Problem 22 · 2013 Math Kangaroo
Stretch
Spatial & Visual Reasoning
path-tracing
A model train set has only identical curved track pieces. Matthias uses 8 of them to make a closed circle (picture on the left). Martin starts his track with the 2 pieces shown on the right. He also wants a closed track using as few pieces as possible. How many pieces will his track use?
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Answer: B — 12
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Hint 1 of 3
Eight identical pieces close a full circle, so each piece bends the track by \(360^\circ \div 8 = 45^\circ\).
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Hint 2 of 3
To come back to the start, all the bends together must add up to one full turn, \(360^\circ\).
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Hint 3 of 3
Martin's two starting pieces curve opposite ways and cancel, so count how many more pieces are needed to still make a full turn.
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Approach: use the fixed bend angle to close a loop
- Each curved piece turns the track \(45^\circ\), because 8 of them make a full circle (\(8 \times 45^\circ = 360^\circ\)).
- To form a closed track the bends must add up to a full turn of \(360^\circ\), but Martin's two opening pieces curve in opposite directions and cancel out.
- Working out the smallest loop that still turns a full \(360^\circ\) starting from that S-shape needs 12 pieces in all, which is choice B.
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