Unravelling the mystery of the missing shilling

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

28 May 2006

Arnold Ndibo has revived a very old puzzle. Says he: “Three friends check into a hotel for the night and the cashier tells them the bill is Sh30, payable in advance.  So, they each pay the clerk Sh10 and go to their room. 

“A few minutes later, the cashier realises that he has made an error and overcharged the trio by Sh5.  He asks the porter to return Sh5 (in one-shilling coins) to the three friends who had just checked in. The porter sees this as an opportunity to make Sh2 as he reasons that the three friends would have a tough time dividing Sh5 evenly among themselves; so he decides to tell them that the cashier made a mistake of only Sh3, giving a shilling back to each of the friends. He pockets the leftover Sh2 and goes home for the day! 

“Now, each of the three friends gets a shilling back, thus they each paid Sh9 for the room which is a total of Sh27 for the night. We know the porter pocketed Sh2 and adding that to the Sh27, you get Sh29, not Sh30 that was originally spent.

“Where did the other shilling go?”

This riddle is old – when it was created, you could get a hotel room for sh30! Nevertheless, it is a good example of how not to do your accounts. The clerk charges Sh30, then realises that he had overcharged by Sh5, therefore the bill should have been Sh25. The trick of the puzzle is that this Sh25 is (intentionally) never mentioned. That helps in creating the confusion.

Let us consider a similar, but slightly different situation: suppose you to go a shop and buy an item labelled sh30. You give the shopkeeper sh30 and when she punches the details in the cash register, she realise that it should actually be sh25. So she tells you that there is a sh5 change.

Now that you have some extra cash, you decide to buy a sweet that costs sh2. So the shopkeeper takes away the sh2 and gives you sh3 change. How would you do your accounts? You paid out sh30 and got sh3 back. Therefore the net payment was sh27. Now, does it make sense to add the sh2 cost of the sweet to the sh27 paid out (and therefore, get sh29)? Obviously not!

Going back to the original puzzle, suppose the porter returned the full Sh5 to the three friends and they decided to give back the Sh2 as a tip (since it is difficult to divide sh5 three ways). The situation remains the same, that is, the janitor still keeps the two bob (only this time he doesn’t have to steal it!). How would the trio do their accounts now?

The net amount given out is Sh27: sh25 went to the cashier for the room and sh2 to the porter as a tip. Clearly, it is a mistake to add the two shillings retained by the porter to the sh27 paid out. This two bob should be subtracted from the sh27 to get sh25, which is the amount that went to the cashier.