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?

$\ 5$

$\ 6$

$\ 7$

$\ 8$

$\ 9$

Solution:

We firstly claim that everything that is either in the center or a corner is odd. This is because every number is next to a consecutive number and therefore has the opposite parity as it.

Therefore, all of the points that are the center or a corner are the same parity. The are $5$ such points, but only $4$ even numbers, so their parity isn't even. Thus, they are all odd. This means the sum of the corners plus the 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

Solution:

Problem 15 in Other Years