Some of the best puzzles I’ve ever seen are the Chess Mysteries, created by Raymond Smullyan. They have given me hours and hours of intellectual stimulation.

Here is a typical example from his book ‘Chess Mysteries of Sherlock Holmes’. Consider the following chessboard. Black is in check-mate from White. Which side (north or south) is white, and which side is white?

The first observation is that this is very different from the normal kind of chess puzzle, which is of the type ‘White to mate in 3 moves’ or similar.

So how would you go about solving the seemingly intractable problem of which side is White?

The only clues you have are the ones from the chess board. Obviously, since black is in checkmate, the last move must have been white’s. So what was white’s last move?

Since Black is in check from the White bishop, either the bishop must have moved to administer check, or some other piece must have moved out of the way to discover check from the bishop. Consider the former case.

If the Bishop moved to administer check, where could it have moved from? Obviously only along the main diagonal. But on any square on that diagonal, it would already have been administering check to the king! That would invalidate Black’s previous move, and therefore it is not a valid possibility.

That leaves only the option that something moved out of the way to administer check. What could it have been? Not the queen, because it could have only moved away from b7 to discover check by the bishop, but at b7, the queen would already have been checking the black king. Not the White bishop on c5, obviously - it is on a black square. So it must have been one of the pawns. Either the pawn on d4 moved there from d5, or the pawn on e5 moved there from e4, or the pawn on h3 moved there from g2 (capturing a black piece). In the first of these cases, White is North; in the second and third, White is South. So which of these is it?

We can eliminate the last option straightaway. If there was a pawn on g2 until one move back, how could the white bishop have gotten to h1 in the first place? So no – that option is verboten.

In order to resolve the question of which pawn moved (of e4 and d5), it is necessary to cast your mind one step further back. Assume that the pawn (whichever it was) was moved out of the way to discover check by the bishop. What could Black’s previous move have been? If it was with the King, where could the King have moved from? It could not have moved into a8 (where it currently is) from a7, because at a7 it was in check from both the queen and the bishop on c5, and there is no prior move by white which could have discovered both checks at the same time. Of course it could not have moved from b7, due to the White King’s presence on c3.

So Black’s King could not have moved to a8 from anywhere! It seems an impossible position and a fruitless quest…

But there is another possibility. Could black’s last move have been, not with the King, but instead with another piece? What other piece, I hear you ask. There’s nothing else on the board! Well, it’s true that there’s no other piece on the board now, but there may have been a piece when Black made the last move. That piece was captured by the Bishop to deliver check.

This doesn’t seem to be any more promising, because where did the Bishop come from to capture this piece X?

Then (if you are the kind of person who solves this kind of problem) you will have an epiphany. Perhaps, before the bishop captured the piece X, it wasn’t a bishop at all!

How is that possible? It’s possible if the Bishop was a pawn that got promoted. And since it captured X on promotion, it must have promoted on h1. So white’s last move was bxa8=B.

And Black’s move before that was Xa8. (X could have been a rook or a queen or a knight).

And if white was moving down the board to promote, then obviously, white was playing North.

Nice, no? Here’s a link to Smullyan’s book, if you are interested in this kind of problems: http://www.amazon.com/dp/0486482014. He also came up with a sequel called Chess Mysteries of the Arabian Knights. Both are fantastic.

I got my first tangible reward for my conscientious blogging yesterday: Dinesh had commented on my Simulators post, recommending a Greg Egan book called Luminous, and I bought it. An introduction to a new book: what could be more valuable? Thanks, Dinesh .

The book contains a short story of the same name. It’s only about 30 pages long, and by normal standards, I would have polished it off in a single sitting. However, a few pages into the story, the author makes such an incredibly audacious claim that I was completely shell-shocked, gob-smacked, and had to keep the book down and think for several hours before I can get back to it.

What Egan claims is: there is no such thing as pure mathematics.

This is as bold as it gets. Mathematics – isn’t its purity something that we were always brought up to believe in? Men may come and men may go, but maths goes on forever.

Even xkcd concurs:

And Egan disagrees. I love this guy!

According to Egan, even the most abstract theorems in Mathematics – Fermat’s Last, for example – have some physical basis. This may be a property of the real world in some cases (for example geometry modeling the real world). But even the so-called ‘abstract mathematics’ – even pure logic – has a physical basis; it exists as a sequence of thoughts in human brains, and therefore as a sequence of neurotransmitters, perhaps, that fire in a particular sequence in the brain. Alternatively, they exist as patterns of charge and current in a semiconductor wafer; but the point is, whatever the manifestation, there is a manifestation.

We are able to see the truth or falsity of mathematical statements without recourse to experiment because of a series of chemical actions and states in the brain.

So if our brains were wired differently, would we perceive mathematical truths differently? Likely, but it’s not that simple. Animals, for instance, could be said to perceive mathematical truths similar to us (though their deductive and inductive capabilities are much lower than ours).

So what would have to be different for us to perceive maths differently?

Egan asks the question a bit differently. Where, he asks, did mathematics come from? Did it exist before the big bang? Or, shortly after the event, did the early quarks and gluons ‘create’ mathematics? Shortly after, he says. And if physics were different, maths would be different.

I am enormously intrigued by Egan’s suggestion. I want to believe it is true – it appeals to my engineering-centric conceptualization of the universe! But I also have my own reasons for believing that pure mathematics is not so pure after all.

The crux of my opinions around this was developed around an investigation of the way we perceive infinity. We are able to assert (and mathematically substantiate) that a statement like the ‘infinite primes’ theorem (that there are infinitely many prime numbers) must be true. Certainly we do not have the physical experience of infinity in order to establish the validity of this theorem by experiment. How, then, are we able to prove it?

The analogy to explain this is that of a chess puzzle. Say we are given a fairly complex chess position, and are asked to prove something about it; for example, to prove that black can never win from that position. Prima facie, it may appear that such a proof is impossible. Since we do not have a limit on the number of moves that are allowed, even if we comprehensively list all possible moves starting from the current position, we cannot eventually exhaust the infinite length of time for which the moves may go on. What is to say that black will not check-mate in the trillion-trillionth move?

Cast this way, the statement cannot be proved. However, there is another way of establishing the proof. On a chessboard, with a finite number of chess pieces, there are only a finite (though large) number of positions possible. Since any valid chess move is one of these positions, a game of chess is then a way of moving from one chess position to another. For a finite number of positions, there are a finite number of interconnections between the positions. To prove something like the ‘black will not checkmate’ hypothesis, therefore, we do not need to reason infinitely; we need only check (and this can be done with a computer in a finite amount of time) whether it is possible to go from the node representing the current position, to one of the nodes representing a white check-mate. If such a path exists, black can win.

The cleverness here is that we have converted the infinity of chess moves into the ‘finity’ of chess positions. This is the same process that works at multiple meta levels in mathematics to allow us to make general statements about infinite sets of mathematical objects. Because underlying the infinity of mathematical operations, there is the equivalent of a ‘finite chessboard’, which is a combination of all the axioms and rules of inference that our minds accept as true, and which we manipulate to go from ‘truth’ to ‘truth’ in logical sequence.

The frightening question is: is this ‘chessboard’ a feature of neuroscience and physics?

Because if it is – and this is what Egan’s story seems to be about – then ‘pure’ mathematics has the same clay feet as the rest of human knowledge, held up by a complex scaffolding of experimental physics.

I’ll get back to finishing Luminous now…