Problem 29 · 2016 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
In each of the five carriages of a train there is at least one passenger. Two passengers are neighbours if they are in the same carriage or in two successive carriages. Each passenger has either exactly 5 or exactly 10 neighbours. How many passengers are on the train?
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Answer: C — 17
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Hint 1 of 3
A passenger's neighbour count = (own carriage size) + (adjacent carriage sizes) − 1, and it must be 5 or 10 for everyone.
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Hint 2 of 3
So for each carriage the sum of it and its neighbours is fixed at 6 or 11; the two end carriages have only one neighbour-carriage.
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Hint 3 of 3
Pin down the middle carriage first, then the two pairs at the ends.
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Approach: fix the carriage sizes from the neighbour counts
- Each passenger's neighbours = (own carriage) + (touching carriages) − 1, so for every carriage the block-sum of it and its neighbours is 6 or 11.
- The two end pairs must sum to 6 (an end carriage plus its single neighbour), and the middle three must sum to 11, which forces the middle carriage to hold 5.
- So the train is (end-pair sum 6) + 5 + (end-pair sum 6) = 6 + 5 + 6 = 17 passengers, e.g. 3, 3, 5, 3, 3.
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