Ponder This Challenge - November 2008 - Ponder^this = ponder...

Ponder This Challenge:

Ponder-this corner reached its 127th month, which is the maximal number that fits into signed byte. To celebrate this, we will praise you--our devoted solvers--share some of your flattering emails, and, of course, give you a special ponder-this question.

First, a salute to your persistence. Here's an example of one devoted follower who sent in solutions from the Intensive Care Unit:

"Let me just take this time to thank you and IBM for this wonderful web site. It has provided me with many hours of puzzle solving pleasure, and also some interesting conversations with my family.

The first three weeks of June this year I spent in the hospital. One of the nights in the ICU my family and I passed some of the time by talking about Ponder This. My daughter read the DNA problem to me at that time, and jokingly I told her I could be the first person to ever solve and answer a Ponder This while in the ICU. I actually sent my entry from my hospital bed about a week later.

Again thanks. Looking forward to next month's puzzle."

Many thanks to all of those who sent in great original and innovative riddles.

Of course it is heartwarming to get compliments like:

*"PS.: Thank you 10^10 times for maintaining this extremely enlightening challenge. Each time I participate, I discover a new world.

That's just great! ;-)"*

Lastly, we are proud to tell you about a scientific paper that one of our solvers published a year ago in the prestigious Physics Review Letter journal. The paper was inspired by April 2007's ponder-this puzzle.

ANTENEODO, C.; MORGADO, W.A.M. Critical scaling in standard biased random walks. Physical Review Letters 99(18): 180602-1-4 (2007):

"We acknowledge Brazilian agencies CNPq and Faperj for partial financial support and IBM research "Ponder This" for having drawn attention to this model."

November's puzzle is about replacing digits (0-9) with letters (ponderthis) where each letter represents a different digit. For example, the only solution for the problem

is

Furthermore, in this case, ponder=628053 and this=1794.

Now for our puzzle. If

$\text{ponder}^\text{this} = \text{ponder}\ldots$

I.e., the result of raising the 6-digits number "ponder" to the power of the 4-digit number "this" starts with the six most significant digits of "ponder", then what is the value (translated back to letters) of the first four digits of the following expression?

Note the decimal points: .dr = 0.dr = (10*d+r)/100; and the absolute value signs (|).

Note that "ponder" and "this" should be six and four digits numbers respectively; i.e. "p" and "t" should not be zero.

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

  Instead of computing the thousands of digits of ponderthis, we can use the following inequality to quickly verify that the first six digits are, indeed, “ponder” (frac stands for the fractional part):

  frac(Log10(ponder)) ≤ frac(this*Log10 (ponder)) < frac(Log10 (ponder+1))

  Out of the all permutations, only three match the condition:

  ponder=504271 this=6983
  ponder=813760 this=2459
  ponder=835402 this=7619

  Here comes the interesting part – although the expression gives three different results, in all three cases, the first four digits, translated back to letters, are “oded”.

