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How does a human calculator do it?

Alexis Lemaire
Root of all easy: Lemaire did this sum in 2004 in less than four seconds

Alexis Lemaire has broken the record for finding the 13th root of a 200-digit number. It's an incredibly hard calculation so how does the "human calculator" do it?

Fancy yourself as a bit of mental arithmetics buff, one of those who relishes totting up the bill after a restaurant meal for 12, one of those who looks down their nose at calculator users?

Well try this for size.

The task is to find the 13th root of 85,877,066,894,718,045,
602,549,144,850,158,599,202,771,247,748,960,878,023,151,
390,314,284,284,465,842,798,373,290,242,826,571,823,153,
045,030,300,932,591,615,405,929,429,773,640,895,967,991,
430,381,763,526,613,357,308,674,592,650,724,521,841,103,
664,923,661,204,223.

The answer's 2396232838850303. Multiply that by itself 13 times and you get the above. Even with a calculator you wouldn't beat Alexis Lemaire doing the calculation in his head.

Alexis Lemaire breaks the record in Oxford
On your marks... Lemaire prepares to break the record at Oxford
Last week, at Oxford's Museum of the History of Science Lemaire broke his own record for the task, taking it down to 77.99 seconds. Mental athlete he may be, but he speaks of his hard work in much the same fashion as any other kind of athlete.

"It is quite difficult. I did a lot of preparation for this. More than four years of work and a lot of training every day. A lot of memorising. I need three things - calculating, memorising and the third on mathematical skills. It is a lot of work and maybe a natural gift."

There is a long-standing fascination with those who can accomplish astounding feats of mental agility. The "ordinary" human wants to know how, but sadly the geniuses and the savants can only offer fragments of insight into how they function, and the scientists who have studied them rarely offer a definitive answer.

Researchers have tried to link problems with the brain either through trauma or malformation to extraordinary mental abilities - one of the theories being that damage to one area prompts compensation in another. Brain scientist Dr Allan Snyder has suggested that everyone may possess such abilities but be unable to access them.

Transforming numbers

Kim Peek, the inspiration for Dustin Hoffman's character in the movie Rain Man, has a malformed brain and a below average IQ and yet is able to rapidly read books, memorising vast swaths of information.

I have these associations between places and numbers - some places are imaginary, I try to vary so I don't confuse the numbers
Alexis Lemaire
Lemaire explains that what he does is about transforming raw numbers into other structures so he can "see" the answer to the problem.

"When I think of numbers sometimes I see a movie, sometimes sentences. I can translate the numbers into words. This is very important for me. The art is to convert memory chunks into some kind of structure.

"I see images, phrases, actions. It's very tactile, sensitive. I have these associations between places and numbers. Some places are imaginary, I try to vary so I don't confuse the numbers. It's important to memorise. I have to be precise."

Lemaire's explanation is similar to that of British savant Daniel Tammet. Tammet set the world record for reciting pi at more than 22,000 digits at the museum in 2004.

To him, each number has a distinct colour and appearance, some beautiful, some not, with each complex calculation making up a landscape.

Icelandic in a week

But his skills also stretch to words, with Tammet having reportedly learned Icelandic in a week after a challenge.

It's safe to assume that Lemaire's brain processes don't involve the words "carry one". But there is an explanation for some of what he does. The memorisation he talks about is a series of algorithms, such as a set to tackle the first five digits of the 200-digit number.

He has refined these processes to mind-boggling lengths. For the much simpler calculation the 13th root of a 100-digit number, the first record was set at 23 minutes in 1970. Now Lemaire can manage the calculation in under four seconds.

And whatever the mental processes that lead him to the answer, the fact that he can do this in seconds and without pen or paper remains awesome to the "ordinary" brain.

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