Problem 28 · 2012 Math Kangaroo
Stretch
Logic & Word Problems
casework
Of 5 lamps each one can be set to “ON” or “OFF”. Each time when the switch of one lamp is changed, not only does the status of that particular lamp change but also that of one other lamp chosen at random. If the same switch is changed several times not always the same other lamp changes. Initially all lamps are set to “OFF”. Then 10 switching operations are carried out. After that one can say that
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Answer: C — definitely not all lamps are switched to “ON”;
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Hint 1 of 2
Each switch flips its own lamp AND one other — so two lamps flip at once.
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Hint 2 of 2
Track the parity (even/odd) of how many lamps are ON.
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Approach: parity of the number of ON lamps
- Every operation flips exactly two lamps, so the number of lamps that are ON changes by an even amount each time.
- Starting from 0 (all OFF, even), the count of ON lamps stays even after any number of operations.
- All five ON would be 5, an odd number, which is impossible — so we can definitely say not all lamps are switched to ON (C).
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