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Several years ago I devised a set of rules to allow you to model a
football match and even a World Cup
. The rules are so simple you can play a full World
Cup in one lesson, and they're accessible enough to appeal to
primary as well as secondary schools. If football, why not
The shorter form known as Twenty20 seems ideal, so here's a game
that, like Twenty20 itself, is both accessible and quick to play.
It also meets my requirements for a mathematical game - it must be
fun to play, involve opportunity for some valid mathematics (plenty
of arithmetic, data-handling, and probability), require decisions
to be made, and use readily available materials (just a single die,
numbered 1 to 6).
Cricket fans will know that in each match the team batting
first will bat for a maximum of twenty six-ball overs, though
should it lose all ten wickets any remaining overs are forfeited.
Its opponents then bat for a maximum of twenty overs in turn,
attempting to beat this score. Generally speaking, a batting team
will hope to get off to a decent start and accelerate more and more
as the innings builds to a climax. Usually this acceleration is
feasible only if sufficient wickets have been conserved in the
earlier stages. Should things go wrong early on there may be a need
for a mid-innings retrenchment - on the other hand, at the end of
the innings runs are far more important than wickets, and later on
a team will take risks it wouldn't dream of taking earlier.
In a cricket game many overs will result in between one and six
runs being scored. This makes it very simple to use a conventional
six-faced die as a means of generating the runs scored over by
over. We must also involve the bowling side as well, so for each
over, both players roll the die. The batting player scores the
number of runs s/he rolls, and if the bowling player rolls a 6 then
the batting team loses one of its ten wickets.
These two simple rules allow an innings to be played through in
just a few minutes. However, it gives a pretty anaemic game, as
neither the number of runs nor the number of wickets is
sufficiently large to make things at all interesting. Worse still,
it's purely mechanical. I want any simulation to require pupils to
make decisions (so does Attainment Target 1, for that matter). The
batting side should be able to decide when to play conservatively,
scoring fairly slowly but with a relatively low risk of losing
wickets. At other times a more aggressive mode may be needed, and
the batsmen will take more chances in an attempt to score extra
This is easily incorporated. We'll call the original style of play
Style 1, but we'll introduce the possibility of Style 2: at any
point the batting player may decide to accelerate, so s/he rolls
the die not once but twice and adds their scores to get the total
score for the over. Of course, more aggressive batting means more
likelihood of losing a wicket, so the bowling player also rolls
twice for that over. If the bowler rolls a 6 on either throw a
wicket is lost (and two 6s mean that two wickets fall).
And since Twenty20 cricket often throws caution to the wind,
there's Style 3 as well, where the batting player chooses to throw
the die three times to get the total number of runs in the over.
Likewise, the bowler throws three times, and each 6 means the fall
of another wicket.
Now if you try this, you'll find that it's much better. However,
it's still not 100% satisfactory. It's too easy for the batting
side to use the super-attacking Style 3 throughout. They're
unlikely to lose all their wickets and may well post a score of
over 200 (in a game of Twenty20 cricket a total of below 100 is
very poor, 150 might be reasonable, and the occasional 200+ marks
an excellent total).
But a good simulation allows you to build in additional features to
model the real situation, and in any cricket team the strongest
batsmen are highest in the batting order and the later batsmen are
weaker - less likely to score runs and more likely to lose
their wickets. So to reflect the weakness of the lower batsmen
we'll incorporate one more rule. Once the batting side has lost
five wickets, further wickets fall whenever the bowling team rolls
either a 5 or a 6.
Play continues, until either twenty overs have been played, or the
batting side has lost all ten wickets. The players now exchange
roles and the ex-bowling side now has twenty overs to try and
overhaul the total made by the first team.
Particularly if you're batting second you're going to have to
evaluate the position each over, and I think the game is worth
playing in its own right.
There's lots of mathematics to be practised and explored in
arithmetic and probability and statistics. There are plenty of
opportunities for enquiry - what is the typical score per over
in each batting style, and how likely are wickets to be lost in
each mode? Then there's the statistical side - how do the
scores we get compare with those in the real thing? (I looked up
the scores of nearly 150 English Twenty20 games last season; scores
ranged from 68 to 239, with a mean average of 159 for the team
batting first and 146 for the side batting second. The average
number of wickets to fall was about 6.2.)
I imagine these simple rules will satisfy most people, but there's
plenty of scope for taking things further. You might like to use a
slightly less blunt instrument for distinguishing between batsmen
- perhaps wickets should fall when the bowler rolls a 6 for
the top four wickets, then on a 5 or 6 for the next three, and on a
4, 5 or 6 for the final three. And wickets fall most easily in the
last couple of overs, so you could introduce a modification to take
care of that. We have batting Styles 1 to 3; is it worth trying a
Style 4? And should the play be directed entirely by the decisions
of the batting side, or should the bowling team have an input into
For further information, a convenient first port of call is the Wikipedia Twenty20