### Exceptional, or ordinary with practice?

How do you solve math problems in your head? Perhaps a better question is, do you solve math problems in your head? With the availability of electronic devices to do it for us, I would not be surprised to learn that many people never try. This is a fascinating question, one I also consider briefly in my new book, Be Different.

I was reading Darold Treffert’s book on savants, and I was intrigued by a few examples of savant thinking. I tried solving some of the problems in his book to get a feel for how “comprehensible” they might be to me, with no recent practice calculating. Here is a simple example:

You have a carriage with a wheel that’s six yards in circumference. How many revolutions will the wheel make while traveling two hundred twenty miles?

This is how I get the answer in my head. I’d be interested in how you might do it:

Six yards is eighteen feet. I see that as a short line.

So one hundred revolutions of a six yard wheel would take me 1,800 feet. That’s a much longer line in my head, one that curves.

Three hundred revolutions would take me 5,400 feet – more than a mile. Now the line has curved back unto itself, making a circle.

How many rotations are there to a mile? Less than three hundred. A mile is a smaller circle. I can see those circles, on inside the other. They do not quite match.

I adjust the length of the longer line that forms the big circle. Try 290 . . . that’s 5,400 less 180, or 5,220. A mile is 5,280. Now I see the line laid flat, like a straight stretch of highway. Two hundred ninety revolutions leaves us sixty feet short of a mile marker. So what’s the fraction?

Three eighteens go into that sixty-foot remainder with the same six remainder. Adding that to the 290, I see the answer is 293 and a third. The six-yard wheel does not fit a one mile line, but it fits perfectly into a three-mile ring. If you put a mark on the wagon wheel, and mark any point where it touches the big circle, those points will touch every time the wheel rolls past. I like that.

If you roll the same wheel around a one-mile ring the points will only touch every third trip around, which is unsettling to me. I like smooth fits, so I will solve the next step using three-mile units.

I can now see the answer: 880 revolutions. A perfect fit. Six yards, three miles, and eight hundred eighty turns.

How many three-mile eight-hundred-eighty revolution units are there in 220 miles? My mind visualizes stacks or piles for this next step. Seventy units reach two hundred ten miles. I quickly see how seventy-three and a third are needed to reach the two-twenty goal.

Stacking seventy-three piles of 880 in my mind takes a little time. Eventually, the stacks add up and I see the result is 64,240. Now I just have to add the third (of 880) and I’m done. To do that, I add three hundred to the pile, making 64,540, and then take back six and two-thirds.

64,533 and 1/3 is the answer to the question.

As a further experiment, I scaled up the distance, to 2450 miles and then 20,315 miles to see if I could keep scaling up the numbers. There must be some limit to that, and it certainly took me longer, but I solved those bigger problems in a few more minutes. Solving the longer distance problems involved one and then two more levels of “stacking” in my mind.

It does not seem that hard to me. I often did similar calculations as a kid, for fun. I’m sure I could do it again, pretty quickly, with some practice.

I test my answer with a calculator. The process to do that is considerably simpler.

I multiply 220 (miles) by 5,280 (feet per mile) to get 1,161,600 – the total distance in feet.

I divide that by 18 (the wheel circumference) to get 64,533.333 – the revolutions turned.

It’s a lot faster to get this answer from a computer, for sure. But is the ability to figure problems like this out in one’s head really exceptional? In today’s world, I would not be surprised if kids never develop these skills. When I grew up, though, pocket calculators did not yet exist and I had to know how solve math problems on my own. Given my own ability, I suspect many people of my generation could solve a problem like this in their heads, but perhaps I am wrong. What do you say?

## Comments

But some of these kids really have no math base at all. When given a much simpler logic problem than yours- How many times would a painter paint the number 4 if he's painting the numbers 400 to 499?- these kids just hope they will randomly guess the correct number. 41? 57? They cannot see that, at minimum it must be 100. And these are neurotypical kids. Actually the ASD kids usually, but not always, have pretty good number sense.

into my students world. THANK YOU, Mark Warner.

So, to answer your question--yes, being able to solve a problem like this in your head is extraordinary, but not because kids nowadays rely too heavily on calculators.

There are always other fun ways to do an arithmetic problem. But what's more fun is a book iike Gradshteyn and Ryzhik's.

Actually I just started reading about Asperger's and it's bothering me a bit, maybe I have it. It's a sort of sinking feeling.