February 2018 - Challenge

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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.

Update 15/2: You earn a '*' for finding the hidden message in one of the solutions.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

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Challenge: 30/01/2018 @ 12:00 PM EST
Solution: 02/03/2018 @ 12:00 PM EST
List Updated: 04/03/2018 @ 12:00 PM EST

People who answered correctly:

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)