Thursday, 29 December 2016

A curious cat and another curious error

The first mathematics book written in English was the snappily-titled, anonymous An introduction for to lerne to recken with the pen, or with the counters accordyng to the trewe cast of Algorisme, in hole numbers or in broken, newly corrected..., first published in the 1530s.  A few years ago the British Library paid £95,000 for a copy of the first edition.  Happily, a facsimile (of the second edition) is now cheaply available in the wonderful series produced by TGR Renascent Books.

The book concludes by presenting lots of interesting problems and their solutions.  One is a version of the Josephus problem, in which fifteen out of thirty merchants are to be cast overboard to save an overloaded galley in a storm: the reader is given a mnemonic for arranging the merchants so that the right fifteen (the Christians rather than the Saracens, as one would expect for the time) are saved.  The mnemonic is a Latin verse, which seems a little odd for a book whose selling point was that it was in English!  Presumably the author was padding his book out with whatever came to hand, not very carefully, as we shall see.

Another example asks about travellers going in opposite directions between London and Paris and when they will meet, or at least I think that is the intention: but since in the book one traveller is going between Paris and London, and the other between Paris and Lyon, they are unlikely to meet unless one of them gets badly lost.  It seems that our author was trying to make the problem more relevant to an English readership but didn't carry his intention through.

The most curious of the problems, to my mind, is "The rule and questyon of a Catte". The problem presented (transcribed from a 1546 edition) is,

"There is a catte at the fote of a tre the lēght of 300 fote / this catte goeth upwarde eche day 17 fote, and descendeth the nyghte 12 fote.  I demaunde in howe ōge tyme that she be at ŷ toppe."

Or in today's spelling, "There is a cat at the foot of a tree of height 300 feet.  This cat goes upward each day 17 feet, and descends each night 12 feet.  I ask, how long a time will she take to reach the top?"  It's good to see that even as far back as the sixteenth century, authors of maths textbooks were presenting realistic real-life problems to their students.

Luckily our author provides the solution:

"Answere.  Take by and abate the nyghte of the day, that is 12 of 17 and there remayneth 5, there fore the catte mounteth eche daye 5 fote / deuyde now 300 by 5 and thereof cometh 60 dayes then she shall be at the toppe.  And thus ye maye do of all other semblable."

That is, "Subtract the night from the day, that is 12 from 17: this gives 5, therefore the cat mounts each day 5 feet. Divide now 300 by 5 and you get 60 days: then she shall be at the top.  And thus you may do all other similar problems."

Which is very neat and useful.  But unfortunately the answer is wrong.  After sliding down 12 feet on the 57th night, the cat will be at a height of 285 feet and will reach the top of the tree after 58 days, not 60.

What is curious about this is that the author seems to have missed the point of the puzzle.  It is interesting, surely, only because it is a trick question, but the author has fallen for the trick. What has happened?

A historian friend with whom I discussed this had a plausible idea.  The author was probably taking his problems from a continental book.  This book may have presented the problem of the cat, and first derived the incorrect solution as above, but then went on to say something like, "But in fact this is not the correct answer, because ..." and explained the trick.  However the unwary writer of An introduction ... didn't read any further, and reproduced the problem with the incorrect solution.  Who knows how many generations of English students were bemused by their cats gaining the top of the tree two days earlier than they had calculated as a result of this carelessness?

Friday, 2 December 2016

Learning from the audience at my Prisoners' Dilemma talk

Today I had the privilege of taking part in an excellent "Mathematics in Action" day at which around 700 school students heard a series of talks about maths.  I was talking about one of my favourite subjects, the Prisoners' Dilemma (PD) (I gave a rather different talk about the same material at Gresham College a few years ago.)  I was in amazing company: the other speakers were David Acheson, James Grime and Hannah Fry, and we were wonderfully compered by Sarah Wiseman.  It was a delight to talk to such an enthusiastic audience.

This was my second such event, and this time I had enough confidence to ask the audience (by show of hands) how they would play the games I was discussing.  I had hoped to win more than my fair share of the "Rock, Paper and Scissors" games, but the audience out-thought me (it probably didn't help my cause that, thanks to a clicker malfunction, my choice was revealed to the audience earlier than I had intended).

But the really interesting thing was when I asked whether the audience would co-operate or defect in the Prisoners' Dilemma.  To my surprise, the vast majority of the audience chose to co-operate: to my even greater surprise, those who defected were loudly hissed by many in the audience!  This made a point that I was coming to (very briefly) at the end of my talk: games like the PD give us insights not just into games but into issues like trust and reputation.  If defecting in a PD results in this kind of opprobrium, then the benefit of a shorter prison sentence may be negated by the damage to one's reputation, and this kind of peer pressure makes co-operation a more profitable choice than defection.

But then when I asked the audience about the "Cold War arms race" PD - should a superpower invest its resources in more and bigger nuclear weapons rather than in health research and education? - the response was different.  People who would co-operate in the standard PD, rather than betray their friend, chose to build up their nuclear arsenals.  Furthermore, there was no hissing.  (To be fair, the way I asked the questions may have had a lot to do with the answers, so I am not claiming that the audience behaviour proves anything at all, only that it is suggestive.)

So basically it appears that we frown upon people who are selfish in their dealings with individuals, but when it is selfishness at national level, our response is quite different.  If this reading is correct, then there ia a real challenge to achieving co-operation between nations because we perceive that kind of co-operation as fundamentally different from people interacting as individuals, and we don't feel the same social behaviour to be nice to other nations as we do when we interact with people.

As Martin Nowak says in his fascinating book Super Co-operators, "… Our analysis of how to solve the [Prisoners'] Dilemma will never be completed.  This Dilemma has no end."