Ponder This Challenge - February 2018 - 3 people throw dice

This month's challenge was inspired by a suggestion from Hamidreza Bidar (thanks!).

Three people play a game. Each one of them has an independent fair die with n_i faces (n_1 faces for the first person, n_2 for the second, and n_3 for the third).
They play a game where each one of them tosses his die once every minute until he gets all the possible outcomes. They all play simultaneously and the game ends when all the players have finished the game.
Find possible n_i such that the expected time till the end of the game, rounded to the nearest second, is 2569.

Solution

• Click here to view the solution

  This month we chose to present Robert Lang's solution (thanks!).

  The challenge is a variant of the coupon collector's problem. Its classical form gives  for n1 = 1, n2 = 1, n3 = 14 an expected time of  (1 +1/2 + 1/3 + … + 1/14) * 14 * 60 =  2731.312… Therefore 13 is an upper bound for n1, n2, n3.
  If we find a formula for the expected duration, the rest can be done by applying the formula a couple of times.

  Let Xn1, Xn2, Xn3 denote the random waiting time until players 1, 2, 3 are done.
  The distribution of these random variables can be found as

  where  are the Stirling numbers of the second kind.

  Now we compute the expectation for the game under consideration:

  We can even avoid the infinite series by exchanging the summations over j and k:

  The results are in minutes and must be multiplied by 60.
  The closest result is 4 – 9 – 13, which gives an expected duration of about 2568.7246.
  The '*' is for finding IBM in the solution 9 – 2 – 13.

