IBM Research | Ponder This | July 2009 challenges

# Ponder This

## July 2009

<<June July August>>

Ponder This Challenge:

What is the probability that the last move in a backgammon (link resides outside of ibm.com) game will be a double? To make things easier, we assume that all the 15 checkers of each player are at their 1-point (just ready to bear-off); so each player, in its turn, removes (bear-off) two checkers for every non-double throw and four checkers for every double.

Hint: The solution is *not* the trivial 1/6.

07/01: Note that there are two players.

07/21: The game is played till its end, even if the result is already determined. For example, when a player has only one checker left, he still throws the dices and game is defined to end with a double if the dices come up equal.

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!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 06/29/2009 @ 3:00 PM EST
Solution: 07/31/2009 @ 3:00 PM EST
List Updated: 08/06/2009 @ 3:00 PM EST

John T. Robinson (06/30/2009 01:40 AM EDT)
Eric Farmer (06/30/2009 02:46 PM EDT)
Pataki Gergely (06/30/2009 03:29 PM EDT)
Michael Brand
(06/30/2009 10:59 PM EDT)
Ioan Marius Popdan (07/01/2009 06:27 AM EDT)
Henry Bottomley (07/01/2009 08:11 AM EDT)
Latchezar Christov (07/01/2009 09:33 AM EDT)
Serge Thill (07/01/2009 09:36 AM EDT)
James MacAdie (07/01/2009 09:45 AM EDT)
Vladimir V. Sedach (07/01/2009 01:06 PM EDT)
Martin Thiim
(07/01/2009 01:40 PM EDT)
Sevag Papazian (07/01/2009 02:01 PM EDT)
Mark Perkins (07/01/2009 02:29 PM EDT)
Alex Wagner (07/01/2009 04:46 PM EDT)
Beryl Schragger (07/01/2009 05:33 PM EDT)
Corey Cerovsek (07/01/2009 06:48 PM EDT)
Karl D'Souza (07/01/2009 07:24 PM EDT)
Reinder Verlinde (07/01/2009 08:01 PM EDT)
Andrew Holster (07/01/2009 09:41 PM EDT)
Dan Wood (07/01/2009 09:58 PM EDT)
Nyles Heise (07/02/2009 12:00 AM EDT)
Joe Fendel (07/02/2009 03:36 PM EDT)
Dan Dima (07/02/2009 05:06 AM EDT)
Bojan Basic (07/02/2009 05:13 AM EDT)
Satish Vedantam (07/02/2009 10:50 AM EDT)
Or Zuk (07/02/2009 11:51 AM EDT)
Mark Mixer (07/02/2009 01:04 PM EDT)
Jonathan Campbell (07/02/2009 02:08 PM EDT)
Jack Kennedy
(07/02/2009 07:03 PM EDT)
Reiner Martin (07/02/2009 07:24 PM EDT)
James Dow Allen (07/03/2009 02:54 AM EDT)
Jochen Hoenicke (07/03/2009 10:47 AM EDT)
Ionel Santa (07/03/2009 10:54 AM EDT)
Cosmin Munteanu (07/03/2009 01:15 PM EDT)
Balakrishnan V (07/03/2009 02:34 PM EDT)
Renato Fonseca
(07/05/2009 06:53 PM EDT)
Ryan Milligan (07/05/2009 11:55 PM EDT)
Jason L.
(07/06/2009 00:27 AM EDT)
Rickard Björklund (07/06/2009 08:23 AM EDT)
Razvan Gheorghita (07/06/2009 09:08 AM EDT)
Daniel Bitin (07/06/2009 01:13 PM EDT)
Kevin Williamson (07/06/2009 02:48 PM EDT)
(07/06/2009 08:14 PM EDT)
Jeff Blohm (07/07/2009 03:31 AM EDT)
Clive Tong (07/07/2009 04:32 AM EDT)
Greg Janee (07/09/2009 02:05 AM EDT)
John G Fletcher (07/09/2009 02:35 AM EDT)
Lukasz Bolikowski (07/09/2009 12:51 PM EDT)
Albert Stadler (07/10/2009 09:08 AM EDT)
Andrea Andenna
(07/11/2009 08:27 PM EDT)
Boris Nikolaus (07/12/2009 08:22 AM EDT)
Asad Usman (07/13/2009 09:54 AM EDT)
Tomek Czajka
(07/14/2009 03:51 PM EDT)
Jimmy Hu
(07/17/2009 03:46 PM EDT)
Robert Szafranski (07/19/2009 07:23 PM EDT)
Gale Greenlee (07/22/2009 03:33 PM EDT)
Satish Kumar K N (07/23/2009 04:23 PM EDT)
Antoine Comeau (07/23/2009 08:54 PM EDT)
David Friedman (07/24/2009 05:32 PM EDT)
Jeremy Cahill (07/29/2009 04:24 AM EDT)
Adriana Clerici (07/29/2009 10:08 AM EDT)
Ehud Schreiber (07/30/2009 03:51 PM EDT)
Andrew Rochon (07/31/2009 04:45 AM EDT)
Travis Craig & Jean-Pascal Harroch (07/31/2009 02:20 PM EDT)

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