AMC 8 · 2023 · #2

Grade 4 geometry-2d

Archetype Paper Fold-and-Cut Symmetry └ Sub-archetype Fold-Cut-Unfold Shape Identification

spatial-visualizationreflection-symmetrypaper-folding physical-representationreflection-unfolding ↑ Prerequisites: spatial-visualizationreflection-symmetry

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Problem

A square sheet is folded in half horizontally and then in half vertically, producing a smaller square that is one quarter the size of the original and is four layers thick. A straight cut is made from the midpoint of the small square's right edge to the midpoint of its bottom edge, removing a tiny triangular corner. When the paper is unfolded, which of the five pictures matches it?

Pick an answer.

(A)

(diagram) square with zig-zag notches along all four edges

(B)

(diagram) square with zig-zag notches along the top edge only

(C)

(diagram) square with an axis-aligned square hole in the center

(D)

(diagram) square with a single upward-pointing triangular hole in the center

(E)

(diagram) square with a small square rotated 45° (diamond) hole in the center

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Create a Physical Representation

Paper-folding-and-cutting problems are exactly what Tool #10 was designed for: fold an actual scrap of paper, make the cut, and unfold it. The answer literally appears in your hand. If no paper is available, walk through the same moves mentally with Tool #17 (Visualize). Either way we finish with Tool #3 (Eliminate) by comparing the resulting shape to the five answer choices.

1STEP 1

Grab a real scrap and fold it the same way — horizontal midline, then vertical — into a four-layer square; both fold lines act as mirrors.

2STEP 2

At the corner that maps to the paper's center, cut between the two edge midpoints — four layers, so one cut frees four identical triangles.

3STEP 3

Unfold the vertical fold: the notch mirrors across that line, forming one small triangular hole on the bottom edge of the top half.

4STEP 4

Unfold the horizontal fold: the hole mirrors again into a matching triangle below, the two meeting at the paper's center.

5STEP 5

The two triangles join into a four-sided figure with equal sides and upright diagonals — a small square tilted 45 degrees: a diamond.

6STEP 6

Match the five choices: only (E) has a single 45-degree-tilted square hole at the paper's center; the rest fail the two-mirror symmetry.

→ (E)

Answer

(diagram) square with a small square rotated 45° (diamond) hole in the center

Two sanity checks. (1) Size: the cut removed only a tiny triangle from a quarter-sized folded square, so the final hole should be small relative to the whole paper — (E)'s tiny center diamond fits, while (A)'s edge-wide damage does not. (2) Symmetry: the cut was located at the center of the original paper, and both fold lines are symmetry axes of the original square, so the final hole must lie at the very center AND be symmetric across both the horizontal and vertical midlines. (E) is the only choice satisfying both conditions.

💡Key takeaway

This AMC 8 problem only needs the Grade 4 idea that "a fold line is a line of symmetry" you already know!

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