Difference between revisions of "2016 AMC 12B Problems/Problem 12"
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==Problem== | ==Problem== | ||
− | All the numbers <math>1, 2, 3, 4, 5, 6, 7, 8, 9</math> are written in a <math>3\times3</math> array of squares, one number in each square, in such a way that if two numbers | + | All the numbers <math>1, 2, 3, 4, 5, 6, 7, 8, 9</math> are written in a <math>3\times3</math> 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 <math>18</math>. What is the number in the center? |
<math>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9</math> | <math>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9</math> |
Revision as of 21:42, 2 July 2016
Contents
Problem
All the numbers are written in a 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 . What is the number in the center?
Solution
Solution by Mlux: Draw a matrix. Notice that no adjacent numbers could be in the corners since two consecutive numbers must share an edge. Now find 4 nonconsecutive numbers that add up to . Trying works. Place each odd number in the corner in a clockwise order. Then fill in the spaces. There has to be a in between the and . There is a between and . The final grid should similar to this.
is in the middle.
Solution 2
If we color the square like a chessboard, since the numbers altrenate between even and odd, and there are five odd numbers and four even numbers, the odd numbers must be in the corners/center, while the even numbers must be on the edges. Since the odd numbers add up to 25, and the numbers in the corners add up to 18, the number in the center must be 25-18=7
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |