Summing it up

By Bobby Neal Winters

Being a mathematician is a special thing.  We mathematicians have been touched by God--touched at least--but it’s not without its cost.  Part of that cost is being set apart as a separate branch of humanity, considered somewhat strange by one’s fellow man.  Another is spending a large part of your life distracted by one problem or another.

This was the case with me the other day as I was coming home from a neighboring state.  I was set apart because I was driving alone, but mainly I was distracted by a problem.  The maddening thing is that now I can’t even remember what the problem was.  This is another one of those things which sets us apart from humanity, a certain forgetfulness in the simple matters of living.

In any case, I was distracted for some extended interval of time, and, as happens so often when we go off on these little mental vacations, the rest of the world keeps going on around us.  In this particular instance, my car kept finding its way along the road, but it didn’t keep going along the right road.  I missed a turn.  When I shook off my distraction, I was off the main highway and in the heart of what we in these parts, in homage to Ned Beatty, refer to as banjo country.

I came to a place where the road widened slightly and to the right side there was an unpainted wooden building with a sign above the door that read “Groceries.”  There was also a sign that said “Cold Pop.”  As it was a warmish day and as I had been riding along with the windows rolled down, the idea of a cold pop appealed to me.

I stepped out of the car onto the gravel and walked to the store.  I couldn’t tell whether the store had ever been painted, but the door, which looked to have been a later addition, had been green at one time. It was a wooden frame with rusted screen wire held to it in a rather haphazard fashion by tacks. When I opened it, it screeched so loudly that the bell above it seemed superfluous.

This was all misleading, however, because when I got inside everything seemed new.  And as I reread that last sentence, I realize it is misleading because it wasn’t new, it was old, but looked new.  The pop machine was an antique, an expertly restored antique, but an antique none the less.  I looked inside and it had been filled with nostalgia brands of soda pop which aren’t available anymore along with currently popular brands in classic packaging.  Before I could extract the pop which and lured me into the store in the first place, I was betrayed by the mathematician in me and became distracted by the rest of the place.

The food shelves had been restored in a fashion similar to the pop machine and were stocked with products in old-style containers.  My eyes ran along them and were drawn immediately to the coffee which was in old-fashioned cans.  I have sorely missed the metal cans that coffee was sold in not so long ago.  My first thought was I wanted to buy some.  My second thought was that in these fancy, classic cans that it would be very expensive.  My third thought was that it wouldn’t hurt to simply see what the price was.

My fourth thought was complete astonishment.  The cans were fairly small but their price was a fraction of what I usually pay.  It don’t remember exactly because the price is not important, but let’s say it was $1.22 for a size I usually pay $5 for.  I must have said “Wow” to myself or something because I heard a voice from the direction of the counter.

“Kin I hep ya, Mister?”

It was an old fellow who was clearly in costume.  He wore a crumpled old felt hat and had a corncob pipe jammed between the jaw teeth on the right side of his mouth.

“Uh...yeah,” I stuttered, “are these prices real?” I asked.

“Yup,” came the answer.

“Is there any limit on the number of cans I can buy?” I asked again, not daring to hope.

“Nope,” was the single syllable reply.

I began to load up my arms.  I don’t know how I managed it, but I got thirteen cans total and, forgetting about the soda pop I’d come in for completely, toted them up to the front.

I sat them down on the counter and he eyed them.

“Thirteen?” he asked.

“Thirteen,” I confirmed.

It was then I noticed that he had several cans in front of him containing different objects.  From two of these he removed numbered disks: one red disk with $1 on it, two green disks each with 10 on them, and two blue disks each with 1 on them.  He laid these out in a row in front of himself and to the right.

He then reached into a can and extracted some black objects that looked like carved scarab beetles.  He laid them out into two rows: one six scarabs long and the other seven, making a total of thirteen.

What happened next happened with a speed that only comes as a result of practice.  I will try my best to explain it to you.  One of the rows of scarab beetles was one longer than the other.  He removed the extra scarab from it and set it my his row of disks.  He then picked up an entire row of scarabs and put them back into their can.

Then he took more of the colored disks from cans and laid out exactly twice as many of each of the different kinds as he had before.

Then he laid out the remaining six scarabs, left over from the first round, into two columns.  Actually, a better way to describe it would be to say he put them into three rows of two.  This was so that he would not have to do any mental activity at all; he just laid out the scarabs mechanically.  As there wasn’t an extra one left over this time, he didn’t set it by the row of disks, but he did take one of the columns and put it back in the scarab can.

He then took numbered disks and doubled the number he’d put down in the previous round, and, as before, laid out his scarabs.  This time there was one row of two and one left over which he put by his pile and put one of the remaining two back in the scarab can.

I could see this process was going to soon stop so I watched carefully.  He doubled the numbered disks one last time and put the lone remaining scarab down next to it pro forma.

Then he put away the numbered disks from the rows without scarabs, counted the remaining ones, and said, “Ya owe me fifteen dollars and eighty-six cents.”

I did a little mental guestimation--as I am lousy at arithmetic--decided it was a fair price in any case, took out the money, and paid the man.

I carried the coffee back out the the car, loaded it and myself up, and drove off.  On the way home, again distracted by what I had seen, it came upon me what the old coot had done.  It is know as the Russian Peasant Algorithm or Egyptian Arithmetic, depending upon whether you are trying to impress Russian peasants or Ancient Egyptians.  

It is a mechanical process which will literally produce the number of objects needed.  In societies that had not yet discovered a way to represent numbers that makes arithmetic convenient, a process such as this one would be quite valuable.  

Lets consider how one could multiply seventeen by thirteen. It is much more fun to do with tokens, of course, but in writing it is more convenient to do it as follows:

17	13 *
8	26
4	52
2	104
1	208 *

Note I’ve used * in place of the scarabs.  The sum of the starred number is 221 which is the product of seventeen and thirteen.

Those of a mathematical bent will, of course, be led to think of binary numbers. One should be aware that the Egyptians where doing this when all the silicon chips in our computers were still sand.

When I ran through all of the cheap coffee I bought that day, I made my way back over to the neighboring state and tried to find that dilapidated old store, but I wasn’t able to.