## 15 March 2014

### The higher math needed to calculate all possible moves in rugby and chess

How many moves are needed? How many moves are possible?

Patrick Comerford

I know of someone who was studying for a degree in math at Maynooth many years ago. He was the first in his family to get to university, and his mother was understandably proud of him.

When she was asked what he was doing in Maynooth, she replied: “A course.”

When asked what the course was, she said: “Sums.”

But I cannot afford to laugh. I got a pass on the pass paper in math in the Leaving Certificate in 1969, and counting beyond four sometimes seems to require higher calculus.

Ireland can win the Six Nations’ Championship later today [15 March 2014] at the showdown in Stade de France in Paris … but only on points difference. And the calculations for this nail-biting day of wall-to-wall rugby involve a level of math that is far higher and is far more difficult than anything I learned while I was at school.

This has happened on four times in the past 41 years, and this was a task that foiled Ireland three out of four years.

So let’s look at the permutations, combinations and calculations if Ireland is to win the title today for the first time since 2009 and for only the time since 1985.

Three sides have six points, England has already won the Triple Crown. In order of their favourable points’ difference, they are:

Ireland (81)

England (32)

France (3)

England plays Italy in the Stadio Olimpico in Rome, starting at 12.30.

If England beats Italy by any margin and later in the day Ireland and France draw in Paris, then it is impossible for England not to win the Six Nations title.

If England beats Italy by 50 points or more, then an Irish win in Stade de France by any margin means Ireland still wins.

If England beats Italy by any margin, and later in the day France wins in Paris, it seems impossible for England not to secure the Triple Crown and the Six Nations title.

I say it seems … because, when it comes to points’ difference, France is 29 points behind England. So, for every point England wins by in Rome, it ratchets up the French target of a 30-point winning margin over Ireland.

If Italy beats England, or the two sides draw, and Ireland wins in Paris, then Ireland wins the Six Nations title.

If Italy beats England, or the two sides draw, and France beats Ireland, then France wins the Six Nations title.

It all explains why Ireland had to pile on all the points possible when Ireland beat Italy last Saturday.

If all that seems difficult to calculate, then consider the calculations that have to be made in playing chess, where the possibilities are boundless, it seems.

In the G2 section of the Guardian, Stephen Moss of Kingston-upon-Thames, Surrey, posed this dilemma:

In Paul Hoffman’s book, King’s Gambit: A Son, a Father and the World’s Most Dangerous Game (published by Hyperion in New York in 2007), he states that: “In practice the possibilities in chess are boundless, although theoretically it is a mathematically finite activity – there are, for example, 988 million positions that can be reached after four moves for white and four for black.” Can that figure possibly be correct? It seems far too big a number after so few moves for each side. And is the often quoted “fact” that there are more possible moves in a chess game than there are atoms in the universe really correct?

His query has drawn a number of interesting answers in the weeks since. I was overcome by the submission from ‘ThomasD’ the Thursday before last [6 March 2013]. He points out:

“There are different answers available depending on how you define the game’s complexity, though the ‘top end’ answer was calculated by Victor Allis as 10123, which compares to estimates of 1081 for the number of atoms in the observable universe.”

More moves in chess than there are atoms in the observable universe? That is almost beyond comprehension. But it certainly illustrates that God’s love and compassion for all is not beyond comprehension.

No wonder I have been a life-long reader of the Guardian.

Meanwhile, how many moves can Brian O’Driscoll make in his last international fixture this afternoon.

It’s wall-to-wall rugby for me today.