Number 95 : February 1999
This is a slightly modified reprint (Iíve removed the more chatty, anecdotal bits) of a letter of mine that appeared in the January 1999 issue of Eaglet, the East Anglian Mensa Newsletter, which is edited by PDG member Annabel McLeod. Annabel featured two puzzles in the October 1998 issue of Eaglet (actually, there were three, but one of them isnít worthy of making an appearance in these august pages). I include the puzzles & responses here for the philosophical issues that arise. They are both, I think, fairly well known, but if you havenít heard of them before and want to do them yourself, youíll have to restrain yourself from looking at the answers too quickly.
Puzzle 1 : The 3 finalists in a competition for a high level post are shown 5 disks, 3 white and 2 black. They are told each will have one disk stuck to his back, and may look at the other twoís backs, but not his own. The 2 disks not used are concealed. The first to emerge and state correctly the colour of the disk on his back, with an explanation of why he can be sure, is the winner. In fact, all 3 white disks are used, with the 2 black disks concealed. Can any winner emerge and if so why ?
Puzzle 2 : A Modern Cana ? I [Patrick OíDonohoe, an East Anglian Mensan, Ed.] once heard the following story from a fundamentalist preacher, back in the bad old days of communism and the cold war : Two anonymous (and perhaps apocryphal) Christian dissidents were attempting to defect from communist Russia to escape the regime's religious persecution. Their route took them through a desolate mountainous region, where (presumably due to appallingly bad planning) their ageing Trabant spluttered and ran out of petrol. Since they were many miles from civilisation and hence fuel, and could not reach their safe destination without the car this left them somewhat in a pickle. After a brief contemplation of their alternatives (most of which included the word "Siberia") they turned to the Boss for inspiration. He soon obliged, and one of the hopefuls realised that they had with them half a gallon of water. With a thankful and reverent air he unscrewed the filler cap and poured the water from its container into the petrol tank- Then both of them knelt beside the car and offered a fervent prayer to Jesus, that He would turn the water into petrol. They then got back into the car and drove away into the Western sunset, freedom, and authorship of this story. So there we have it: proof that miracles still happen in the 20th century. Or could there perhaps be a more rational explanation ... ? Annabel then asks : The water-into-wine story is familiar to us all. Do you think there is any connection ? If so, do you think this interpretation detracts from or supports the Christian view ?
Puzzle 1 : The first puzzle is one of my favourites. Itís an exercise in recursive logic. Person A, who can see 2 white disks, says "what if I had a black disk on my back ? In that case, person B would see one white disk and one black disk and would say to himself "if I had a black disk on my back, person C would see 2 black disks, and would know he had a white disk; but person C hasnít said anything, so I must have a white disk", but person B has kept quiet, so I (person A) must have a white disk". QED.
I wouldnít have bothered discussing this if it hadnít been for Ian Stewartís article (Mathematical Recreations - Monks, Blobs & Common Knowledge) in Augustís Scientific American. This has a variant of the problem whereby a crowd of monks have blobs secretly painted on their foreheads in the middle of the night and are told in the morning "at least one of you has a blob on his forehead, ... do you ?". This is a more general form of the problem (isomorphic to an "n finalists, n white disks and n-1 black disks, I think). I hadnít realised that the problem generalises in this way. In a sense it does (check it !), and in another it doesnít. The moral of Ian Stewartís article is that the monks arenít told anything they donít already know - but they have to think recursively to a situation where the "no news" could, indeed be "news" (as it would be most obviously if there were only two monks) !
One of the deficiencies in the 3/2 problem is that it relies on all three finalists being of similar intelligence. If one of them is a dunce who takes half an hour to work out from seeing a white and a black disk that he must have a white disk, then nothing much can be deduced from a non-response. When we get to the 4/3 game - given that some people (and not just dunces) canít work out the 3/2 game however long you give them, very little indeed could be deduced. And even with highly intelligent people, how would you know how long to give the other finalists before theyíd had long enough to do one less recursion than you ? Ian Stewartís article tries to get round this by having a bell rung every 10 seconds, so that youíre only allowed to claim a blob or non-blob every 10 seconds. I donít think this works, but maybe Iím being thick - maybe someone who read the article could convince me ?
Puzzle 2 : In response to Patrick OíDonohoeís "A Modern Cana" - the answer is, presumably, that the residual petrol in the tank that was below the exit point floats on top of the water, and can therefore now find its way to the engine. I canít quite see the same revitalising effect occurring with the sludge at the bottom of a wine vat, but you never know. The interesting point, though, at least for sceptics like me, is that Iíd be tempted to dismiss the "petrol" story out of hand as a fabrication, especially if told it credulously as a "proof of divine action". Maybe we ignore some facts about the world because they are relayed to us by "nutters" ? Getting back to the Russian story, it is just about possible to construe the petrol event as divine intervention even if it isnít a miracle - the dissidents (presumably) didnít know that what they were doing had a logical basis, and God moves in mysterious ways, and may have put the idea in their heads ...