Competition · AMC preparation · step 4 of 4

AMC 8 · 2020 · #9

Grade 3 geometry-3dcounting

Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position

spatial-visualizationsystematic-enumerationcombinations-basic caseworksystematic-enumeration ↑ Prerequisites: spatial-visualization

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Problem

Akash's birthday cake is a 4 × 4 × 4 inch cube with icing on the top face and all 4 side faces but no icing on the bottom. The cake is sliced into 64 unit cubes (1 × 1 × 1 inch each). How many of these small cubes end up with icing on exactly 2 of their faces?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Create a Physical Representation

Because the icing covers 5 of the 6 faces, the usual symmetric "corners have 3, edges have 2, faces have 1" rule does not apply — the missing bottom flips some categories. Tool #10 (build a physical cube of 4 × 4 × 4 unit cubes, or sketch it) makes it visible which positions touch exactly two iced faces. Tool #7 (Identify Subproblems) splits the count into three location types — top edges, vertical edges, and bottom corners — that each contribute to the "exactly two iced" count. Tool #2 (Systematic List) then counts each type without missing or double-counting any cube.

1STEP 1

Label the cube positions

Sort the 64 pieces by position — corners, edge-middles, face-centers, interior. A piece is iced only where it touches a frosted outer face.

corners=8, edge middles=12 × 2 = 24, face centers=6 × 4=24, interior=2 × 2 × 2=8

2STEP 2

Split into three cases

Split the count into the three sub-cases that can give exactly two iced faces: the top edges, the vertical edges, and the bottom corners.

cases = {top edges, vertical edges, bottom corners}

3STEP 3

Count the top edges

Top edges: each of the 4 top edges has 2 middle pieces touching the iced top plus one iced side, so exactly two faces are iced — 8 pieces.

4 top edges × 2 middle cubes = 8 cubes

4STEP 4

Count the vertical edges

Vertical edges: each of the 4 vertical edges has 2 middle pieces touching two iced sides, exactly two faces again — 8 pieces.

4 vertical edges × 2 middle cubes = 8 cubes

5STEP 5

Count the bottom corners

Bottom corners: the 4 bottom corners each touch two iced sides and the bare bottom, giving exactly two iced faces — 4 pieces.

4 bottom corners × 1 = 4 cubes

6STEP 6

Add the three counts

Add the three groups to reach the total number of pieces frosted on exactly two faces.

8 + 8 + 4 = 20 → (D)

Combining sub-problem answers with a single addition is the wrap-up step of any multi-step word problem in Grade 3.

Exactly twenty of the sixty-four unit cubes end up with icing on exactly two of their faces.

▸ Why?

The two-iced cubes split into three groups that never overlap and leave none out — eight middle cubes along the top edges, eight middle cubes along the vertical edges, and four cubes at the bottom corners — so the total is those three group sizes added together.

▸ Why?

Every two-iced cube lands in exactly one of these three location groups with no cube shared and no cube missed, so adding the three group sizes rebuilds the whole count.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

Each of the four top edges holds two middle cubes that each touch the iced top and one iced side, giving four equal groups of two, which is eight.

▸ Why?

A middle cube on a top edge lies on exactly two iced outer faces (the top and one side), and its iced faces pair one-for-one with those outer faces, so it has exactly two iced faces.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

Four edges each carrying two such cubes is four equal groups of two, and four groups of two is eight.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

Each of the four vertical edges holds two middle cubes that each touch two iced side faces, giving four equal groups of two, which is eight.

▸ Why?

A middle cube on a vertical edge lies on exactly two iced side faces and no other iced face, and its iced faces pair one-for-one with those two outer faces, so it has exactly two iced faces.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

Four edges each carrying two such cubes is four equal groups of two, and four groups of two is eight.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

Each of the four bottom corners touches two iced side faces while its bottom stays bare, so each of these four cubes has exactly two iced faces.

▸ Why?

A bottom-corner cube lies on two iced side faces plus the un-iced bottom, and its iced faces pair one-for-one only with the iced outer faces it lies on, so exactly two of its faces are iced.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

3.OA.D.8Identify Subproblems

Answer

Sanity check the parts. The full big cube has 4 · 4 · 4 = 64 unit cubes. Of these, 8 are deep inside (no icing), 4 · 4 = 16 sit in the centers of the 4 side faces (each iced once), 4 centers of the un-iced bottom have 0 iced faces, 4 centers of the top have 1 iced face, and the 20 we counted have 2 iced faces, while the 4 top corners have 3 iced faces. The remaining bottom-edge middles (4 · 2 = 8 cubes) have 1 iced face. Tallying: 8 + 16 + 4 + 4 + 20 + 4 + 8 = 64. The bookkeeping closes, so 20 is consistent. Also 20 is choice (D), which is one of the offered answers.

💡Key takeaway

This AMC 8 problem only needs Grade 3 multiplication and addition you already know — count edges and corners, group them with ×, then add!

Label the cube positions
Split into three cases
Count the top edges
Count the vertical edges
Count the bottom corners
Add the three counts

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Label the cube positions

Split into three cases

Count the top edges

Count the vertical edges

Count the bottom corners

Add the three counts

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