Climb, count and figure it out: The Weekly Puzzle by Dilip D’Souza

Work out not just one answer but two, and the puzzling way they’re connected.

Two puzzles for today. And a meta-puzzle at the end.

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1) On top of a remote hill is a small temple. In a house at the bottom of the hill lives a priest. Once a year, he sets out on a pilgrimage to the temple, using the only path that leads there. This is how it goes.

At sunrise one day, he starts walking up the hill. He stops for lunch, stops now and then for a rest, and reaches the temple just as the sun sets.

He stays up there for a few days, lost in prayer. At sunrise again one day, he sets out for home. Stops for rests, stops for lunch and reaches home a little before sunset because he walks a little faster going downhill than uphill.

Question: Is there a spot on the hill he reaches at the same time on the clock on both days? I’m not asking where or when, just whether there is such a point.

2) I hand you a bag that contains 100 marbles. Each has a random number written on it. You take out two marbles at random. You add their numbers and write that total on a new marble. You put this marble in the bag and throw away the two you removed; so now there are 99 marbles in the bag.

Repeat until there’s only one marble left in the bag.

Question: What’s the number on it?

3) Meta-puzzle question: What’s common to these two puzzles?

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The answers

1) Yes there is such a spot. Imagine the two journeys being done by two different priests on the same day. At some point, they would meet and pass each other. That’s the point we are searching for. There will be such a spot. Put another way, regardless of how the priest strolls up and down the hill, the fact that such a spot exists is invariant.

2) The number on the last marble is the sum of all 100 original numbers. Why? Let’s say the two marbles you remove are marked 14 and 21. You’ve reduced the sum of the marbles in the bag by the sum of 14 and 21, ie 35. But then you put back a marble with 35 written on it. Thus the sum of all the marbles in the bag remains the same. Keep this up and your last marble will have that total. The way to look at this is to realise that the sum of the marbles in the bag is invariant.

3) This idea of invariance is the link between these two puzzles, and is fundamental to plenty of serious inquiry in mathematics and science.