The 100-Digit Challenge Problems

1. Evaluate .

2. A photon starts at (1/2, 1/10) and travels east with speed 1. If there are reflecting circles of radius 1/3 centered at each integer lattice point, how far is the photon from the origin at time 10?

3. What is the norm of the following infinite matrix viewed as an operator on , the space of infinite square-summable sequences:

?

4. What is the global minimum of

?

5. If p(z) is the cubic polynomial that best approximates 1/Gamma[z] on the unit disk in the sup-norm (i.e., worst case), what is the error in this approximation?

6. An unbiased random walk on the integer lattice returns to the origin with probability 1. For which ε is this probability 1/2, where *ε* biases the east-west portion of the walk (i.e., , , 1/4, and 1/4 are the east, west, north, and south probabilities)?

7. Let A be the 20000*×*20000 matrix having the primes (2, 3, 5, …, 224737) along the main diagonal, and with whenever |i-j| is a power of 2, and 0 elsewhere. What is the upper left entry of the inverse of A?

8. When is the temperature at the center of a *2**×**2* square equal to 1, assuming the starting temperature is 0 everywhere except on one side, where it is 5, and the temperatures on the sides do not change? Use where u(x, y, t) is the temperature at time t.

9. Which α in [ 0, 5] maximizes ?

10. What is the probability that an infinitely small Brownian motion particle in the plane that starts at the center of a 10*×**1* rectangle hits one of the short sides before one of the long sides? "Brownian motion" means that the particle follows a 2-dimensional random walk with infinitesimal step length until it hits the boundary of the rectangle.