On Friday we posed the following conundrum.

Three people went out for a Christmas meal in a very nice restaurant. They had an excellent dinner with some quality wines and were presented with a bill for £300. (We did say it was a very nice restaurant!). They each paid their share of £100 and prepared to leave.

At this point the restaurant manager realised that there had been a terrible mistake, and the bill should have been £250. She summoned the waiter and gave him £50 in ten pound notes to give to the three customers as a refund. (Anyone would think it is Christmas – oh wait, it is)

Now the waiter was a bit disgruntled with these particular customers, as they hadn’t given him a tip despite paying a large sum of money for the meal. Furthermore, how were they going to split £50 between the three of them?

So the waiter pocketed two of the ten pound notes as a tip and handed the customers a ten pound note each. The customers left and everyone seemed to be happy with the situation – the customers had received a refund, the waiter had his tip, and the restaurant manager had been paid the correct amount. But should they have been so happy?

Each of the three customers had paid £90 for their share of the meal which makes £270. And the waiter has £20 which makes £290 in total. But the diners paid £300, so where is the missing ten pound note?

As with many puzzles of this sort, like many pieces of “magic”, the trick is to spot the misdirection. In this case the £270 that the customers paid includes the £20 in the waiter’s pocket. So adding the two sums together is meaningless, rendering the comparison with the original £300 equally meaningless.

Consider the following sequence:

- Customers pay £300 – the restaurant has £300
- The manager gives £50 to the waiter – the restaurant has £250, the waiter has £50 and the customers have still paid £300
- The waiter gives the customers £30 – the restaurant still has £250, the waiter has £20 and the customers have paid £270

And this all balances.