I can multiply any three-digit number by 1667 in my head, usually in less than 10 seconds.

Also, to make it even more impressive, I come up with the digits of the result in proper order (not in reverse!)

How can I do this?

First off, I think the use of "singnificant number", or even "significant digit" (as in the comments) is incorrect, since such usage normally involves rounding errors.

The key to the problem is that 1667 is quite near to 1000 * (1 2/3), and in fact is exactly (1000 * (5/3)) + 1/3.

Once we have everything in fractions, since any number of the form nnn is divisible by 3 (nice fact- if a number's digits add up to a multiple of 3, then the number itself is a multiple of 3), it is easy to construct an algorithm which takes advantage of this in order to keep things at a managable level of complexity.

Anyway, the formula (which is easily computed in the head) that will give you the correct number is:

(nnn/3) * 5 * 1000 + nnn/3

Again, all numbers of the form nnn (i.e. 111, 222, 333... 888, 999) are divisible by three. Thus, for 555, we get:

555/3 = 185
185*5 = 925
925*1000 = 925000
925000 + 555/3 = 925000 + 185 = 925185.

With some concentration, this could easily be done in the head in under 10 seconds.

[Presumably the author meant by "significant number", the fact that we can give the 925 part first. If not, then this will presumably only count as a partial solution.]

Posted by RoyCook on 2003-10-05 12:58:01