(1) My high school teacher thought I needed a challenge, so he assigned a classic. Given N points in general position on a circle, if you connect each pair then how many pieces is the circle's interior divided into? The solutions for small N are 1, 2, 4, 8, ?. I "invented" difference equations and found the 4th-degree polynomial solution.

(2) One or two years later I was asked to construct (or prove the existence of) an infinite sequence of A, B, C with no equal adjacent subsequences. A novel solution came to me in a flash while swinging on a swing.

To clarify the problem, if you start A.B.C.B.A.C.B.C.A.B.C.B.A.C.B you seem to be off to a good start.

You have equal subsequences

__A.B.C.B.A.C.B__.C.

__A.B.C.B.A.C.B__ but they are NOT adjacent; there's a 'C' in between.

However, you've painted yourself into a corner:

You cannot add an 'A':

A.B.C.B.A.C.B.C.A.B.

__C.B.A__.

__C.B.A__
You cannot add a 'B':

A.B.C.B.A.C.B.C.A.B.C.B.A.C.

__B__.

__B__
You cannot add a 'C':

__A.B.C.B.A.C.B.C__.

__A.B.C.B.A.C.B.C__
But I was — and still am — an "idiot savant" :

Originally Posted by

**Wiploc**
I remember the first time I was able to tie my own shoe.

I was last in my Kindergarten class to successfully tie my shoes.