The Game of Cricket - Parity

Featuring Mr Mxyzptlk, Russian number theorists and surprise tests

Cricket is a sport, yes. But it’s also a game, and in this irregular series of as yet unknown length, I plan to delve deeper into some of the gamier aspects of the sport.

(What’s the difference between a sport and a game, you may well ask. Well, I may get around to spelling that out at some point, but I think as I delve into these various areas that I’m defining as part of the game, you’ll quickly get a sense of where my boundaries are. And, if not, then, heck, maybe that’s a fun little game in itself.)

Cricket is a one-dimensional game. I mean that in both a literal and untruthful sense.

Untruthful because cricket is obviously played on extremely large, two-dimensional grounds. Plus, the ability, and ever-increasing tendency, for batters to hit the ball into the air and (ideally) out of those grounds adds a third dimensional aspect to the game. Cricket is also infamously played across lengthy stretches of time, adding a fourth dimensional aspect to it, especially for those of you versed in general relativity. (Furthermore, could notorious mischief-making Superman villain Mr Mxyzptlk play cricket? If so, for those of you comic book nerds versed in the home location of this spell-check-defying fiend, there’s a fifth-dimensional element there as well.)

Having said all that, cricket is also one-dimensional in a very literal sense. Because, ultimately, the fielding aspect and the time aspect and the fifth-dimensional imp aspect are all secondary to the fundamental contest between batter and bowler. And that contest takes place along a single dimension.

Now, obviously, even that simplified claim is disgustingly untruthful. The stumps have width. The batter can move across their crease or back away. The bowler can swing or spin or cut the ball, moving it off the pitch or through the air. There are, it turns out, technically other dimensions still in play.

But the crux of the claim is true. After all, nobody can actually draw a one-dimensional line. Even the finest of pencils contains some width to it. But that doesn’t prevent geometers (those ambitious dynamos!) from abstracting to the essence of a line, the shortest distance between two points. (And no, I’m not going to get bogged down in spatial variants where this isn’t true. Pipe down, non-Euclidean geometry fans. You know who you are.) And the contest between batter and bowler, at its heart, takes place across just the one dimension, along the line that connects opposite ends of the pitch.

Now, cricket isn’t unique in this. Consider, for example, a penalty shot in soccer (or ‘football’ as the overly literal-talking folk in pretty much every other country in the world call it). Or, a better example, a pitch in baseball (aka ‘throwball’). In both those cases, two opponents face off from a fixed distance apart, with the initiator of the action trying to get a ball past whatever the opposite of an action-initiator is. The bat and ball confrontation in cricket is, in this sense, the same as those contests.

There is, however, a key aspect in which cricket differs from those sports, and which transforms the one-dimensionality at the heart of cricket into something much more interesting.

That key aspect? The non-striker.

Because the idealised straight line of the pitch is not just the dimension on which the bat and ball confrontation is contested. It’s also the dimension on which runs are scored. And this secondary function of the dimension is what opens up a whole heap of compelling gameplay aspects to cricket.

The most fundamental of those aspects is that the line ends. (Yes, from a purely hypothetical perspective, a game with one dimension of scoring could take place over an infinite line, but it would make a quick single somewhat challenging, that’s for sure.) This means that once the batter reaches the end of the line, the only way it which it makes any game sense to even theoretically score more runs is to change direction and head back.

It’s easy enough to imagine game choices for cricket that stick to this linear run-scoring mechanism while still only requiring one batter. Just off the top of my head, you could:

• define a ‘run’ as being a Bagginsesque ‘there and back again’ style process, in which the lone batter has to not just reach the other end but also safely return. In other words, what we currently think of as two runs. There’d be some fun aspects to this. A boundary would be known as a ‘two’. A massive tonk into the crowd? That’s a huge ‘three’, and no mistake. Also? Bradman? Averaging under fifty. Not so special now, are we, you old show-off? (Answer: yes, of course he still is.)

• alternatively, singles could still be taken as we currently understand them, but upon completion of a single, the bowler is asked to bowl from the other end. This variant of the game would introduce a completely different set of tactics that might be fun to explore. For example, exactly how ruinous might over rates become? Could you take enough quick singles to completely knock your opponents out of the World Test Championship? Fingers crossed.

But cricket as we know it did not take either of those single batter gaming options. Instead, it took the far more interesting option of adding a second batter into the mix, thereby opening up for our consideration the wonderful world of parity.

In mathematics, ‘parity’ is just a way of talking about whether numbers are even or odd. (And, by ‘numbers’, I mean, of course, whole numbers (or ‘integers’). Unwhole numbers (eg 99.94, or, indeed, 49.97) are, as the palindrome so gently reminds us, never odd or even.)

(Why do mathematicians use bothersome words such as ‘parity’ or ‘integer’ or ‘bothersome’, when other, more commonplace words could just as easily be employed? Frankly, it’s mostly to keep the riffraff out, but I’m not one to discriminate against you all. Come on in. Riff and raff to your collective hearts’ content.)

Mathematicians lose their shit over odd and even numbers. This is wild, considering it’s a concept so utterly trivial that even a kindergartener gets the gist of it pretty quickly. Odd and even numbers are, however, at the heart of the branch of mathematics known as number theory, since they’re the simplest example of ‘modular arithmetic’. (Modular arithmetic is basically the mathematics of remainders. ‘Seven divided by four equals one remainder three’, for example. So, number theorists waste a lot of time considering the ways, in this particular case, that seven and three are effectively the same number, at least in terms of remainders when divided by four. To be fair, it’s probably not as pointless and boring as that makes it sound. But heck, it couldn’t possibly be, could it?)

