09:43 am, 14 Sep 06

### eight queens boggler answered

[problem statement]

Monica explains:

Monica explains:

When I was young I was taught the multiplication table. Fine. I waited for the other shoe to drop but noone ever taught me the "division table". So I figured it out myself by doing all the long divisions by hand (this was before calculators) and I memorized all fractions from 1/1 to 12/12 .

Decades later, I was teaching a data structures and algorithms class at my old univeristy. In the class book (probably Wirth's, but I'm not sure) was a complete listing of all solutions to the 8-queens problem expressed in a compact form as 8-digit numbers. One of them jumped out at me as containing the magic sequence "428571" which are the digits that form the expansion of 3/7 = 0.428571428571.. . .

So I quickly figured out it was 255 / 7 that gave us 36.428571 as a valid solution to the 8-queens problem. I was so fond of this discovery that I made it my corporate logo when I started Syntience Inc. in 2004.

3smallishmagihukumajwzgraydonSo he made us practice all the decimal expansions p/q for p, q in [1,15] or so, and also all the squares and cubes up to 30 or so, plus regular cases that occur beyond due to recurring factors. It seemed silly at the time, but it illustrated an important point that I'd never had a teacher illustrate to me before. And indeed, solving systems of linear and quadratic equations

didbecome much simpler thereafter. I like to think it even helped make number theory feel a bit more natural, since I had more of a concrete "image" of the factors within numbers, and maybe even made studying the construction of the reals easier because I had a concrete "image" of rationals as finite repeating strings.The same rule holds for learning languages -- you have to drill on vocabulary so that you won't stumble on each word while trying to absorb patterns of grammar -- but unfortunately I only learned this lesson after already dropping French.

flipzaggingI demand my money back.