AMC 10 · 2025 · #23

Grade 7 number-theory

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Lattice Constraint Analysis

linear-diophantinesystems-of-equationsgcd convert-to-algebra ↑ Prerequisites: gcd

📏 Long solution 💡 3 insights

Problem

A rectangular grid has 141 rows and 91 columns, so it holds 141 × 91 = 12,831 squares, and each square has room for two numbers. Horace writes 1 through 12,831 row by row: 1 to 91 across row 1, 92 to 182 across row 2, and so on down to row 141. Vera writes the same numbers column by column: 1 to 141 down column 1, 142 to 282 down column 2, and so on across to column 91. How many squares get two copies of the same number?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Comparing two whole grids of 12,831 numbers is hopeless to do square by square, so Tool #4 (Introduce a Variable) names a single square by its row r and column c and asks only about that one square. Tool #13 (Convert to Algebra) writes what Horace wrote there and what Vera wrote there as two formulas in r and c; a "same number" square is exactly where the two formulas are equal, which turns the whole puzzle into one equation. Tool #5 (Look for a Pattern) finds that once you have one matching square, the next one is a fixed step away — right 9 columns and down 14 rows. Tool #2 (Make a Systematic List) then just walks that step from the first match to the last and counts how many fit on the grid.

1STEP 1

Name one square by row and column

Name one square by its row r (top is 1) and column c (left is 1) instead of chasing 12,831 numbers: 1 ≤ r ≤ 141, 1 ≤ c ≤ 91.

1 ≤ r ≤ 141, 1 ≤ c ≤ 91

2STEP 2

Write Horace's number there

A row holds 91 numbers, so Horace fills 91(r-1) of them above row r, then counts c more into it: he writes 91(r-1)+c.

H = 91(r-1) + c

3STEP 3

Write Vera's number there

A column holds 141 numbers, so Vera fills 141(c-1) of them left of column c, then counts r down it: she writes 141(c-1)+r.

V = 141(c-1) + r

4STEP 4

Set the two numbers equal

A match means the two agree, so 91(r-1)+c = 141(c-1)+r. That gives 90r = 140c - 50, which tidies to 9r + 5 = 14c.

91(r-1)+c = 141(c-1)+r ⟹ 90r = 140c - 50 ⟹ 9r + 5 = 14c

5STEP 5

Find the first match and the repeating step

(1,1) works, since 9+5 = 14. As gcd(9,14) = 1, the smallest step keeping both sides balanced (each gains 126) is r by 14, c by 9.

9(r+14)+5 = 9r+5+126, 14(c+9) = 14c + 126

6STEP 6

List the matches and count them

Stepping from (1,1) gives (15,10), (29,19), …, stopping exactly at (141,91): r = 1 + 14k for k = 0 to 10, so 11 squares — choice (C).

r = 1 + 14k, k = 0,1,…,10 → 11 squares → (C)

Answer

The first match is the top-left corner (1,1) and the last is the bottom-right corner (141,91) — a satisfying picture, since both people obviously write 1 in the top-left and 12,831 in the bottom-right. Between them the matches step evenly by 14 rows and 9 columns. Check the count another way: the row jumps from 1 to 141, a total climb of 140, in steps of 14, giving 140/14 = 10 steps, hence 10+1 = 11 squares; the column jumps from 1 to 91, a climb of 90 in steps of 9, giving 90/9 = 10 steps, the same 11. Both coordinates agree, so 11 is solid and matches (C).

💡Key takeaway

Label a square by row and column, write each person's number as a formula, set them equal, and the matching squares turn out to be evenly spaced dots marching corner to corner — just count the stops to get 11.

Name one square by row and column
Write Horace's number there
Write Vera's number there
Set the two numbers equal
Find the first match and the repeating step
List the matches and count them