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Friday Puzzle -- Beggars and their pennies

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Friday Puzzle -- Beggars and their pennies

#1

Friday Puzzle -- Beggars and their pennies

Alex Y

Five beggars sat down in a circle, and each piled up, in a heap before him, the pennies he had received that day: and the five heaps were equal.

Then spake the eldest and wisest of them, unfolding, as he spake, an empty sack.

"My friends, let me teach you a pretty little game! First, I name myself ‘Number One,’ my left-hand neighbour ‘Number Two,’ and so on to ‘Number Five.’ I then pour into this sack the whole of my earnings for the day, and hand it on to him who sits next but one on my left, that is ‘Number Three.’ His part in the game is to take out of it, and give to his two neighbours, so many pennies as represent their names (that is, he must give four to ‘Number Four’ and two to ‘Number Two’); he must then put into the sack half as much as it contained when he received it; and he must then hand it on just as I did, that is, he must hand it to him who sits next but one on his left—who will of course be ‘Number Five.’ He must proceed in the same way, and hand it on to ‘Number Two,’ from whom the sack will find its way to ‘Number Four,’ and so to me again. If any player cannot furnish, from his own heap, the whole of what he has to put into the sack, he is at liberty to draw upon any of the other heaps, except mine!"

The other beggars entered in to the same with much enthusiasm: and in due time the sack returned to ‘Number One,’ who put into it the two pennies he had received during the game, and carefully tied up the mouth of it with a string. Then, remarking "it is a very pretty little game," he rose to his feet, and hastily quitted the spot. The other four beggars gazed at each other with rueful countenances. Not one of them had a penny left!

How much had each at first?

P.S. Lewis Carroll has the answer in pounds, shillings and pence, but I will accept the answer in US$ if you prefer. ;)

Re: Friday Puzzle -- Beggars and their pennies

#2

Observation

Larry Barrett

The rule is that each beggar receiving the bag first takes out pennies to give to his neighbors according to their number. Then he puts into the bag 1/2 of the amount that was in the bag when he received it, bfore passing it on.

Lewis Carroll said " and in due time the sack returned to ‘Number One,’ who put into it the two pennies he had received during the game, and carefully tied up the mouth of it with a string. "

If he was playing by the rules, he would have first taken out of the bag pennies for his neighbors, 2 for Number Two and 5 for Number Five. He would then need to put back in the bag 1/2 of what was in it when he received it, using his pile (2 pennies) and taking the rest from anyone else. So if the game ended there, he would have taken back 2 pennies from Number Two and 5 from Number Five. These, added to his 2 pennies, makes 9 and that must be 1/2 of what was in the bag when he received it (presumably 18). So the bag at that point would contain 27 pennies. And if no one else had any, then the bag held everything they started with. So each would have started with 5 2/5 pennies.

What is wrong with this logic?

Re: Friday Puzzle -- Beggars and their pennies

#3

The end of the game was not defined up front

Alex Y

Nothing wrong with your logic. That is one possible definition of the end of the game. LC's implicit "rule" for the end of the game is that "this continues until everyone has passed the bag exactly once"

I do find a problem with a solution that involves fractional pennies, though! ;)

Re: Friday Puzzle -- Beggars and their pennies

#4

Re: The end of the game was not defined up front

Larry Barrett

I made an error in arithmetic. The bag must have had 18 pennies in it when it came to Number One. He took out 2 for #2 and 5 for #5, leaving 11 in the bag. He then put back in 9, 2 from #2, 5 from #5, and the 2 that he had. The bag then had 20 in it. So each beggar started with 4; no fractional pennies this time.

Re: Friday Puzzle -- Beggars and their pennies

#5

That's a more elegant problem

Alex Y

than the one LC posed. I agree, and a simple change to the problem would be to state that #1 followed the same steps re giving to neighbors and adding to bag, then walked away with the bag, leaving the others with nothing.

The actual problem is much harder computationally, but no more interesting, IMO.

Re: Friday Puzzle -- Beggars and their pennies

#6

But does it work?

Alex Y

Beggar #3 receives a bag containing 4 pennies. How can he remove and give 2 to #1 and 4 to #4?

So, is there a solution to the LB version, or are we back to trying to solve the LC version?

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