2025 AMC 12B Problem 19

A rectangular grid of squares has

141

rows and

91

columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from

1

through

141 \times 91 = 12{,}831

into the squares. Horace fills the grid horizontally: he puts

1

through

91

in order from left to right into row

1,

puts

92

through

182

into row

2

in order from left to right, and continues similarly through row

141.

Vera fills the grid vertically: she puts

1

through

141

in order from top to bottom into column

1,

then

142

through

282

into column

2

in order from top to bottom, and continues similarly through column

91.

How many squares get two copies of the same number?

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Solution

In row

r,

column

c,

Horace writes

(r-1)\cdot 91 + c

and Vera writes

(c-1)\cdot 141 + r.

Setting these equal gives

90r - 140c = -50,

i.e.

9r = 14c - 5.

This requires

c \equiv 1 \pmod 9,

c = 1, 10, 19, \ldots, 91

— that is

11

values, and each yields a valid

r

between

1

and

141.

11

squares match. Thus, the correct answer is C.