Bell Boy Problem

[jack]

Three men go to stay in a motel in a single room (their personal relationship is irrelevant to the problem). They are charged £300 and split it £100 each. Then the manager realises that they were massively overcharged, and the actual cost should only have been £25 total. The manager gives £275 to bellboy (so, not as much of a dive as I was imagining?).

But the bellboy, noticing 275 doesn't divide evenly into three, pockets £272, returning £1 to each of the men.

Now each man has paid £99 to stay in the room and 3 x £99 = £297. The bellboy has pocketed £272. £297 + £272 = £569 - so where is the missing £269?

When you put it like that, the maths is clearly screwed up. But when the original cost is £30, apparently it's easy to add the stolen money again instead of subtracting it.

Comments

[simont]

drswirly had a version of this problem that tries to confuse you further, by coming in two stages.

The first time through, in the usual way, they are charged £30 which should have been £25, the bellboy keeps £2 of the £5 refund, and the mis-accounting says "so they've each spent £9, and the bellboy has £2, which adds up to £29 – but where's the missing £1?"

The second time, one of the three men has decided there must have been foul play, so he refuses to go back to that motel. So now there are only two men, sharing an identical room. Again, they are mistakenly charged £30 (this time splitting it only two ways), and again, the manager gives the bellboy £5 to refund to them. This time, the bellboy keeps £3 and gives each man £1 back. "So now they've each spent £14 and the bellboy has £3, which makes £31 – aha, we've found the missing £1!"

[damerell]

Argh! Years ago when I was a Scout, one of the other boys' parents told me the original of this one. I hadn't yet learned it's unwise to let idiots know you're cleverer than them, and the result was that this particular idiot produced this endless series of hoary old brainteasers, all of which I already knew the answer to because they were hoary and old.

Normally this was just moderately annoying - I'd say "it takes two men two hours to dig the holes", or whatever - but in this case although the fallacy is plainly obvious, this chap was too stupid (or obtuse, but I suspect stupid) to understand the explanation...