Solvers

• Uoti Urpala (02/02/18 05:39 AM IDT)
• *Tyler Mullen (02/02/18 06:38 AM IDT)
• Evgenii Zheltonozhskii (02/02/18 10:03 AM IDT)
• Reiner Martin (02/02/18 12:33 PM IDT)
• *Joseph DeVincentis (02/02/18 12:51 PM IDT)
• Guillaume Escamocher (02/02/18 02:08 PM IDT)
• Eden Saig (04/02/18 11:10 PM IDT)
• Shirish Chinchalkar (04/02/18 11:40 PM IDT)
• JJ Rabeyrin (05/02/18 12:14 AM IDT)
• Gary M. Gerken (05/02/18 12:34 AM IDT)
• *Balakrishnan Varadarajan (05/02/18 03:33 AM IDT)
• *Alper Halbutogullari (05/02/18 04:48 AM IDT)
• Lorenz Reichel (05/02/18 07:58 AM IDT)
• *Sean Egan (05/02/18 10:23 AM IDT)
• *Ritesh Ranjan (05/02/18 10:27 AM IDT)
• Guangzhong Sun (05/02/18 02:08 PM IDT)
• *Rudy Cortembert (05/02/18 02:31 PM IDT)
• *Mathias Schenker (05/02/18 04:04 PM IDT)
• Christopher Marks (05/02/18 04:18 PM IDT)
• *Rob Pratt (05/02/18 05:22 PM IDT)
• *Oleg Vlasii (05/02/18 07:32 PM IDT)
• *David Friedman (05/02/18 10:25 PM IDT)
• Guy Hollington (05/02/18 11:03 PM IDT)
• Steven Langerwerf (05/02/18 11:16 PM IDT)
• *Hugo Pfoertner (06/02/18 01:16 AM IDT)
• Liubing Yu (06/02/18 01:41 AM IDT)
• Joaquim Neves Carrapa (06/02/18 02:05 IDT)
• Kang Jin Cho (06/02/18 06:22 AM IDT)
• *Li Li (06/02/18 06:38 AM IDT)
• *David Greer (06/02/18 11:26 AM IDT)
• Graham Hemsley (06/02/18 06:26 PM IDT)
• *Vivek Karunakaran (07/02/18 06:37 AM IDT)
• Tom Sirgedas (07/02/18 12:20 AM IDT)
• Ken Dsouza (07/02/18 06:26 PM IDT)
• Madiyar Aitbayev (07/02/18 07:46 PM IDT)
• Yan-Wu He (08/02/18 07:06 AM IDT)
• *Andreas Stiller (09/02/18 02:08 AM IDT)
• Rickard Björklund (09/02/18 04:20 AM IDT)
• *Muralidhar Seshadri (09/02/18 05:18 AM IDT)
• *Kang Jin Cho (09/02/18 08:12 AM IDT)
• Jonas Vistrand (09/02/18 11:41 AM IDT)
• Gunilla Bergendahl (09/02/18 11:46 IDT)
• *Benjamin Phillabaum (09/02/18 02:36 PM IDT)
• Dieter Beckerle (09/02/18 09:24 PM IDT)
• Ian Sleightholme (09/02/18 11:59 PM IDT)
• Azamat Berdyshev (10/02/18 06:54 AM IDT)
• Taylor Firman (11/02/18 01:43 AM IDT)
• *Atul Gupta (11/02/18 02:05 AM IDT)
• Henio Grimouth (11/02/18 01:23 PM IDT)
• Dan Dima (11/02/18 05:39 PM IDT)
• John Snyder (12/02/18 06:39 PM IDT)
• Yash Pote (13/02/18 11:48 AM IDT)
• Jordan Venderley (14/02/18 02:47 AM IDT)
• Clive Tong (14/02/18 10:17 AM IDT)
• Shouky Dan & Tamir Ganor (14/02/18 04:21 PM IDT)
• *Roger Duthie (14/02/18 11:13 PM IDT)
• *Shouky Dan & Tamir Ganor (15/02/18 01:58 PM IDT)
• *Dan Dima (15/02/18 06:20 PM IDT)
• Peter Gerritson (16/02/18 02:20 AM IDT)
• *Lakshman inkulu (16/02/18 03:35 AM IDT)
• *Luke Pebody (16/02/18 09:01 AM IDT)
• *Thomas Egense (16/02/18 09:06 AM IDT)
• Rogerio Ponce da Silva (16/02/18 08:11 PM IDT)
• *Daniëlle van den Berg (17/02/18 02:00 PM IDT)
• *Bhilahari Jeevanesan (17/02/18 19:50 PM IDT)
• *Dieter Beckerle (17/02/18 08:56 PM IDT)
• *Soumitra Agarwal (18/02/18 01:00 AM IDT)
• Todd Will (18/02/18 12:29 PM IDT)
• Dan Salajan (18/02/18 10:55 PM IDT)
• *Bharat Kumar Aryasomayajula (19/02/18 11:56 AM IDT)
• *Robert Lang (19/02/18 04:12 PM IDT)
• Cynthia Beauchemin (19/02/18 09:58 PM IDT)
• Toke Eskildsen (19/02/18 11:42 PM IDT)
• *Feyzullah Egriboyun (20/02/18 01:52 AM IDT)
• *Phil Benamy (20/02/18 07:20 AM IDT)
• Gidi Navrotsky (21/02/18 02:26 AM IDT)
• Chris Shannon (21/02/18 09:35 AM IDT)
• *Karim Alaoui (22/02/18 10:32 AM IDT)
• *Erik Wünstel (23/02/18 03:48 PM IDT)
• *Hizami Anuar (24/02/18 06:03 PM IDT)
• *Motty Porat (26/02/18 04:16 AM IDT)
• Oscar Volpatti (26/02/18 04:22 PM IDT)
• *Bert Dobbelaere (26/02/18 07:08 PM IDT)
• *Nyles Heise (27/02/18 09:06 AM IDT)
• Claudio Hüni (27/02/18 08:50 PM IDT)
• *Gumpina Rama Krishna Chaitanya (28/02/18 04:57 PM IDT)
• Radu-Alexandru Todor (01/03/18 01:50 AM IDT)
• *Sai KISHORE Datti (01/03/18 05:21 AM IDT)
• *Sasi teja alluri (01/03/18 05:25 AM IDT)
• Bora Naveen (01/03/18 12:23 PM IDT)
• *Kamil Jarosz (03/03/18 10:06 AM IDT)