Problem 15 · 2014 Math Kangaroo
Hard
Number Theory
divisibilitypath-tracing
In the figure, the heart and the arrow are arranged as pictured. At the same moment the heart and the arrow begin to move. The arrow moves around the figure 3 spaces clockwise and the heart 4 spaces anticlockwise, and then they stop. This process repeats over and over again. After how many repetitions does the arrow find itself for the first time in the same triangle as the heart?
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Answer: E — That will never happen
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Hint 1 of 2
After r repetitions the arrow has moved 3r spaces one way and the heart 4r spaces the other way, around 7 triangles.
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Hint 2 of 2
Look at the gap between them modulo 7: each repetition changes it by a fixed amount that never makes it zero.
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Approach: track the gap between them around the ring of 7 triangles
- The figure is a ring of 7 triangles. In one repetition the arrow moves 3 spaces one way and the heart 4 spaces the other way.
- So in each repetition the arrow gains \(3+4 = 7\) spaces on the heart, which is a whole lap around the 7 triangles.
- Gaining exactly 7 spaces each time returns them to the same relative positions, so the gap between them never changes.
- Since they do not start in the same triangle, they can never meet.
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