Odd and even numbers, then, are the specific case of remainders when you divide by two. From that utterly trivial starting point, the number theorists build a towering edifice of all kinds of startling proofs, a quite disconcerting number of which involve prime numbers for reasons that remain shrouded in mystery. (Well, not really - but you do have to think reasonably hard about it, which is close enough to being shrouded in mystery for most people’s purposes.)

For example, in 1931, a Soviet mathematician named Lev Schnirelmann expended a quite silly amount of energy to prove that every integer can be represented by the sum of no more than roughly 300,000 primes. (A prime number, as you probably recall, is the most delicious cut of number, because it’s only divisible by one and itself! Mmmmm….)

Six years later, his mate (they may, in fact, not have known one another at all) Ivan Vinogradov proved that every sufficiently large odd number can be represented as the sum of just three primes. Which you might think is a pretty dramatic improvement, until you learn that ‘sufficiently large’, in this case, means ‘we’re not exactly sure, maybe greater than a number with roughly four million digits?’).

Anyway, as is so often the case, I’ve found myself distracted by the antics of 1930s Russian number theorists. The point is this: odd and even numbers, and the general concept of parity, have ramifications that aren’t immediately obvious, and cricket’s decision to throw a second batter into the middle opens up the game of cricket to lots of those fun consequences.

In cricket’s case, whether an odd or even number of runs are scored from a particular delivery determines which of the two batters is on strike for the next delivery. A lot of the time that doesn’t matter, especially in the case of two batters more or less equally capable of facing the bowler.

Where things get more interesting, and where the game aspects of cricket come to the fore, is, of course, when the batters are of vastly different ability.

Farming the strike requires the precise deployment of odd and even valued scoring shots to ensure the more skilful batter is facing the bowlers as much as possible.

Often this simplifies to singles and dot balls, interspersed with occasional fours and sixes. (Twos might be scurried by particularly cunning practitioners, but a perfectly timed three is too precise an ambition to ever practically consider.)

However, the parity aspects of the game of cricket are not based simply on the idea that retaining the strike requires even-valued scoring shots (including dots). There is another parity-switching wrinkle, and that is the idea that the end of the over also switches which batter is on strike.

It’s this latter aspect that makes things intriguing. Without it, the better batter could simply defend most balls, hitting the occasional loose delivery to and/or over the boundary. Repeat until that better batter gets out and/or wins/saves the game.

The requirement to find a single at the end of the over instead charges the contest in a way that could so easily have been lost. It enriches the game of cricket immeasurably. Moreover, the necessity to find a single allows cricket to regularly revel in a variation of one of the great unsolved paradoxes - the surprise test paradox.

The surprise test paradox is this: A teacher tells their class that at some point in the upcoming week (Monday through Friday) they will give them a surprise test.

‘Aha,’ the more cunning students reason. ‘But that means the test can’t come on Friday, because if we get that far into the week, it won’t be a surprise. After all, if we don’t have a test by the end of class Thursday, we’ll know for a fact it must be on Friday.’

‘Furthermore,’ they will deduce. ‘By the same logic, it can’t take place on the Thursday. Because we’ve already ruled out Friday as a possible day for the surprise test, that means by the time we reach Thursday, it’ll have to be on that day, which now also wouldn’t be a surprise.’

The students then similarly work their way backwards, ruling out Wednesday, and, hence, Tuesday and, ergo, Monday. Turns out they can’t receive a surprise test at all!

Which is why it’s so startling when they do have one on the Tuesday.

Logicians - a class of mathematicians even more prone than number theorists to getting bogged down in matters that seem blatantly obvious - haven’t quite cracked the best way to resolve this paradox, despite all their fancy axioms and Latin-named reasoning techniques (‘modus ponens’? Grow up, logicians!).

And a darn good thing, too. Because the better batter choosing when in the over to take the single and farm the strike is at least partially analogous to the surprise test paradox.

Usually the field is out in the early parts of the over to prevent those maximally-valued even scores (ie, fours and sixes). While the field is out, it’d be easy to take a single and rotate the strike. But doing so too early puts the poorer batter on strike for a larger proportion of the over.

On the other hand, attempting to take the single later in the over when the field is in is much more difficult and brings with it the risk that the better batter might fail, putting the tailender on strike for the entirety of the following over.

Ideally, then, the better batter goes Full Spanish Inquisition, taking the single when the fielding side doesn’t expect it, but not so early that they don’t get the benefits of shielding the tailender.

It’s not exactly the same as the surprise test paradox, but it shares enough traits with it to add yet another frisson of entertainment to strike-farming that a less paradox-fuelled game might lack.

And, look, none of this is new. Anybody who’s watched any amount of cricket knows all of this already - it’s what makes a batter shielding a tailender such a fun thing to watch. But what I’m asking you to appreciate is why it’s fun. (After all, only by careful introspection and analysis can we have proper fun.) It’s fun because, way back at the dawn of cricket, correct choices were made with how best to actualise the one-dimensional aspect of cricket run-scoring. Choices that involved parity.

Or, to paraphrase two of the great fun-loving philosophers in history: ‘Parity on, dudes*!’

* Also, dudettes