Ponder This Challenge - May 2005 - Rectangular grid / glass cubes

Ponder This Challenge:

Puzzle for May 2005

This month's puzzle comes from Gale Greenlee.
Both parts need to be solved.

We have received 100 correct solutions for the May puzzle.
The list of correct solvers is now closed.
The solution will be posted soon.

Part 1:

Imagine a rectangular grid of compartments, of size 68 by 122. That's 8,296 compartments. Suppose we had 8,290 black marbles and were to place them in compartments, at most one marble per compartment, in such a way that each row and each column would have an even number of marbles. How many ways could this be done?

Part 2:

Imagine 6 little clear glass cubes with dimensions of 1x1x1. Imagine 90 more just like them except these 90 have a pretty little bright red sphere at their center; the spheres have a radius of 1/10. So, all together we have 96 little 1x1x1 cubes.

Now let's arrange these 96 into a block with dimensions of 4x4x6 so that in any column (up and down) or row (left to right) or (front to back) there is an even number of small cubes with the red centers. Zero is an even number.

How many ways would there be to do this?

The first 100 people who answer correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

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

Solution

• Click here to view the solution

  Answer:

  Part 1:

  6 * 68C3 * 122C3 = 6 * 50116 * 295240 = 88777487040.

  Select three of the 68 rows (in 68C3=50116 possible ways), and three of the 122 columns (in 122C3=295240 ways); in the resulting 3x3 pattern, there are six ways to arrange 6 empty compartments and 3 black marbles with one marble in each row and each column. All other compartments receive black marbles.

  Part 2:

  Each "occupied" plane (a plane with nonzero number of clear cubes) has to have at least four clear cubes. So there can't be two horizontal occupied planes (parallel to x-y plane). So some vertical line has an odd number of clear cubes, hence an odd number of red cubes.

