinations-basicspatial-visualization systematic-enumerationcaseworksymmetry-argument ↑ Prerequisites: probability-basiccombinations-basic

📏 Long solution 💡 3 insights

Problem

Two identical blank cubes are painted separately. For each cube, each of its six faces is independently coloured black or white with probability one half. Find the probability that the two finished cubes can be rotated so that they look exactly the same.

Pick an answer.

(A)

$\frac{9}{64}$

(B)

$\frac{289}{2048}$

(C)

$\frac{73}{512}$

(D)

$\frac{147}{1024}$

(E)

$\frac{589}{4096}$

How to solve

Strategy Organize Information in More Ways

Tool #15 (Organize in More Ways) is the key move: instead of listing 4096 pairs, sort the 64 paint jobs into groups where everything in a group is a rotation of everything else. Once the group sizes are known, the whole probability is one short formula. Tool #7 (Subproblems) splits the sorting job by how many faces are black, since rotation cannot change that count. Tool #17 (Spatial) and Tool #10 (Physical Model) settle the only hard sorting questions — the 2-black and 3-black cases — by picturing or holding a cube. Tool #2 (Systematic List) keeps the group sizes accounted for so they add to 64. Tool #5 (Pattern) uses black-white symmetry to get the 4-, 5-, and 6-black cases free. Tool #3 (Eliminate) separates the final value from the near-miss answer choice.

1STEP 1

Count the outcomes

Count all pairs of paint jobs.

2⁶ = 64, 64 × 64 = 4096

2STEP 2

Group by rotation

The chance is a sum of squared group sizes.

P = Σ_groups s/64 · s/64 = 1/4096Σ_groups s²

3STEP 3

Sort by black count

Make buckets by black count.

C(6, 0), …, C(6, 6) = 1, 6, 15, 20, 15, 6, 1 (sum 64)

4STEP 4

Zero and one black face

Each of these is a single group.

k = 0: {1} k = 1: {6}

5STEP 5

Two black faces

Opposite or adjacent gives two groups.

k = 2: {3, 12}, 3 + 12 = 15 = C(6, 2)

6STEP 6

Three black faces

They split into corner and strip layouts.

k = 3: {8, 12}, 8 + 12 = 20 = C(6, 3)

7STEP 7

Mirror the rest

Four and up follow by symmetry.

1 + 6 + 3 + 12 + 8 + 12 + 12 + 3 + 6 + 1 = 64

8STEP 8

Square, add, simplify

Squaring and adding gives one forty-seven over one thousand twenty-four.

P = (1 + 36 + 9 + 144 + 64 + 144 + 144 + 9 + 36 + 1)/4096 = 588/4096 = 147/1024

Answer

147/1024

Two checks. First, the group sizes must partition all 64 paint jobs, and 1 + 6 + 3 + 12 + 8 + 12 + 12 + 3 + 6 + 1 = 64. ✓ Second, the size of the answer is sensible: 147/1024 ≈ 0.144, comfortably above the 1/64 ≈ 0.016 you would get if the cubes had to match face-for-face with no rotating allowed, and far below 1/2 — rotating helps a lot but most pairs still differ. Watch the traps: choice (E) is 589/4096, exactly one more than the correct numerator 588, so an arithmetic slip of 1 in the sum of squares lands there; choice (A) is 9/64 = 576/4096, only 12 short of the true numerator, so it is close enough to look right after a sloppy count. The exact value 588/4096 = 147/1024 confirms (D).

💡Key takeaway

Sort all 64 paint jobs into groups that are rotations of each other; the chance both cubes land in the same group is the sum of the group sizes squared over 4096, which is 588/4096 = 147/1024.

Count the equally likely outcomes
Sort paint jobs into rotation groups
Split the buckets by black count
The 0-black and 1-black cases
The 2-black case splits in two
The 3-black case splits in two
Mirror the counts for 4, 5, 6
Square, add, and simplify