Brief Solutions

Here are solutions to the problems based on the work of the Macalester team of Danny Kaplan and Stan Wagon, with additional notes regarding better solutions when we know them. In most cases there are better solutions than ours to the 10-digit problem (it is arguable, of course, but our approach was the right one, or close to it, for #1, #2, #8, and #9). One hundred digits are given in all cases but one; thanks to F. Bornemann and J. Boersma's team) for the extra digits in cases we did not have them. Mathematica code giving a complete solution is included in all cases except #2, which, while not difficult, requires the most programming. As far as our own solutions go, we can say that we had 14 digits to all the problems, with 14 or 15 occurring on #3, #5, #7, and #10. Using our work and that of others, all problems except #3 yield 100 digits with no problem (when one uses the proper algorithm, which can be hard to find). Problem 3 is more difficult.

On the other hand, it is tricky to come up with a good measure of difficulty, as different people have different expertise. For our team, the last four problems to be solved were #8, #6, #10, and #5, in that order, with #5 being last (Trefethen called #5 "exceptionally difficult"; three of the teams scoring 99 missed the tenth digit on #5). The other six we could handle without much research or code, but the last four required library work or heavy programming.

As far as getting 500 digits in reasonable time, that can be done for #2, #4, and #8 by our methods; for #5, #6, #7, and #10 by others (and probably #1 and #9). But we must not get too caught up in the search for digits; the real point of this exercise is the search for algorithms. Note that all problems can be solved using machine precision, except #2, for which it is probably impossible to avoid high precision.

1 Divide and Conquer

2 A Sea of Mirrors

3 A Singular Problem

4 How Low Can You Go?

5 Lost in Hyperspace

6 To Be Fair, It Must Be Biased

7 Inverting an Almost-Zero Matrix

8 It Gets Hot, but When?

9 Surprise

10 Think Outside the Box

Created by Mathematica (June 27, 2004)