This could almost be a DOME entry, but I’m probably the only person who calls it this.

“To quote one of the modern, white-ball good coaches, Simon Katich: in a chase, you subtract the amount of balls [remaining] from the amount of runs [required] and divide it by six. That’s the amount of sixes needed.”

  • Carlos Brathwaite, commentating on The Hundred final, August 21st 2021.

In the 2016 T20 World Cup final against England, the West Indies required 19 runs off of six balls, and Brathwaite faced Ben Stokes.

The formula says: subtract 6 from 19 (that’s 13) and divide the result by six (to get two and a bit) – if it held true, the West Indies would need a couple of sixes off of the last over (and seven runs from the remaining four balls).

As it turned out, Brathwaite hit the first two balls for six. Then the next two as well, winning the match.

You would have to surmise that Carlos Brathwaite is someone who knows a bit about run chases.

When I was half-watching The Hundred final, my ears pricked up – here was an expert cricketer unapologetically explaining the arithmetic he would do at the crease in-between facing 90mph bouncers rattling around his head.

His co-commentator, Michael Vaughan (a fine batsman himself) then asked him to go through it again so he could apply the calculation for the situation in the match: Birmingham Phoenix needed 56 runs off off 21 balls – a difference of 35 – and divided it by 6 (“about six”).

There was nothing showy about it, nothing dismissive – Vaughan treated it as genuinely interesting, Brathwaite explained it patiently and clearly, and it felt like a natural conversation between one expert and another about a tool they could use, a psychological way of turning a difficult run chase into a series of achievable tasks. (Only one of the BBC2 team, a little later on, said “I’m rubbish at maths”, which for me ought to result in a long walk back to the pavilion.)

It was a lovely bit of maths-positive coverage, and I applaud it. But, of course, it got me wondering: where does it come from?

In this analysis, I’m going to call $B$ the number of balls remaining, $R$ the number of runs required, and $S$ the number of sixes required.

I reckon the thinking is something like:

  • you would normally expect to score about a run a ball
  • if you did that, you’d score a $B$ more runs, leaving you $R-B$ runs short
  • if you hit $\frac{R-B}{6}$ sixes, you would catch that up.

It’s a nice, simple and logical approach to it – although the mathematical argument behind it isn’t quite right.

As @tony_mann pointed out, it breaks down slightly in some cases – for example, if you need fourteen off of the final two balls, the formula says you need two sixes to win. That’s clearly not the case, you would fall two runs short. So what’s gone wrong?

The trouble is, you’re counting the balls you hit for six twice. Let’s algebra it up:

  • You have $B$ balls remaining
  • You hit $B-S$ of them for one run each and $S$ of them for six
  • You score $(B-S) + 6S$ runs, or $B + 5S$.
  • If $R = B + 5S$, then $S = \frac{R-B}{5}$

I reckon you ought to divide $R-B$ by five instead of six (if nothing else, it’s a much easier calculation to do in your head – a later shot had Brathwaite working it out on his phone((+1 for appropriate use of technology, I think)), but that seems like something you’d not be able to do in, say, a T20 World Cup final.)

I’ll leave the final two points about this to Tony (paraphrased slightly):

  • Might dividing by five demoralise the batter by making the number of sixes seem bigger?
  • How many wickets or dot balls does an attempt at a six cost?

* Many thanks to @tony_mann and @karenshancock for the conversations that led to this post.