AMC 10 · 2004 · #16

Grade 3 geometry-2d
Archetype Casework by Position / Vertex Type └ Sub-archetype Count of Type
systematic-enumerationsymmetry-argument caseworkcomplementary-counting ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 2 insights 📊 Diagram
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Problem
A 5×5 grid holds squares of every size from 1×1 up to 5×5, all drawn along the grid lines. The single cell at the exact center of the grid is shaded black. Count how many of these squares cover that black center cell.

Pick an answer.

(A)
12
(B)
15
(C)
17
(D)
19
(E)
20

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

There are only five sizes of square, so the clean move is to split the count by size (Tool #7, Identify Subproblems) and tally each size separately in an organized list (Tool #2, Make a Systematic List) so nothing is missed or double-counted. A quick diagram (Tool #1) pins the black cell at the third column and third row, which stays fixed while we slide each square over it. For a given size, the trick is to see that 'covers the center' just means the square can start in only a few positions left-to-right and the same few up-to-down, so the placements form a small array. Counting those position-arrays for the five sizes produces the sequence 1,2,3,2,1 of one-direction positions (Tool #5, Look for a Pattern), whose squares 1,4,9,4,1 are the per-size counts to add.

1STEP 1

Locate the center square

The black cell is the exact middle of the 5×5 grid — column 3, row 3 — and every square we count must sit on top of it.

center cell = (column 3, row 3)
2STEP 2

Turn 'covers the center' into a sliding count

Fix a k×k square by its left and bottom edges. If h starting columns keep it over the center, so do h rows — giving h × h placements.

N_k = h_k × h_k = h_k^ 2
3STEP 3

Count positions for each size

Center-covering starts per direction: 1, 2, 3, then only 2 for 4×4 (no column 6) and 1 for 5×5 — squaring gives 1, 4, 9, 4, 1.

h_k: 1,2,3,2,1 → N_k: 1,4,9,4,1
4STEP 4

Add up all the sizes

The five size groups do not overlap, so add them: 1 + 4 + 9 + 4 + 1 = 19 — choice (D).

1+4+9+4+1 = 19 → (D)
Answer
19
The counts 1,4,9,4,1 are symmetric, which makes sense: a size-k square and a size-(6-k) square are hemmed in by the grid in mirror-image ways, so their position counts match (1⇔1, 2⇔2). The total 19 is also less than the grand total of all squares on the grid, 25+16+9+4+1 = 55, as it must be, since only some squares cover the center. And 19 is one of the listed choices, (D).
💡Key takeaway

To count how many squares cover one spot, slide each size left-right and up-down, multiply the two counts of positions that still cover the spot, then add the sizes up — and remember the grid's edges leave the biggest squares fewer places to sit.

  • Locate the center square
  • Turn 'covers the center' into a sliding count
  • Count positions for each size
  • Add up all the sizes

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