»  Solutions to puzzles in my National Review Online Diary

  November 2008

—————————

The Math Corner in this month's diary included two brainteasers.

[1]   Find a 5th degree polynomial with two equal minimums and two equal maximums.
[2]   Say something intelligent about rating the relative strength of sports teams based on competitive results among them, where not all teams play all others.

—————————

Solutions

For the first, shame on me for having forgotten, or never known about (please don't ask) the Chebyshev polynomials, which are what you get in trigonometry when you write out cos(nX) in terms of cos(X). Read all about 'em here.

The second was solved by Arpad Elo in the 1950s. His solution became the Elo rating system for chess players. Bob Runyan later extended it to games with actual numerical scores (as opposed to just win-lose-tie). Many thanks to reader Joe for all that.