Difference between revisions of "1995 AHSME Problems/Problem 15"
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Then at point 4, it moves to point 1. | Then at point 4, it moves to point 1. | ||
| − | We can see that at this point, the bug will cycle between | + | We can see that at this point, the bug will cycle between 1, 2, and 4 |
More specifically, we can see that all numbers congruent to 0 (mod 3) will have the bug on point 4 on that step number. | More specifically, we can see that all numbers congruent to 0 (mod 3) will have the bug on point 4 on that step number. | ||
Latest revision as of 20:54, 2 August 2016
Problem
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point
Solution
Let us see how the bug moves.
First, we see that if it starts at point 5, it moves to point 1.
At point 1, it moves to point 2.
At point 2, since 2 is even, it moves to point 4.
Then at point 4, it moves to point 1.
We can see that at this point, the bug will cycle between 1, 2, and 4
More specifically, we can see that all numbers congruent to 0 (mod 3) will have the bug on point 4 on that step number.
Thus, we can conclude that the answer is 
See also
| 1995 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
