The missing money - hotel room problem

I'm re-reading The Infinite Book by John D Barrow at the moment; a little dry in places; a bit too much theology and some badly placed diagrams in relation to the accompanying text, but still an interesting read. One of the quotes used is a famous puzzle paradox involving people staying in a room. I'll paraphrase here, but you can check Scopes for a full set of details.

3 people stay in one room at a hotel. It costs $30 so they each pay $10. The hotel accidentally forgot to take a discount into account - the room should only have cost $25. The manager hands $5 to a bellhop to return to the guests. The bellhop, who hasn't been tipped, decides to take $2 out of the $5 and returns $1 each to the guests.

Each guest has paid $9 for the room totalling $27; the bellhop has $2 so where's the remaining $1?


Scopes shows how this missing dollar doesn't really exist, but the flaw here is that it deals with dividing 3 into 10 and it's easy to assume the resulting confusion is due to the remainders in place (can't split $5 exactly by 3). So here's a different take on it.

2 people stay in one room at a hotel. It costs £20 so they each pay £10. The hotel accidentally forgot to take a discount into account - the room should have cost only £14. The manager hands £6 to a bellhop to return to the guests. The bellhop, who hasn't been tipped, decides to take £2 out of the £6 and returns £2 each to the guests.

Each guest has paid £8 for the room totalling £16; the bellhop has £2 so where's the remaining £2?

Even worse now we have a whole £2 missing. So to take a leaf from Scopes let's show how this works in terms of equations. Let's start at the beginning with the amount the guests pay and the amount the hotel has

(10+10)=20

Easy. But the amount the hotel has is wrong it should be £14 so:

(10+10)=14+(3+3)

Now let's say we have an honest bellhop who returns all the money so the money has to travel from the right-hand side (hotel) to the left-hand side (guests) and doing so reverses the sign

(10+10)-(3+3)=14 or (10-3)+(10-3)=14

Each guest pays £7 and the hotel has £14. Now for our petty larcency; the money hasn't gone to the guests so it stays on the hotel side of the equation

(10+10)=14+(2+2)+2

Some of the money is returned

(10+10)-(2+2)=14+2 or (10-2)+(10-2)=14+2

The mathematical flaw is taking the bellhop's money on the right and moving it to the guest's side of the equation without changing the sign. The money has been removed from the guest not from the hotel; the hotel has the amount of money it thinks it should have; the guests don't know any different and everyone's happy until the guests ask for a new receipt.