AMC 10B/12B 2025 (Problem 23/19)

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\times91=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?

\text{(A)}\;7

\text{(B)}\;10

\text{(C)}\;11

\text{(D)}\;12

\text{(E)}\;19