2016 AMC 12B Problem 12

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

Answer: C

Concepts:parity logical deduction

Difficulty rating: 1590

Small hint:

Color the grid like a checkerboard; consecutive numbers must lie on opposite colors, so they alternate

Big hint:

The five same-colored cells (four corners plus center) must hold the five odd numbers, which sum to

25

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.