At the Outer Limits of Mental Math

​In 1669, right around this time of year, the mat​hematician John Wallis found himself once again laying awake in bed, unable to sleep. Wallis, who is credited with crucial contributions to fields ranging from calculus to music theory, did what he usually did in such circumstances. He did math. Wallis didn't just count sheep, he squared sheep.

One night, he would write later in a letter, "in bed, without pen, ink or paper or anything equivalent, I did by memory extract the square root of 30000,00000,00000,00000,00000,00000,00000,00000, which I found to be 1,73205,08075,68077,29353, etc. and did the next day commit it to writing." In a subsequent journal ​article extolling the virtues of night-math, Wallis noted that he'd had some additional assistance from a persistent cough, which aggravated his insomnia.

In that same article, which was written several years later, Willis pondered whether or not he still had the capability for doing such a mental calculation. Memory, of course, fades.

I am utterly incapable of mental math. I rely on an arsenal of dry erase boards, notepads, and Wolfram Alpha. Mostly, I need to see it—the numbers and symbols, acquiring form.

In the grand scheme of things I could be considered disadvantaged. The human brain is, on average, capable of some truly impressive feats of calculation. Nothing like Wallis, of course, but our oversized lumps of grey matter aren't oversized for nought.

It helps if we put it in computing terms, where metrics for computational capabilities are well-established. A 1989 article​ in Foresight Update sought to do just this. "Just as we ask how many mips or megaflops an IBM PC or a Cray can perform, we can ask how many operations the human brain can perform," wrote Ralph C. Merkle, now a faculty member at Singularity University. "Neither the mip nor the megaflop seems quite appropriate, though; we need something new." That new thing is synapse operations per second.

It's not the only possibility, as Merkle goes on to explain. We can look at energy expended per operation also, or we can come up with some telling ratios by looking at the volumes of information delivered by the retina to the brain for processing. The overall idea is that we can determine processing capabilities by looking at the brain's energy limits.

"By estimating the distance between synapses we can in turn estimate how many synapse operations per second the brain can do," Merkle wrote. "This estimate is only slightly smaller than one based on multiplying the estimated number of synapses by the average firing rate, and two orders of magnitude greater than one based on functional estimates of retinal computational power."

Finally, Merkle wrote, "It seems reasonable to conclude that the human brain has a raw computational power between 10^13 and 10^16 operations per second."

Those numbers are highly theoretical and probably untestable. Merkle's range is well beyond the capabilities of the average contemporary CPU, but the problem with the brain is that we don't get to program it how we like, however much we like to think otherwise. A nice new Intel CPU won't have quite the same responsibilities of an ad hoc human brain.

Chris Westbury, a cognitive neuroscientist at the University of Alberta, attempts to put the same question in rather more real-​world terms. Westbury offers two competing figures. One puts things completely in the real-world, probably to a fault. This comes from estimating the average decision speed of the brain (choosing between this or that), which is about two decisions per second. That doesn't even rate.

A better metric is then just comparing the clock speed of a single neuron to the clock speed of a computer processor. Westbury concludes that, based on these figures, a neuron is bested by a typical CPU 5 million times over. That's not terribly meaningful either, for obvious reasons. We as humans have quite the parallel processing setup.

For a less theoretical view of human mental calculation, we have competitions. The first Mental Calculation World Cup was held in 2004 in Annaberg-Buchholz, Germany. The tasks are super intense. In one minute, how many days have elapsed since some random date in the 1600s? Or, add 10 10-digit numbers 10 times in seven minutes. Th​is year saw a world record from the German "calculator" Wenzel Grüß: factorizing 20 randomly selected five-digit numbers in 10:03 minutes. The cup itself went to 13-year-old Grant​h Thakkar from India.

These feats are only slightly less astonishing when you consider that most extreme mental math is done algorithmically. It's possible in a way to program the brain with routines or functions, each of which take a very large problem and whittle it down into more manageable problems. For example, if some calculation or part of a calculation requires multiplication by five, we can instead multiply by 10 (just move the decimal point over) and take half of that. Multiplying any two digit numbers can be broken down into combinations of powers of 10 followed by combinations of much smaller and more manageable leftover values. (It helps to imagine any given number as a combination of this form: 10r + s. So, 93 would be 9r + 3, a less intimidating perspective.)

There are lots of these sorts of tricks for the simple reason that "calculator" not too long ago meant human being. CERN, home to the Large Hadron Collider, once kept a unit of human calculators in its employ. Math luminaries like John von Neumann and Bernhard Reimann had day jobs as calculators. Wallis himself was a cryptographer.

In a history of hum​an calculators found at the University of St Andrews website, J J O'Connor and E F Robertson note something pretty important. Most of the mental math super-geniuses throughout history hit their peak about age 10. Moreover, education seems to actually limit or diminish mental math abilities. "This may be due to the simple fact that such calculating abilities require continual practice for many hours each day and education occupies too much time to allow this to continue," the pair suggests.

We can infer something pretty important from that, I think. This is an ability easily lost, a fragile thing. None of us has to be a mental calculation wizard, but still. As the mantra of the Dune "mentats" goes, "It is by will alone I set my mind in motion."