AMC 8 · 2000 · #8

Grade 3 arithmeticgeometry-3d

Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known

set-partitionspatial-visualizationmulti-digit-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

Three standard dice (faces 1 through 6) are stacked. Of the 18 total faces, 7 are visible and 11 are hidden (the backs, bottoms, and the faces pressed between adjacent dice). The visible faces show 1, 1, 2, 3, 4, 5, 6. Find the total number of dots on the hidden faces.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Find an Invariant

We are never told which face of each die points where, and we cannot see eleven of them — so trying to identify each hidden face is the wrong path. The invariant (Tool #11) is the sum of the six faces on a single die: 1+2+3+4+5+6 = 21 no matter how the die is turned. Tool #9 (Solve an Easier Problem) tells us to first solve for one die, then scale to three. Once we know the grand total of dots on all three dice, the hidden total is just total minus visible — a single subtraction.

1STEP 1

One die's six faces sum to 21, and this total never changes when you turn the die.

1 + 2 + 3 + 4 + 5 + 6 = 21

2STEP 2

Three dice give 3 × 21 = 63 dots across all eighteen faces.

3 × 21 = 63

3STEP 3

The seven visible faces 1, 1, 2, 3, 4, 5, 6 add up to 22.

1 + 1 + 2 + 3 + 4 + 5 + 6 = 22

4STEP 4

Every face is seen or hidden, so hidden dots = 63 − 22 = 41.

63 - 22 = 41 → (D)

Answer

Check the size of the answer. There are 11 hidden faces, each showing a number between 1 and 6, so the hidden total must lie between 11 × 1 = 11 and 11 × 6 = 66. Our answer 41 sits comfortably in that range. The average hidden face value is

\frac{41}{11}

≈ 3.7, very close to a die's average face value of

\frac{21}{6}

= 3.5 — exactly what we should expect when the hidden faces are a mostly random mix. The arithmetic also checks: 22 + 41 = 63 = 3 × 21.

💡Key takeaway

The six faces of a die always add to 21, no matter how it is turned. Three dice hold 63 dots in all, so the hidden dots are just 63 - 22 = 41 — one subtraction once you spot the invariant.

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