Difference between revisions of "2024 AMC 8 Problems/Problem 17"
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A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a <math>3</math> x <math>3</math> grid attacks all <math>8</math> other squares, as shown below. Suppose a white king and a black king are placed on different squares of a <math>3</math> x <math>3</math> grid so that they do not attack each other. In how many ways can this be done? | A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a <math>3</math> x <math>3</math> grid attacks all <math>8</math> other squares, as shown below. Suppose a white king and a black king are placed on different squares of a <math>3</math> x <math>3</math> grid so that they do not attack each other. In how many ways can this be done? | ||
− | (A) | + | <math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32</math> |
==Solution 1== | ==Solution 1== |
Revision as of 12:39, 27 January 2024
Contents
Problem
A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a x
grid attacks all
other squares, as shown below. Suppose a white king and a black king are placed on different squares of a
x
grid so that they do not attack each other. In how many ways can this be done?
Solution 1
Corners have spots to go and
corners so
.
Edges have
spots to go and
sides so
in total.
is the answer.
~andliu766
Solution 2
We see that the center is not a viable spot for either of the kings to be in, as it would attack all nearby squares.
This gives three combinations:
Corner-corner: There are 4 corners, and none of them are touching orthogonally or diagonally, so it's
Corner-edge: For each corner, there are two edges that don't border it,
Edge-edge: The only possible combinations of this that work are top-bottom and left-right edges, so for this type
Multiply by two to account for arrangements of colors to get ~ c_double_sharp
Video Solution 1 (super clear!) by Power Solve
Video Solution 2 by Math-X (First understand the problem!!!)
~Math-X
Video Solution 3 by OmegaLearn.org
Video Solution 4 by SpreadTheMathLove
https://www.youtube.com/watch?v=Svibu3nKB7E
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
See Also
2024 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.