AMC 10 · 2025 · #19

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 grid has 141 rows and 91 columns. Horace writes 1 through 12,831 row by row (left to right, top to bottom); Vera writes the same numbers column by column (top to bottom, left to right). A square gets "two copies of the same number" when Horace's number and Vera's number land in that same square. Count how many squares this happens in.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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 each square by row and column.

1 ≤ r ≤ 141, 1 ≤ c ≤ 91

2STEP 2

Write Horace's number there

Horace's number runs 91 per row plus the column.

H = 91(r-1) + c

3STEP 3

Write Vera's number there

Vera's runs 141 per column plus the row.

V = 141(c-1) + r

4STEP 4

Set the two numbers equal

Equating gives nine r plus five equals fourteen c.

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

5STEP 5

Find the first match and the repeating step

Solutions repeat every fourteen rows.

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

6STEP 6

List the matches and count them

Counting inside the grid gives 11.

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