MathWorld Headline News

A Hundred-Dollar Challenge

By Eric W. Weisstein

February 4, 2002--The January/February 2002 issue of the magazine SIAM News contains an interesting challenge to readers. In it, Nick Trefethen proposes 10 computational problems, each of which has a single real number as its answer. The author has offered a $100 award to the person or group that manages to calculate the greatest number of correct digits by May 20, 2002. Points are 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 solution of random walks.

The complete list can be downloaded in PDF and is reproduced in MathWorld as well as below for convenience. Solutions should be sent to lnt@comlab.ox.ac.uk.

1. What is $\lim_{\epsilon\to 0} \int_\epsilon^1 x^{-1}\cos(x^{-1}\ln x)\,dx$ ?

2. A photon moving at speed 1 in the

plane starts 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. 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. What is the global minimum of the function

$\begin{displaymath}\mathop{\rm exp}\nolimits (\sin(50x)) + \sin(60e^y) + \sin(70... ...\sin(80y)) - \sin(10(x + y)) + {\textstyle{1\over 4}}(x^2+y^2)?\end{displaymath}$

5. 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. 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. 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. 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. 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. 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?

References

Trefethen, N. "A Hundred-Dollar, Hundred-Digit Challenge." http://www.siam.org/siamnews/01-02/challenge.pdf