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 is the number in the center?

$\ 5$

$\ 6$

$\ 7$

$\ 8$

$\ 9$

Answer: C

Concepts:parity logical deduction

Difficulty rating: 1420

Solution:

Color the array like a checkerboard, with the center and four corners one color and the four edge squares the other color. Consecutive numbers occupy adjacent squares, so the path $1,2,\ldots,9$ alternates colors. Therefore all five odd numbers occupy one color class and all four even numbers occupy the other.

The color class consisting of the center and four corners has five squares, so it contains the five odd numbers. Hence the sum of the corners and center is $1+3+5+7+9=25.$

Since the corners have a sum of $18,$ the center has a value of $25-18=7.$

Thus, the correct answer is C .

2016 AMC 10B Problem 15

Answer: C

Difficulty rating: 1420

Solution:

Problem 15 in Other Years