Problem 28 · 2017 Math Kangaroo
Stretch
Logic & Word Problems
Geometry & Measurement
symmetrycasework
30 dancers are standing in a circle facing the centre. The dance instructor shouts “Left” and many of them turn 90° to the left. Unfortunately, some are confused and turn right, so that some dancers are now directly facing each other. All of the ones that are facing each other are shaking their head. It turns out that 10 dancers shake their head. Then the dance instructor says “Turn around” and all of them turn 180° to look in the opposite direction. Again, all of the ones that are directly facing each other shake their head. How many dancers are shaking their head second time round?
Show answer
Answer: A — 10
Show hints
Hint 1 of 2
Two dancers facing each other still form a special pair after both turn 180°.
Still stuck? Show hint 2 →
Hint 2 of 2
Turning everyone around swaps who faces whom, but the count is preserved by symmetry.
Show solution
Approach: track neighbouring pairs facing each other versus back-to-back before and after the turn-around
- After turning, each dancer looks clockwise or anticlockwise; a neighbouring pair faces each other when both look toward the gap between them, and is back-to-back when both look away from it.
- Going once around the circle, every switch from clockwise-runs to anticlockwise-runs is matched by a switch back, so the number of facing gaps always equals the number of back-to-back gaps.
- The first round has 10 head-shakers, i.e. 5 facing gaps, hence also 5 back-to-back gaps; turning everyone 180° reverses all directions, so those 5 back-to-back gaps become the new facing gaps.
- That gives 5 facing pairs again, so 10 dancers shake their heads the second time, choice A.
Mark:
· log in to save