AMC 8 2003 Problem 23 - Cat and Mouse
Source:
July 23, 2011
Problem Statement
In the pattern below, the cat (denoted as a large circle in the figures below) moves clockwise through the four squares and the mouse (denoted as a dot in the figures below) moves counterclockwise through the eight exterior segments of the four squares.[asy]defaultpen(linewidth(0.8));
size(350);
path p=unitsquare;
int i;
for(i=0; i<5; i=i+1) {
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
}
path cat=Circle((0.5,0.5), 0.3);
draw(shift(0,1)*cat^^shift(4,1)*cat^^shift(7,0)*cat^^shift(9,0)*cat^^shift(12,1)*cat);
dot((1.5,0)^^(5,0.5)^^(8,1.5)^^(10.5,2)^^(12.5,2));label("1", (1,2), N);
label("2", (4,2), N);
label("3", (7,2), N);
label("4", (10,2), N);
label("5", (13,2), N);
[/asy]If the pattern is continued, where would the cat and mouse be after the 247th move?
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(1,0)*cat);
dot((0,0.5));
[/asy]
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(1,1)*cat);
dot((0,0.5));
[/asy]
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(1,0)*cat);
dot((0,1.5));
[/asy]
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(0,0)*cat);
dot((0,1.5));
[/asy]
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(0,1)*cat);
dot((1.5,0));
[/asy]