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Today, I’ll discuss one of my favourite paradoxes, and unravel it. It’s called the Waiter Paradox.

What is a Paradox?

The dictionary definition says that a paradox is “A seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true”.

The Waiter Paradox

The following is taken from the Paradox section of www.curiouser.co.uk (it’s called the Missing Pound Paradox here, but I always thought that the Waiter Paradox sounded more catchy).

“Three ladies go to a restaurant for a meal. They receive a bill for £30. They each put £10 on the table, which the waiter collects and takes to the till. The cashier informs the waiter that the bill should only have been for £25 and returns £5 to the waiter in £1 coins. On the way back to the table the waiter realises that he cannot divide the coins equally between the ladies.

As they didn’t know the total of the revised bill, he decides to put £2 in his own pocket and give each of the ladies £1.

Now, each of the ladies paid £9. Three times 9 is 27. The waiter has £2 in his pocket. Two plus 27 is £29. The ladies originally handed over £30.

Where is the missing pound?”

Where indeed?

Resolution

The “Paradox” here stems from a misrepresentation of the facts, leading to some bad counting. Let’s do some simple book-keeping.

We’ll call the ladies $L$ , $M$ and $N$ . The waiter will be $W$ and the cashier will be $C$ . The amounts of money held by all parties at any time will be stored like this: $(L,M,N,W,C)$

Ignore all the rest of the money involved, like other money that the ladies and waiter are carrying, and the rest of the money in the cashier’s till. It’s not important because we only want to keep track of £30. At the start, each lady has £10.

$(10,10,10,0,0)$

The waiter collects the money.

$(0,0,0,30,0)$

The cashier takes £25 for the meal and returns £5 to the waiter.

$(0,0,0,5,25)$

The waiter returns £1 to each lady.

$(1,1,1,2,25)$

We started with £30 in circulation and we end this way too. Clearly there’s no missing pound here. So where does the paradox come from? It’s true that each lady paid £9. It’s true that 3 times £9 is £27, and it’s true that the waiter has £2 at the end.

It’s also true that £2 plus £27 is £29, but why are we adding these numbers together? The ladies have paid £27 in total, of which £25 is with the cashier and £2 is with the waiter. So the £2 held by the waiter is already included in the £27. Why would we add it on again? What we should be adding is the money that the ladies have not paid; the £3 that they got back in change. This makes £30. There’s no paradox, just bad book-keeping.

More Paradoxes: The Suprise Test Paradox

On Friday, a teacher says to their class that at some point during the next week, there will be a surprise test. However, the students start thinking about this:

The teacher can’t wait until next Friday for the exam, because then the test won’t be unexpected. So it can’t be Friday.
Since we’ve removed Friday from the possible test days, the same logic applies to Thursday.
By repeating this logic, we removed all the possible test days.
So, it turns out that the teacher can’t give a surprise test at all.

However, the teacher sets a test on Wednesday and all the students are surprised! Teachers can certainly give surprise tests, as anybody who has experienced one will tell you. So where did the students go wrong? The Surprise Test Paradox is subtle and much harder to pin down than it first appears. Have a think about it.

I’d like to take this opportunity to announce that I’ll be writing a surprise blog article on this paradox, at some point before the end of 2014. See you next week!

More paradoxes may be found at http://www.curiouser.co.uk/ and http://www.paradoxes.co.uk/.