Question

Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?

$16$

$24$

$25$

$26$

$35$

Solution:

If the lower cells contain $a,$ $b,$ and $c,$ the middle row will have $a+b$ and $b+c$ .

This means that the top row will have $a+2b+c$ . To minimize this, put $b=1$ in the middle and $2,3$ in the outer cells. This yields a top number of $7$ .

To maximize it, put $b=9$ in the middle and $7,8$ in the outer cells. This yields a top number of $33$ . The desired difference is $33-7=26$ .

Thus, D is the correct answer.

2006 AMC 8 Problem 22

Solution:

Problem 22 in Other Years