  ---

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

Solvers

• The list of correct solvers is now closed. (None)
• Daniel Bitin (05.02.2005 @10:44:20 AM EDT)
• John Hart (05.02.2005 @11:38:33 AM EDT)
• Eugene Vasilchenko (05.02.2005 @11:38:37 AM EDT)
• Joseph DeVincentis (05.02.2005 @11:49:06 AM EDT)
• Mark Perkins (05.02.2005 @01:36:24 PM EDT)
• Lauri Ylinen (05.02.2005 @03:51:32 PM EDT)
• Joe BGI SF (05.02.2005 @04:03:03 PM EDT)
• Fendel (05.02.2005 @04:03:03 PM EDT)
• Aditya Prasad (05.02.2005 @06:12:59 PM EDT)
• Christopher Howe (05.02.2005 @10:33:48 PM EDT)
• Vishal Mamania (05.03.2005 @03:24:46 AM EDT)
• Guler Amos (05.03.2005 @04:30:27 AM EDT)
• Wolfgang Kais (05.03.2005 @05:13:12 AM EDT)
• Amit Chattopadhyay (05.03.2005 @05:50:02 AM EDT)
• Narendra Agrawal (05.03.2005 @06:06:41 AM EDT)
• Dan Dima (05.03.2005 @07:03:12 AM EDT)
• Benjamin Lee (05.03.2005 @11:02:53 AM EDT)
• Smith (05.03.2005 @11:02:53 AM EDT)
• George Chen (05.03.2005 @11:43:58 AM EDT)
• Jeff Shute (05.03.2005 @04:47:00 PM EDT)
• Bill Schwennicke (05.03.2005 @05:34:19 PM EDT)
• Vlad Gotlib (05.03.2005 @10:36:53 PM EDT)
• Gary M Gerken (05.04.2005 @12:09:59 AM EDT)
• Dharmadeep Muppalla (05.04.2005 @01:03:23 AM EDT)
• Chang-Seo Park (05.04.2005 @02:19:19 AM EDT)
• Kim Se Kwon (05.04.2005 @02:53:56 AM EDT)
• Robert Benea & Teresa Madrid (05.04.2005 @03:03:29 AM EDT)
• Patrick J. LoPresti (05.04.2005 @11:25:42 AM EDT)
• David McQuillan (05.04.2005 @11:54:54 AM EDT)
• Clive Tong (05.04.2005 @12:27:10 PM EDT)
• Bryan Knight (05.04.2005 @03:07:00 PM EDT)
• Cesar Sanchez (05.04.2005 @06:57:21 PM EDT)
• IQSTAR (05.04.2005 @11:18:25 PM EDT)
• Alvaro Martinez Echevarria (05.05.2005 @12:09:46 AM EDT)
• Sean (Fuxiang) Yu (05.05.2005 @09:17:37 AM EDT)
• Tarun Saxena (05.05.2005 @02:00:13 PM EDT)
• John G. Fletcher (05.06.2005 @02:09:58 AM EDT)
• Dejan Stakic (05.06.2005 @05:07:22 AM EDT)
• Amr Eid (05.06.2005 @07:44:48 AM EDT)
• Rethnakaran Pulikkoonattu (05.06.2005 @07:53:00 AM EDT)
• Vinoad Senguttuvan (05.06.2005 @05:06:19 PM EDT)
• John Dalbec (05.08.2005 @12:25:33 AM EDT)
• Chooky (05.08.2005 @12:27:51 AM EDT)
• Edna Mantsour (05.08.2005 @02:18:23 AM EDT)
• Dipak K Jha (05.09.2005 @01:05:00 AM EDT)
• Onno Waalewijn (05.09.2005 @03:43:00 AM EDT)
• Raicho Tchalkov (05.09.2005 @06:51:24 AM EDT)
• Alex Wice (05.09.2005 @09:57:41 AM EDT)
• John Tromp (05.09.2005 @10:06:00 AM EDT)
• Dinesh Layek (05.09.2005 @10:32:00 AM EDT)
• Alan Lemm (05.09.2005 @12:42:42 PM EDT)
• Karthik Kothandaraman (05.09.2005 @03:50:05 PM EDT)
• Kiran Bhogadi (05.09.2005 @08:55:00 PM EDT)
• Rodney Martis (05.10.2005 @12:56:06 AM EDT)
• Daniel Chong Jyh Tar (05.10.2005 @07:21:15 AM EDT)
• Serg Korensky (05.10.2005 @08:02:21 AM EDT)
• Abhijit Toley (05.10.2005 @08:38:56 AM EDT)
• Balakrishnan V (05.10.2005 @01:45:10 PM EDT)
• Jamie Broomall (05.10.2005 @03:33:30 PM EDT)
• Dick King (05.10.2005 @08:03:06 PM EDT)
• Eric Valpey (05.10.2005 @09:02:01 PM EDT)
• Frans Buijsen (05.11.2005 @05:30:59 AM EDT)
• Joe Vanore (05.11.2005 @12:23:25 PM EDT)
• Jayavel Sounderpandian (05.12.2005 @01:25:00 AM EDT)
• Udo Klein (05.12.2005 @04:11:30 AM EDT)
• Karl Dsouza (05.12.2005 @08:35:57 AM EDT)
• Hubert Pujol (05.12.2005 @09:42:50 AM EDT)
• Alan O'Donnell (05.12.2005 @11:51:19 AM EDT)
• Frank Vernaillen (05.12.2005 @09:57:46 PM EDT)
• Artur Jackson (05.13.2005 @08:35:41 AM EDT)
• David Friedman (05.13.2005 @05:31:13 PM EDT)
• Meghana Mande (05.14.2005 @12:43:33 AM EDT)
• Aaron Marcus (05.16.2005 @04:58:18 AM EDT)
• Arnab Bose (05.16.2005 @08:39:39 AM EDT)
• Lexing Xie (05.17.2005 @03:26:56 PM EDT)
• Reejis Stephen & Dion Harmon (05.17.2005 @04:17:14 PM EDT)
• Bill Webb (05.17.2005 @09:52:39 PM EDT)
• John Ritson (05.18.2005 @01:34:38 AM EDT)
• Vineet Dwivedi (05.18.2005 @05:34:41 AM EDT)
• Keith Campbell (05.18.2005 @10:24:18 AM EDT)
• Doodoong (05.18.2005 @02:23:19 PM EDT)
• Lilesh S Ghadi (05.19.2005 @02:10:19 AM EDT)
• Mayank Bathiya (05.19.2005 @03:21:55 AM EDT)
• Mark Pilloff (05.19.2005 @04:32:46 AM EDT)
• Mark Brader (05.19.2005 @04:49:41 AM EDT)
• Divyesh Dixit (05.19.2005 @06:07:48 AM EDT)
• Danny Kodicek (05.19.2005 @06:55:42 AM EDT)
• Matthew Charlap (05.19.2005 @09:30:42 AM EDT)
• Bryce Herdt (05.19.2005 @04:12:08 PM EDT)
• Damir Brasnic (05.19.2005 @07:52:11 PM EDT)
• Ted Schuerzinger (05.19.2005 @11:32:10 PM EDT)
• Sumana M A (05.20.2005 @08:21:56 AM EDT)
• Steven Noble (05.22.2005 @01:24:16 AM EDT)
• Mikael Johansson (05.22.2005 @05:24:57 PM EDT)
• Trivik (05.23.2005 @09:32:59 AM EDT)
• Peter Mattsson (05.23.2005 @11:40:35 AM EDT)
• Wei Liu (07:01:51 PM EDT)
• Zdenek Spacek (06:16:40 AM EDT)
• Hagen von Eitzen (07:05:00 AM EDT)
• Libin Shen (12:06:49 PM EDT)
• Gurpreet Singh Pahwa (11:42:25 AM EDT)
• Jeffrey Pickhardt (11:43:27 AM EDT)