Difference between revisions of "2003 AMC 8 Problems/Problem 23"

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==Problem==
 
==Problem==
In the pattern below, the cat (denoted as a cat in the figures below) moves clockwise through the four squares,  and the mouse (denoted as a mouse in the figures below) moves counterclockwise through the eight exterior segments of the four squares.
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In the pattern below, the cat moves clockwise through the four squares,  and the mouse moves counterclockwise through the eight exterior segments of the four squares.
  
 
<center>
 
<center>
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==Solution==
 
==Solution==
  
Break this problem into two parts: where the cat will be after the 247th move, and where the mouse will be.
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Break this problem into two parts: where the cat will be after the <math>247^{th}</math> move, and where the mouse will be.
  
The cat has four possible configurations which are repeated every four moves. <math>247</math> has a remainder of <math>3</math> when divided by <math>4</math>. This corresponds to the position the cat has after the 3rd move, which is the bottom right corner.
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The cat has four possible positions in 1 cycle which are repeated every four moves. <math>247</math> has a remainder of <math>3</math> when divided by <math>4</math>. This corresponds to the position the cat has after the 3rd move, which is the bottom right corner.
  
Similarly, the mouse has eight possible configurations that repeat every eight moves. <math>247</math> has a remainder of <math>7</math> when divided by <math>8</math>. This corresponds to the position the rat has after the 7th move, which can easily be found by writing two more steps to be the bottom edge on the left side of the grid.
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Similarly, the mouse has eight possible positions in 1 cycle that repeat every eight moves. <math>247</math> has a remainder of <math>7</math> when divided by <math>8</math>. This corresponds to the position the rat has after the 7th move which is bottom left corner.
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The only arrangement with the mouse in that position and the cat in the bottom right square is <math>\boxed{\textbf{(A)}}</math>.
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==Video Solution==
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https://youtu.be/RCUzhVOi7XI
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~DSA_Catachu
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https://www.youtube.com/watch?v=OxtaQkcJDfU
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==Video Solution #2==
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https://youtu.be/xMy3wMFF3KQ Soo, DRMS, NM
  
The only configuration with the mouse in that position and the cat in the bottom right square is <math>\boxed{\textbf{(A)}}</math>.
 
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2003|num-b=22|num-a=24}}
 
{{AMC8 box|year=2003|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:24, 4 November 2022

Problem

In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.

2003amc8prob23a.png

If the pattern is continued, where would the cat and mouse be after the 247th move?

2003amc8prob23b.png

Solution

Break this problem into two parts: where the cat will be after the $247^{th}$ move, and where the mouse will be.

The cat has four possible positions in 1 cycle which are repeated every four moves. $247$ has a remainder of $3$ when divided by $4$. This corresponds to the position the cat has after the 3rd move, which is the bottom right corner.

Similarly, the mouse has eight possible positions in 1 cycle that repeat every eight moves. $247$ has a remainder of $7$ when divided by $8$. This corresponds to the position the rat has after the 7th move which is bottom left corner.

The only arrangement with the mouse in that position and the cat in the bottom right square is $\boxed{\textbf{(A)}}$.

Video Solution

https://youtu.be/RCUzhVOi7XI ~DSA_Catachu

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

Video Solution #2

https://youtu.be/xMy3wMFF3KQ Soo, DRMS, NM


See Also

2003 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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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