Free Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart

him (you don’t need to be an expert logician at that stage) and puts his hand up after the first ring.
    Since his 1-monk logic is correct, so is the 2-monk logic, then the 3-monk . . . all the way to the 100-monk logic. So this puzzle is a striking example of the Principle of Mathematical Induction. This says that if some property of whole numbers holds for the number 1, and if its validity for any given number implies its validity for the next number, no matter what those numbers may be, then it must be valid for all numbers.
    That’s the usual story, but there’s more. So far I’ve assumed that every monk has a blob. However, very similar reasoning shows that this requirement is not essential. Suppose, for example, that 76 monks out of a total of 100 have blobs. Then, if everyone is logical, nothing happens until just after the 76th ring, when all the monks with blobs put up their hands simultaneously, but none of the others.
    At first sight, it’s hard to see how they can work this out. The trick lies in the synchronisation of their deductions by the bell, and the application of common kowledge. Try two or three monks first, with different numbers of blobs, or cheat by peeking at the answers on page 291.

Pickled Onion Puzzle
    Three weary travellers came to an inn, late in the evening, and asked the landlord to prepare some food.
    ‘All I got is pickled onions,’ he muttered.
    The travellers replied that pickled onions would be fine, thank you very much, since the alternative was no food at all.
The landlord disappeared and eventually came back with a jar of pickled onions. By then, all the travellers had fallen asleep, so he put the jar on the table and went off to bed, leaving his guests to sort themselves out.
    The first traveller awoke. Not wishing to make a pig of himself, and not knowing what anyone else had already eaten, he took the lid off the jar, threw away an onion that looked bad, ate one-third of the onions that remained, put the lid back on the jar, and went back to sleep.
    The second traveller awoke. Not wishing to make a pig of himself, and not knowing what anyone else had already eaten, he took the lid off the jar, threw away two onions that looked bad, ate one-third of the onions that remained, put the lid back on the jar, and went back to sleep.
    The third traveller awoke. Not wishing to make a pig of himself, and not knowing what anyone else had already eaten, he took the lid off the jar, threw away three onions that looked bad, ate one-third of the onions that remained, put the lid back on the jar, and went back to sleep.
    At this point the landlord returned and removed the jar, which now contained six pickled onions.
    How many were there to start with?
     
    Answer on page 292

Guess the Card
    The Great Whodunni has an endless supply of mathematical card tricks. This one allows him to identify a specific card, chosen from 27 cards taken from a standard pack.
    Whodunni shuffles the 27 cards, and lays them out in a fan so that his victim can see all of them.
    ‘Choose one card, mentally, and remember it,’ he tells him. ‘Turn your back, write down the card, and seal it in this envelope, so that we can verify your choice at the end.’

    Now Whodunni deals out the 27 cards, face up, into three piles of 9 cards each, and asks the victim to say which pile the chosen card is in.
    He picks up the piles, stacks them together without shuffling, then deals them into three piles and asks for the same information.
    Finally, he picks up the piles, stacks them together without shuffling, then deals them into three piles and asks for the same information for a third time.
    Then he picks out the chosen card.
    How does the trick work?
     
    Answer on page 293

And Now with a Complete Pack
    Whodunni can do even better. In just two deals, he can correctly identify a card chosen from the full 52-card pack.
    First, he deals the cards in 13 rows of 4 cards, and asks which row the card is in.
    Then he reassembles