Solutions to the SIAM 100-Dollar, 100-Digit Challenge

This file was exported from a Maple worksheet.

Team: Gaston Gonnet, Informatik, ETH, Zurich, and Robert Israel, Mathematics, UBC.
Most of these solutions are the ones that actually produced the results we sent in, but I (R.I.) have changed a few of them. All these solutions use Maple.

Each problem had an answer that was a single real number, of which 10 correct digits were required.

The complete results are at Our team was one of 20 first prize winners. See our prize.

Problem 1:

What is limit(int(x^(-1)*cos(x^(-1)*log(x)),x = epsilon .. 1),epsilon = 0)  ?


Problem 2:

A photon moving at speed 1 in the x-y  plane starts at t = 0  at (x, y) = (.5, .1)  heading due east.  Around every integer lattice point ``(i,j)  in the plane, a circular mirror of radius 1/3  has been erected.  How far from the origin is the photon at t = 10 ?


Problem 3:

The infinite matrix A  with entries a[1,1] = 1 , a[1,2] = 1/2 , a[2,1] = 1/3 , a[1,3] = 1/4 , a[2,2] = 1/5 , a[3,1] = 1/6 , etc., is a bounded operator on l^2 .  What is || A ||?.


Problem 4:

What is the global minimum of the function

exp(sin(50*x))+sin(60*exp(y))+sin(70*sin(x))+sin(sin(80*y))-sin(10*(x+y))+(x^2+y^2)/4 ?


Problem 5:

Let f(z) = 1/GAMMA(z) , where GAMMA(z)  is the gamma function, and let p(z)  be the cubic polynomial that best approximates f(z)  on the unit disk in the supremum norm abs(abs(`.`))[infinity] .  What is abs(abs(f-p))[infinity] ?


Problem 6:

A flea starts at (0,0) 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 (0,0) sometime during its wanderings is 1/2.  What is epsilon ?


Problem 7:

Let A  be the 20,000 x 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[i,j]  with abs(i-j) = 1 ,2,4,8,...,16384.  What is the (1,1) entry of A^(-1) ?


Problem 8:

A square plate [-1, 1]  x [-1, 1]  is at temperature u = 0 .  At time t = 0  the temperature is increased to u = 5  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 u = 1  at the center of the plate?


Problem 9:

The integral I(alpha) = int((2+sin(10*alpha))*x^alpha*sin(alpha/(2-x)),x = 0 .. 2)  depends on the parameter alpha .  What is the value alpha*epsilon*[0, 5]  at which I(alpha)  achieves its maximum?


Problem 10:

A particle at the center of a 10 x 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?