MathWorld Headline News

Hundred-Dollar Challenge Winners Announced

By Eric W. Weisstein

May 25, 2002--The January/February 2002 issue of the magazine SIAM News contained an interesting challenge to readers. In it, numerical analyst Nick Trefethen proposed 10 computational problems, each of which has a single real number as its answer. The author offered a $100 award to the person or group that managed to calculate the greatest number of correct digits by May 20, 2002. Points were to be awarded on the basis of one point for each correct digit for a maximum of 10 correct points per problem.

The hundred-dollar, hundred-digit challenge problems range in subject from numerical integration to global minimization to the solution of random walks.

The winners of the competition were announced today on Trefethen's website. Ninety-four teams from 25 countries entered the competition. Of these, 20 teams scored a perfect 100 points, and five teams scored 99 points. Trefethen will randomly select three of the teams with perfect scores and present them with the prize of $100 each.

One of the teams achieving a perfect score was made up of Mathematica developers and collaborators. The members of "Team Mathematica" consisted of Paul Abbott of the University of Western Australia and Brett Champion, Yifan Hu, Daniel Lichtblau, and Michael Trott of Wolfram Research, Inc., who completely solved all problems using Mathematica. The following features of Mathematica were crucial and repeatedly used in the solution of the 10 problems: fast machine arithmetic, arbitrary-precision significance arithmetic, interval arithmetic, dense and sparse linear algebra, symbolic integration, sequence transformations and extrapolation, and high-precision special-function evaluations.

In fact, for most of the 10 problems, it is straightforward to calculate as many digits as desired using Mathematica's arbitrary-precision arithmetic.

The following table gives the descriptions and the numerical answers to all 10 problems.

#	Answer	Problem
1	0.3233674316	What is $\lim_{\epsilon\to 0} \int_\epsilon^1 x^{-1}\cos(x^{-1}\ln x)\,dx$ ?
2	0.9952629194	A photon moving at speed 1 in the - plane starts at at heading due east. Around every integer lattice point in the plane, a circular mirror of radius has been erected. How far from the origin is the photon at ?
3	1.274224152	The infinite matrix $\mathsf{A}$ with entries $a_{11} = 1$ , $a_{12} = 1/2$ , $a_{21} = 1/3$ , $a_{13} = 1/4$ , $a_{22} = 1/5$ , $a_{31} = 1/6$ , etc., is a bounded operator on $\ell^2$ . What is $\Vert\mathsf{A}\Vert$ ?
4	-3.306868647	What is the global minimum of the function
5	0.2143352345	Let $f(z)=1/\Gamma(z)$ , where $\Gamma(z)$ is the gamma function, and let be the cubic polynomial that best approximates on the unit disk in the supremum norm $\Vert\cdot\Vert _\infty$ . What is $\Vert f-p\Vert _\infty$ ?
6	0.06191395447	A flea starts at on the infinite 2D integer lattice and executes a biased random walk: At each step it hops north or south with probability 1/4, east with probability $1/4 + \epsilon$ , and west with probability $1/4 - \epsilon$ . The probability that the flea returns to sometime during its wanderings is 1/2. What is $\epsilon$ ?
7	0.7250783462	Let $\mathsf{A}$ be the $20{,}000\times 20{,}000$ matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, ..., 224737 along the main diagonal and the number 1 in all the positions $a_{ij}$ with $\vert i-j\vert = 1$ , 2, 4, 8, ..., 16384. What is the (1, 1) entry of $\mathsf{A}^{-1}$ ?
8	0.4240113870	A square plate $[-1, 1]\times [-1, 1]$ is at temperature . At time the temperature is increased to along one of the four sides while being held at u = 0 along the other three sides, and heat then flows into the plate according to $u_t = \Delta u$ . When does the temperature reach at the center of the plate?
9	0.7859336743	The integral $I(\alpha)=\int_0^2 [2 + \sin(10\alpha)]x^\alpha \sin(\alpha/(2- x))\,dx$ depends on the parameter $\alpha$ . What is the value $\alpha\in[0, 5]$ at which $I(\alpha)$ achieves its maximum?
10	0.3837587979 x 10^-6	A particle at the center of a $10\times 1$ rectangle undergoes Brownian motion (i.e., 2D random walk with infinitesimal step lengths) till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?