Comments on: Il Buono, il Brutto, il Cattivo

By: Rani

Rani — Mon, 06 Jul 2009 00:48:25 +0000

The ugly and the good are sometimes given (in this order!) as homework in Discrete Math introductory courses.
The bad is known as http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem and was first proved in 1935.

By: yaniv

yaniv — Mon, 06 Jul 2009 06:05:49 +0000

*SPOILER*

Rani,

You are welcome to still write down the answers to the Ugly and the Good. Although the Bad is much harder, it could be given in Discrete Math as well, as its proof relies entirely on the pigeonhole principle too.

What about the Unrelated?

By: yaniv

yaniv — Mon, 06 Jul 2009 06:24:54 +0000

BTW, Liron Raz asked me the Good, the Bad and the Ugly, and Oren Sarig asked me the Unrelated.

Thanks guys!

By: Nadav

Nadav — Tue, 07 Jul 2009 12:21:14 +0000

Il Non Collegato Solution:

call the Polynomial f. f(x(i)) = +-1 for each i=1,…,7
Suppose f=g*h, two Polynomials of degree 1 at least.
so one of them is of degree 3 at most. WLOG it’s g.
since the polynomials are of integer coefficients, and x(i) are integers too, f(x(i))=g(x(i))*h(x(i)), and all g(x(i)) must divide +-1, so g(x(i)) = +-1 for each i = 1,…,7
so we now have g, a polynomial of degree at most 3, getting +-1 on 7 integers.
one of the values +1 or -1 is received 4 times at least. WLOG it’s 1
so we now have g-1, a polynomial of degree at most 3, getting 0 on 4 integers at least.
so g-1 == 0 every place.
contradiction (g’s degree is at least 1)