The Jan–Feb issue of SIAM News contained the following challenge from L. N. Trefethen of Oxford University (http://www.siam.org/siamnews/01-02/challenge.pdf has full text), with a deadline of May 20, 2002.

Each October, a few new graduate students arrive in Oxford to begin research for a doctorate in numerical analysis. In their first term, working in pairs, they take an informal course called the "Problem Solving Squad." Each week for six weeks, I give them a problem, stated in a sentence or two, whose answer is a single real number. Their mission is to compute that number to as many digits of precision as they can. Ten of these problems appear in this issue. I would like to offer them as a challenge to the SIAM community. Can you solve them?

Scoring will be simple: You get a point for each correct digit, up to ten for each problem, so the maximum score is 100 points.

Hint: They're hard!

— Nick Trefethen, Oxford University
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I worked hard on these, since the problems were very attractive to me, and recruited a colleague, Danny Kaplan, to help. We got all ten, each to at least 14 digits. For some it was easy to get 500 digits; for others, it was difficult for us to get even 10 digits. If you have a better solution to any of these, do tell. In any case, we learned a lot working on these, as some very interesting ideas, numerical, symbolic, and algorithmic, show up in the solutions.

Results

There were 94 entries from 25 countries, 20 of which achieved perfect scores of 100 points. Five teams got 99 points. Our Macalester College team (Danny Kaplan and Stan Wagon) was one of the 20. Full results are at http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/. Winners with web pages on the problems are:

Three of the winners and I have written a book for SIAM that contains detailed information on the solution of all these problems, and on the followup challenge raised by Trefethen: compute the values to 10,000 digits. We have succeeded in getting 10000 digits for nine of the ten problems. The book also contains an Appendix with 22 additional problems in the style of the original 10. For more information on the book see: http://www.siam.org/books/100digitchallenge/

The 100-Digit Challenge Problems

Brief Solutions

Created by Mathematica (June 27, 2004)