Read Online Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart - Free Book Online Page B
Authors: Ian Stewart
Tags: General, Mathematics
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appreciate.
Brothers Aelfred, Benedict and Cyril are asleep in their cell, when the novice Legpulla sneaks in and paints a blue blob on the top of each of their shaven heads. When they awake, each notices the blob on the other’s head. Now, the monastery rules are clear: it is impolite to say anything that will cause direct embarrassment to another member of the order, but it is also impolite to conceal anything embarrassing about yourself. And impoliteness is not permitted under any circumstances. So the monks say nothing, and their demeanour gives no hint of what they have seen.
Each vaguely wonders whether he, too, has a blob, but dare not ask, and there are no mirrors in their cell, nothing reflective at all. And so things remain until the Abbot enters, frowns, and informs them (neatly avoiding direct embarrassment) that ‘At least one of you has a blue blob on his head.’
Of course, all three monks know that. So does the information make any difference to them?
If you’ve not met this puzzle before, it helps to start with a simpler version, just two monks, Aelfred and Benedict. Each can see the other’s blob, but has no idea what his own head might bear. After the Abbot’s public announcement, Aelfred starts thinking. ‘I know Benedict has a blob, but he doesn’t, because he can’t see the top of his own head. Dear Lord, do I have a blob? Hmmm . . . Suppose I don’t have a blob. Then Benedict will see
that I don’t, so he will immediately deduce from the Abbot’s remark that he must have a blob. But he hasn’t shown any sign of embarrassment. Oh dear, I must have a blob.’ Benedict comes to a similar conclusion.
Without the Abbot’s remark, these deductions don’t work, yet the Abbot tells them nothing - apparently - that they don’t know already. Except ... Each knew that at least one monk (the other one) had a blob, but they didn’t know that the other monk knew that at least one monk had a blob.
Got that? Very well - what happens with three monks? Again, they can all deduce that they have blobs, but only after the Abbot’s announcement (see the answers on page 291). The same goes when there are four, five, or more monks, if all of them have blobs on their heads. Indeed, suppose there are 100 monks. Each bears a blob, each is unaware of that, and each is an amazingly rapid logician. To avoid timing issues, suppose that the Abbot has a bell. ‘Every ten seconds,’ he tells them, ‘I will ring this bell. That will give you time to carry out the necessary logic. Immediately after I ring, all monks who can deduce logically that they have a blob must put their hands up.’ He waits ten minutes, ringing his bell from time to time, but nothing happens. ‘Oh, yes, I forgot,’ he says. ‘Here is one extra piece of information. At least one of you has a blob.’
Now nothing happens for 99 rings, and then all 100 monks simultaneously raise their hands after the 100th ring.
Why? Monk number 100, say, can see that the other 99 all have blobs. ‘If I do not have a blob,’ he thinks, ‘then the other 99 all know this. That takes me out of the reckoning altogether. So they are making whatever series of deductions you get with 99 monks when I don’t have a blob. If I’ve sorted out the 99-monk logic right, then after 99 rings they will all put up their hands.’ He waits for ring 99, and nothing happens. ‘Ah, so my assumption is wrong, and I must have a blob.’ Ring 100, up goes his hand. Ditto for the other 99 monks.
Ah, yes . . . but maybe monk 100 was wrong about the 99-monk logic. Then it all falls apart. However, the 99-monk logic
(with the hypothetical assumption that monk 100 is blobless) is the same. Now monk 99 expects the other 98 to put up their hands at the 98th ring, unless monk 99 has a blob. And so it goes, recursively, until we finally get down to a single hypothetical monk. He sees no blobs anywhere, is startled to discover that somebody has one, immediately deduces it must be
Brothers Aelfred, Benedict and Cyril are asleep in their cell, when the novice Legpulla sneaks in and paints a blue blob on the top of each of their shaven heads. When they awake, each notices the blob on the other’s head. Now, the monastery rules are clear: it is impolite to say anything that will cause direct embarrassment to another member of the order, but it is also impolite to conceal anything embarrassing about yourself. And impoliteness is not permitted under any circumstances. So the monks say nothing, and their demeanour gives no hint of what they have seen.
Each vaguely wonders whether he, too, has a blob, but dare not ask, and there are no mirrors in their cell, nothing reflective at all. And so things remain until the Abbot enters, frowns, and informs them (neatly avoiding direct embarrassment) that ‘At least one of you has a blue blob on his head.’
Of course, all three monks know that. So does the information make any difference to them?
If you’ve not met this puzzle before, it helps to start with a simpler version, just two monks, Aelfred and Benedict. Each can see the other’s blob, but has no idea what his own head might bear. After the Abbot’s public announcement, Aelfred starts thinking. ‘I know Benedict has a blob, but he doesn’t, because he can’t see the top of his own head. Dear Lord, do I have a blob? Hmmm . . . Suppose I don’t have a blob. Then Benedict will see
that I don’t, so he will immediately deduce from the Abbot’s remark that he must have a blob. But he hasn’t shown any sign of embarrassment. Oh dear, I must have a blob.’ Benedict comes to a similar conclusion.
Without the Abbot’s remark, these deductions don’t work, yet the Abbot tells them nothing - apparently - that they don’t know already. Except ... Each knew that at least one monk (the other one) had a blob, but they didn’t know that the other monk knew that at least one monk had a blob.
Got that? Very well - what happens with three monks? Again, they can all deduce that they have blobs, but only after the Abbot’s announcement (see the answers on page 291). The same goes when there are four, five, or more monks, if all of them have blobs on their heads. Indeed, suppose there are 100 monks. Each bears a blob, each is unaware of that, and each is an amazingly rapid logician. To avoid timing issues, suppose that the Abbot has a bell. ‘Every ten seconds,’ he tells them, ‘I will ring this bell. That will give you time to carry out the necessary logic. Immediately after I ring, all monks who can deduce logically that they have a blob must put their hands up.’ He waits ten minutes, ringing his bell from time to time, but nothing happens. ‘Oh, yes, I forgot,’ he says. ‘Here is one extra piece of information. At least one of you has a blob.’
Now nothing happens for 99 rings, and then all 100 monks simultaneously raise their hands after the 100th ring.
Why? Monk number 100, say, can see that the other 99 all have blobs. ‘If I do not have a blob,’ he thinks, ‘then the other 99 all know this. That takes me out of the reckoning altogether. So they are making whatever series of deductions you get with 99 monks when I don’t have a blob. If I’ve sorted out the 99-monk logic right, then after 99 rings they will all put up their hands.’ He waits for ring 99, and nothing happens. ‘Ah, so my assumption is wrong, and I must have a blob.’ Ring 100, up goes his hand. Ditto for the other 99 monks.
Ah, yes . . . but maybe monk 100 was wrong about the 99-monk logic. Then it all falls apart. However, the 99-monk logic
(with the hypothetical assumption that monk 100 is blobless) is the same. Now monk 99 expects the other 98 to put up their hands at the 98th ring, unless monk 99 has a blob. And so it goes, recursively, until we finally get down to a single hypothetical monk. He sees no blobs anywhere, is startled to discover that somebody has one, immediately deduces it must be
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