1993 AMC 8 Problem 14

Question

The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers

1,

2,

3.

Then

A + B =

\begin{array}{|c|c|c|} \hline 1 & \phantom{X} & \phantom{X} \\ \hline \phantom{X} & 2 & A \\ \hline \phantom{X} & \phantom{X} & B \\ \hline \end{array}

2

3

4

5

6

Each row and each column must contain

1,

2,

3

exactly once

Once two entries of a row or column are known, the third is forced; fill in step by step

Filling the grid so each row and column has

1,

2,

3,

the top row becomes

1,

3,

2,

the middle row

3,

2,

A,

and the bottom row

2,

1,

B.

The middle row forces

A = 1,

and the last column

2,

1,

B

forces

B = 3.

So

A + B = 1 + 3 = 4.

Thus, the correct answer is C .

Problem 14 in other years