Hundred-dollar, Hundred-digit Challenge problems

The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by Nick Trefethen (2002). A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. The deadline for the contest was May 20, 2002. In the end, 20 teams solved all of the problems perfectly within the required precision, and an anonymous donor aided in producing the required prize monies. The challenge and its solutions were described in detail in the book (Folkmar Bornemann, Dirk Laurie & Stan Wagon et al. 2004).

The problems

edit

From (Trefethen 2002):

  1.  
  2. A photon moving at speed 1 in the xy-plane starts at t = 0 at (x, y) = (0.5, 0.1) 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?
  3. The infinite matrix A with entries   is a bounded operator on  . What is  ?
  4. What is the global minimum of the function  
  5. Let  , where   is the gamma function, and let   be the cubic polynomial that best approximates   on the unit disk in the supremum norm  . What is  ?
  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  , east with probability  , and west with probability  . The probability that the flea returns to (0, 0) sometime during its wanderings is  . What is  ?
  7. Let A be the 20000×20000 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   with  . What is the (1, 1) entry of  ?
  8. A square plate   is at temperature  . At time  , the temperature is increased to   along one of the four sides while being held at   along the other three sides, and heat then flows into the plate according to  . When does the temperature reach   at the center of the plate?
  9. The integral   depends on the parameter α. What is the value of α in [0, 5] at which I(α) achieves its maximum?
  10. A particle at the center of a 10×1 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?

Solutions

edit
  1. 0.3233674316
  2. 0.9952629194
  3. 1.274224152
  4. −3.306868647
  5. 0.2143352345
  6. 0.06191395447
  7. 0.7250783462
  8. 0.4240113870
  9. 0.7859336743
  10. 3.837587979 × 10−7

These answers have been assigned the identifiers OEISA117231, OEISA117232, OEISA117233, OEISA117234, OEISA117235, OEISA117236, OEISA117237, OEISA117238, OEISA117239, and OEISA117240 in the On-Line Encyclopedia of Integer Sequences.

References

edit
  • Bailey, D. H.; Borwein, J. M. (2003-09-22). "Sample Problems of Experimental Mathematics" (PDF).
  • Bornemann, F. (2002-11-05). "Short Remarks on the Solution of Trefethen's Hundred-Digit Challenge" (PDF).
  • Bornemann, Folkmar; Laurie, Dirk; Wagon, Stan; Waldvogel, Jörg (2004). The SIAM 100-digit challenge: A study in high-accuracy numerical computing. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). ISBN 978-0-89871-561-3. MR 2076374. Review (June 2005) from Bulletin of the American Mathematical Society.
  • Leslie, M. (Ed.) (2002). "NetWatch: Decimal Decathlon". Science. 295 (5559): 1431d–1431. doi:10.1126/science.295.5559.1431d.
  • Trefethen, Nick (2002). "A Hundred-dollar, Hundred-digit Challenge" (PDF). SIAM News. 35 (1): 65.
  • Weisstein, Eric W. "Hundred-Dollar, Hundred-Digit Challenge Problems". MathWorld.