2016 AMC 10B Problem 15

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

Small hint:

Adjacent consecutive numbers have opposite parity

Big hint:

The center and four corners must all have the same parity

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 .