AMC 8 · 2023 · #4

Grade 4 geometry-2d
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Coordinate Spiral Position
perfect-squaresprime-numberspattern-recognitionspatial-visualization pattern-recognitionprimality-testsystematic-enumeration ↑ Prerequisites: perfect-squaresprime-numbers
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Problem
The numbers 1 through 49 are written on a 7 × 7 grid in a spiral that starts at the center with 1. The four shaded squares sit on the same diagonal as the number 7 (the diagonal that runs from the top-left corner of the grid to the bottom-right corner). We need to figure out which numbers land in those four shaded squares and then count how many of them are prime.

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
3
(E)
4

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

How to solve
Strategy Look for a Pattern

We do not need to fill in all 49 cells. The spiral has a clean pattern (Tool #5): each new layer ends at a perfect square (9, 25, 49), and these odd perfect squares always sit on the same diagonal as 7. So we can sketch just enough of the grid (Tool #1, Draw a Diagram) to see which way the spiral walks after each odd square, then list (Tool #2) the four shaded numbers by counting one or two cells along the spiral path from the nearest odd square. Finally, we check primality of those four small numbers — no algebra (Tool #13) and no big calculation needed.

1STEP 1

Each spiral layer ends on a perfect square; the odd squares 1, 9, 25, 49 all lie on the 7-diagonal, giving anchors for the shaded cells.

1 = 1², 9 = 3², 25 = 5², 49 = 7² all lie on the 7-diagonal (top-left to bottom-right)
2STEP 2

From 25 (the 5×5 corner), step back along the bottom edge: 25, 24, 23. So the lower-right shaded cell holds 23.

25, 24, 23 (two cells back along the bottom edge of the 5× 5 layer)
3STEP 3

Same idea for the other three cells (anchors 16, 36, 49): the four shaded numbers are 19, 23, 39, 47.

Four shaded values = {19, 23, 39, 47}
4STEP 4

Check primality: 19, 23, 47 are prime; 39 = 3 × 13 is composite — so three of the four are prime.

19 (prime), 23 (prime), 39 = 3 × 13 (composite), 47 (prime) → 3 primes
5STEP 5

The primes among the shaded numbers are 19, 23, 47 — three of them — matching choice (D).

#{primes among 19, 23, 39, 47} = 3 → (D)
Answer
3
The answer is between 0 and 4 because there are only four shaded cells, so 3 is in range. Each of the candidate primes (19, 23, 47) is small enough to verify by hand — we only had to check divisibility by 2, 3, 5 (and 7 for 47), since √(49) = 7 bounds the prime-search for any number below 50. The one composite, 39, jumps out because its digit sum 3 + 9 = 12 is a multiple of 3. The result is consistent with both the picture (the four shaded cells indeed live on the 7-diagonal between 9, 25, and 49) and with the AMC 8 difficulty for an early problem.
💡Key takeaway

This AMC 8 problem only needs Grade 4 pattern-finding and prime checking you already know!

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