The Corner

Mental Arithmetic

Several readers have e-mailed in with the story of the German math whiz who

has set a record by calculating the 13th root of a 100-digit number in his

head. The story can be read here.

Without wishing to detract at all from Herr Mittring’s achievement, I note

that this kind of thing isn’t so astounding as it at first seems, and in

fact is probably within the range of things anyone could do if he set out

doggedly to do it.

Look: The average 100-digit number — that would be around 5,000 trillion

trillion trillion trillion trillion trillion trillion trillion, of course –

has a base-10 logarithm of 99.69897, so its 13th root has a log of 7.66915.

That root is therefore around 47 million. So you only have to figure out 8

digits.

Memorized tables will get you 3 or 4 digits instantly. E.g. you can divide

up all the 100-digit numbers to get the first 2 digits of their 13th roots:

Numbers beginning 10000000 thru 12654373 have 13th roots beginning with 41.

Numbers beginning 12654377 thru 17182636 have 13th roots beginning with 42.

Numbers beginning 17182641 thru 23167793 have 13th roots beginning with 43.

… etc., thru to

Numbers beginning 93874803 thru 99999965 have 13th roots beginning with 49.

This is of the order of things that you can quite easily memorize. With a

bit of serious effort you — or me, or anyone — could memorize the 3-digit

equivalent, a list of 81 items. (I.e. for 13th roots beginning 412 to 492.

The smallest number whose 13th power has 100 digits is 41,246,264; the

biggest is 49,238,825.)

You can also very easily memorize the right-most digits of 13th powers. For

numbers ending in 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 they are, believe it or not:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9. For example, the 13th power of 247 is

12,736,801,848,653,359,358,345,383,963,927. Again, you could extend this to

two or more digits (though it gets tricky very quickly).

Once you have a good stock of memorized base points like this, a bit of fast

trial & error will get you there.

(I’ve assumed here that the 13th root is a whole number. In this kind of

competition, they invariably are.)

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