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# Ponder This

## July 2004

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Ponder This Challenge:

Puzzle for July 2004.

This month's puzzle comes from Aditya K Prasad, who heard it from a friend. It is similar to our November 2002 puzzle (from Martin Gardner's February 1979 Scientific American column), but with an important difference. To be considered for publication, please supply answers for BOTH parts of the puzzle.

Part 1:
There is a round table divided into 4 equal quadrants, with one cup in each quadrant. The quadrants are labeled with letters (A, B, C, D) that do not move. Initially, each cup is randomly face-up or face-down. You are blindfolded and put in front of the table. On each turn of the game, you instruct a genie to flip the cups in whichever positions you choose (e.g., you may say "flip the cups in A and B"), possibly choosing no cups or possibly all four. The genie complies. At this point, if all the cups on the table are face-up, the genie will tell you that you have won the game and are free to go. If not, he rotates the cups randomly (possibly not rotating them) and you play another turn. Give a strategy to win this game in a finite number of moves (the solution is not unique).

Remark: the outcome of "rotation" the four cups is one of the four possible positions: the cups originally at (A,B,C,D) can be at (A,B,C,D), (B,C,D,A), (C,D,A,B), or (D,A,B,C). It is not an arbitrary permutation.

Notice that you cannot examine the current orientation of any cup at any time. This contrasts with the earlier puzzle.

Part 2:
Suppose that instead of 4, there are n divisions and cups. For which n is it possible to guarantee a win? Prove your answer is correct.

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: 07/01/2004 @ 9:00 AM ET
Solution: 08/02/2004 @ 8:30 AM ET
List Updated: 07/01/2004 @ 9:00 AM ET

### People who answered correctly:

Luke Pebody (7.06.2004 @01:16:42 PM EDT)
Joe BGI SF Fendel (7.07.2004 @12:12:23 PM EDT)
Jonathan Derryberry (7.07.2004 @01:24:17 PM EDT)
Hagen von Eitzen (7.08.2004 @04:01:52 AM EDT)
Evgeniy Tarassov (7.08.2004 @09:51:41 AM EDT)
A.F.Jarvis (7.08.2004 @11:11:10 AM EDT)
Nilesh Dalvi (7.09.2004 @02:26:17 AM EDT)
James Dow Allen (7.09.2004 @04:10:55 AM EDT)
Bill Roscoe (7.09.2004 @08:39:17 AM EDT)
Atif Hussain (7.09.2004 @10:44:17 AM EDT)
Patrick J. LoPresti (7.09.2004 @12:26:25 PM EDT)
Shiva P Kasiviswanthan (7.09.2004 @02:45:44 PM EDT)
Mark J. Tilford (7.11.2004 @03:46:14 PM EDT)
Michael Brand (7.12.2004 @04:45:38 AM EDT)
David McQuillan (7.12.2004 @01:48:09 PM EDT)
John G. Fletcher (7.13.2004 @09:34:06 PM EDT)
Daniel Bitin (7.14.2004 @04:26:42 AM EDT)
Rocke Verser (7.14.2004 @06:56:29 AM EDT)
Mark Sandler (7.14.2004 @04:48:21 PM EDT)
Clive Tong (7.15.2004 @03:57:34 PM EDT)
David Woodruff (7.16.2004 @06:29:37 AM EDT)
Jian Li (7.17.2004 @11:32:20 AM EDT)
Bart De Vylder (7.19.2004 @08:49:02 AM EDT)
Peter Huggins (7.20.2004 @02:31:19 PM EDT)
Colin Bell (7.20.2004 @04:55:55 PM EDT)
Derek Kisman (7.21.2004 @04:52:27 AM EDT)
Agricul Turemistake (7.22.2004 @01:00:34 AM EDT)
Alex and Igor Naverniouk (7.22.2004 @03:21:34 AM EDT)
Tomas G. Rokicki (7.22.2004 @04:38:35 AM EDT)
Hao Wu (7.23.2004 @11:55:02 AM EDT)
Wolfgang Kais (7.25.2004 @10:36:48 AM EDT)
Dharma Deep (7.25.2004 @10:56:32 AM EDT)
Mayank M Kabra (7.27.2004 @12:14:37 AM EDT)
Yoo Je-Sung (7.27.2004 @07:27:28 AM EDT)
Amit Sinha (7.28.2004 @03:53:00 PM EDT)
Dan Dima (7.29.2004 @09:19:36 AM EDT)
Behdad and Behnam Esfahbod (7.29.2004 @07:37:24 PM EDT)
Coserea Gheorghe (7.31.2004 @11:16:35 AM EDT)

Attention: If your name is posted here and you wish it removed please send email to the ponder@il.ibm.com.