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2024 AMC 8 Problems/Problem 17: Difference between revisions

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==Problem==
==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 <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 other words, not right next to each other). In how many ways can this be done?
 
<asy>
unitsize(29pt);
import math;
add(grid(3,3));
 
pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)};
 
for (int i=0; i<a.length; ++i) {
    pair x = (1.5,1.5) + 0.4*dir(225-45*i);
    draw(x -- a[i], arrow=EndArrow());
}
 
label("$K$", (1.5,1.5));
</asy>
 
<math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32</math>


(A) <math>20</math>  (B) <math>24</math>  (C) <math>27</math>  (D) <math>28</math>  (E) <math>32</math>


==Solution 1==
==Solution 1==


Corners have <math>5</math> spots to go and <math>4</math> corners so <math>5 \times 4=20</math>.
If you place a king in any of the <math>4</math> corners, the other king will have <math>5</math> spots to go and there are <math>4</math> corners, so <math>5 \times 4=20</math>.
Sides have <math>3</math> spots to go and <math>4</math> sides so <math>3 \times 4=12</math>
If you place a king in any of the <math>4</math> edges, the other king will have <math>3</math> spots to go and there are <math>4</math> edges so <math>3 \times 4=12</math>.
<math>20+12=32</math> in total.
That gives us <math>20+12=32</math> spots for the other king to go into in total.
<math>\boxed{\textbf{(E)} 32)}</math> is the answer.
So <math>\boxed{\textbf{(E)} 32}</math> is the answer.
~andliu766, captaindiamond868 (minor edits)
~AliceDubbleYou


~andliu766
==Solution 2==


==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.


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:  
This gives three combinations:  


Corner-corner: There are 4 corners, and none of them are touching orthogonally or diagonally, so it's <math>\binom{4}{2}=6</math>
Corner-corner: There are 4 corners, and none of them are touching orthogonally or diagonally, so it's <math>\binom{4}{2}=6</math>


Corner-edge: For each corner, there are two edges that don't border it, <math>4\cdot2=8</math>
Corner-edge: For each corner, there are two edges that don't border it, <math>4\cdot2=8</math>  
 
Edge-edge: The only possible combinations of this that work are top-bottom and left-right edges, so <math>2</math> for this type
 
 
<math>6+8+2=16</math>
 
==Solution 3 (Complementary Counting)==
 
We see that the center is off-limits for the kings. Additionally, the kings cannot be next to each other. There are <math>24</math> ways for the kings to be placed adjacently: <math>16</math> corner-edge permutations and <math>8</math> edge-edge permutations. In total, there are <math>8P2=56</math> ways to arrange the kings conditions aside. Therefore, our answer is <math>56-24=</math><math>\boxed{\textbf{(E)} 32}</math>
 
~abirgh


Edge-edge: The only possible combinations of this that work are top-bottom and left-right edges, so <math>2</math>
==Video Solution by Central Valley Math Circle (Goes through full thought process)==
https://youtu.be/eQ6UUmZuhhY


<math>6+8+2=16</math> Multiply by two to account for arrangements of colors to get <math>E) 32</math>
~mr_mathman


==Video Solution 1 by Math-X (First understand the problem!!!)==
==Video Solution 1 by Math-X (First understand the problem!!!)==
https://youtu.be/nKTOYne7E6Y
https://youtu.be/BaE00H2SHQM?si=Q2e8OfkuzKZXmoau&t=4624


~Math-X
~Math-X
==Video Solution (A Clever Explanation You’ll Get Instantly)==
https://youtu.be/5ZIFnqymdDQ?si=_SHs3ZXS1ZKF4-v2&t=2381
~hsnacademy
==Video Solution 2  (super clear!) by Power Solve==
==Video Solution 2  (super clear!) by Power Solve==
https://youtu.be/SG4PRARL0TY
https://youtu.be/SG4PRARL0TY
Line 39: Line 73:
==Video Solution 4 by SpreadTheMathLove==
==Video Solution 4 by SpreadTheMathLove==
https://www.youtube.com/watch?v=Svibu3nKB7E
https://www.youtube.com/watch?v=Svibu3nKB7E
==Video Solution by NiuniuMaths (Easy to understand!)==
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
https://www.youtube.com/watch?v=quWFZIahQCg
==Video Solution by Interstigation==
https://youtu.be/ktzijuZtDas&t=1922
==Video Solution by Dr. David==
https://youtu.be/vO1rcJZzIrQ
==Video Solution by WhyMath==
https://youtu.be/psiAz9Owyzk
==Video Solution by Daily Dose of Math (Simple, Certified, and Logical)==
https://youtu.be/OwJvuq6F7sQ
~Thesmartgreekmathdude
== Video solution by TheNeuralMathAcademy ==
https://youtu.be/f63MY1T2MgI&t=1695s
==See Also==
{{AMC8 box|year=2024|num-b=16|num-a=18}}
{{MAA Notice}}
[[Category:Introductory Combinatorics Problems]]

Latest revision as of 22:58, 20 October 2025

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 $3$ x $3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3$ x $3$ grid so that they do not attack each other (in other words, not right next to each other). In how many ways can this be done?

[asy] unitsize(29pt); import math; add(grid(3,3)); pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)}; for (int i=0; i<a.length; ++i) { pair x = (1.5,1.5) + 0.4*dir(225-45*i); draw(x -- a[i], arrow=EndArrow()); } label("$K$", (1.5,1.5)); [/asy]

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32$


Solution 1

If you place a king in any of the $4$ corners, the other king will have $5$ spots to go and there are $4$ corners, so $5 \times 4=20$. If you place a king in any of the $4$ edges, the other king will have $3$ spots to go and there are $4$ edges so $3 \times 4=12$. That gives us $20+12=32$ spots for the other king to go into in total. So $\boxed{\textbf{(E)} 32}$ is the answer. ~andliu766, captaindiamond868 (minor edits) ~AliceDubbleYou

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 $\binom{4}{2}=6$

Corner-edge: For each corner, there are two edges that don't border it, $4\cdot2=8$

Edge-edge: The only possible combinations of this that work are top-bottom and left-right edges, so $2$ for this type


$6+8+2=16$

Solution 3 (Complementary Counting)

We see that the center is off-limits for the kings. Additionally, the kings cannot be next to each other. There are $24$ ways for the kings to be placed adjacently: $16$ corner-edge permutations and $8$ edge-edge permutations. In total, there are $8P2=56$ ways to arrange the kings conditions aside. Therefore, our answer is $56-24=$$\boxed{\textbf{(E)} 32}$

~abirgh

Video Solution by Central Valley Math Circle (Goes through full thought process)

https://youtu.be/eQ6UUmZuhhY

~mr_mathman

Video Solution 1 by Math-X (First understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=Q2e8OfkuzKZXmoau&t=4624

~Math-X

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=_SHs3ZXS1ZKF4-v2&t=2381 ~hsnacademy

Video Solution 2 (super clear!) by Power Solve

https://youtu.be/SG4PRARL0TY

Video Solution 3 by OmegaLearn.org

https://youtu.be/UJ3esPnlI5M

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

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=quWFZIahQCg

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=1922

Video Solution by Dr. David

https://youtu.be/vO1rcJZzIrQ

Video Solution by WhyMath

https://youtu.be/psiAz9Owyzk

Video Solution by Daily Dose of Math (Simple, Certified, and Logical)

https://youtu.be/OwJvuq6F7sQ

~Thesmartgreekmathdude

Video solution by TheNeuralMathAcademy

https://youtu.be/f63MY1T2MgI&t=1695s

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

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