Question

All the numbers $1,2,3,4,5,6,7,8,9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18.$ What number is in the center?

$5$

$6$

$7$

$8$

$9$

Solution:

Color the grid like a checkerboard so the four corners and the center share one color. Since consecutive numbers occupy adjacent (opposite colored) squares, the numbers alternate parity along the chain, so the five same-colored cells contain the five odd numbers $1,3,5,7,9,$ which sum to $25.$ The four corners add to $18,$ so the center is $25-18=7.$

Thus, the correct answer is C.

2016 AMC 12B Problem 12

Solution:

Problem 12 in Other Years