Solvers

• Sangxia Huang (11/01/2008 04:31 AM EDT)
• Dan Dima (11/01/2008 07:15 PM EDT)
• Michel Speiser (11/01/2008 07:36 PM EDT)
• Gary M. Gerken (11/01/2008 08:48 PM EDT)
• Harold Gutch (11/01/2008 09:44 PM EDT)
• Javier Rodríguez (11/01/2008 11:07 PM EDT)
• John T. Robinson (11/02/2008 01:08 AM EDT)
• Nyles Heise (11/02/2008 04:10 AM EDT)
• Guler Amos (11/02/2008 08:23 AM EDT)
• Bojan Basic & Novi Sad (11/02/2008 08:30 AM EDT)
• Matthew Bennett (11/02/2008 12:41 AM EDT)
• Balakrishnan V (11/02/2008 04:33 PM EDT)
• Vladimir Sedach (11/02/2008 05:27 PM EDT)
• Roee Shenberg (11/03/2008 07:32 AM EDT)
• Albert Stadler (11/03/2008 08:36 AM EDT)
• Andreas Eisele (11/03/2008 10:41 AM EDT)
• Philippe Fondanaiche (11/03/2008 11:25 AM EDT)
• Eugene Vasilchenko (11/03/2008 11:55 AM EDT)
• Michael van Fondern (11/03/2008 01:59 PM EDT)
• Rickard Björklund (11/03/2008 03:45 PM EDT)
• Yury X Volvovskiy (11/03/2008 12:09 PM EDT)
• Tom Sirgedas (11/03/2008 09:54 PM EDT)
• Henry Bottomley (11/03/2008 10:06 PM EDT)
• Are Hjørungnes (11/04/2008 06:56 AM EDT)
• Frank Vernaillen (11/04/2008 08:27 AM EDT)
• Lukasz Bolikowski (11/04/2008 08:54 PM EDT)
• Balacescu Victor (11/04/2008 07:54 PM EDT)
• Martin Giese (11/04/2008 09:13 AM EDT)
• Mark Perkins (11/04/2008 09:54 PM EDT)
• James Dow Allen (11/05/2008 02:14 AM EDT)
• Martin Thiim (11/05/2008 05:52 AM EDT)
• Ken Crounse (11/05/2008 05:28 PM EDT)
• Ashish Mishra (11/05/2008 08:15 PM EDT)
• C A Subramanian (11/06/2008 07:25 AM EDT)
• Farid Alberto Lian Martinez (11/06/2008 09:29 AM EDT)
• Steffen Wolf (11/06/2008 09:53 AM EDT)
• Sasha Ivrii (11/06/2008 10:34 AM EDT)
• Arthur Breitman (11/06/2008 10:47 AM EDT)
• Harald Bögeholz (11/06/2008 12:50 PM EDT)
• Arno Candel (11/06/2008 01:41 PM EDT)
• Michael Liepelt (11/06/2008 04:17 PM EDT)
• Nick Miller (11/06/2008 05:57 PM EDT)
• K j Slag (11/07/2008 02:08 AM EDT)
• D Adrian Tanner (11/07/2008 01:22 PM EDT)
• Daniel Bitin (11/07/2008 02:08 PM EDT)
• Alex Wagner (11/07/2008 05:19 PM EDT)
• Dario Serino (11/07/2008 12:16 PM EDT)
• Lawrence Hon (11/08/2008 03:37 PM EDT)
• Jesse Kolman (11/09/2008 02:36 AM EDT)
• Fred Schalekamp (11/09/2008 06:19 PM EDT)
• Sylvain Becker (11/10/2008 07:58 AM EDT)
• Andreas Stiller (11/10/2008 10:03 PM EDT)
• Uri Yanover (11/10/2008 04:29 PM EDT)
• Rohit Babbar and Rahul Babbar (11/11/2008 11:43 AM EDT)
• Donald T. Dodson (11/11/2008 01:47 PM EDT)
• Graeme McRae (11/11/2008 02:36 PM EDT)
• Andrea Andenna (11/11/2008 04:20 PM EDT)
• Ben Weir (11/11/2008 06:49 PM EDT)
• Boris Nikolaus (11/12/2008 08:53 PM EDT)
• Victor Chang (11/13/2008 12:30 AM EDT)
• Mario Loriedo (11/13/2008 04:00 PM EDT)
• Shan Sha Lin (11/13/2008 08:38 PM EDT)
• Heikki Leskinen (11/14/2008 04:08 PM EDT)
• James Riordan (11/14/2008 04:14 PM EDT)
• Li Han (11/14/2008 05:11 PM EDT)
• Stéphane Higueret (11/15/2008 11:05 AM EDT)
• Clerici Adriana (11/15/2008 02:09 PM EDT)
• Nagy Zoltan (kirk) (11/16/2008 01:22 AM EDT)
• Fabio Filatrella (11/18/2008 02:38 PM EDT)
• Aviv Nisgav (11/19/2008 04:39 PM EDT)
• Ioan Marius Popdan (11/20/2008 06:43 AM EDT)
• Robert Benea (11/20/2008 03:19 PM EDT)
• Álvaro Martínez Echevarría (11/20/2008 06:04 PM EDT)
• Alan O'Donnell (11/21/2008 01:07 PM EDT)
• John G. Fletcher (11/21/2008 04:34 PM EDT)
• Stephan Keil (11/21/2008 08:23 PM EDT)
• Luis M. Lacoma (11/23/2008 10:13 AM EDT)
• William Webb (11/23/2008 02:30 PM EDT)
• Clive Tong (11/24/2008 06:10 AM EDT)
• Alexandru Nicau (11/25/2008 02:47 PM EDT)
• Reiner Martin (11/25/2008 07:37 PM EDT)
• Adam Daire (11/25/2008 08:33 PM EDT)
• Botond Vasarhelyi (11/26/2008 10:11 AM EDT)
• John Tromp (11/26/2008 03:05 PM EDT)
• Stephen Merriman (11/27/2008 05:20 AM EDT)
• Varun Ahluwalia (11/27/2008 09:14 AM EDT)
• Jiri Navratil (11/29/2008 02:47 AM EDT)
• Michael Wegener (11/29/2008 02:25 PM EDT)
• Michael Schaaf (11/29/2008 03:17 PM EDT)
• Surya Kareenahalli (12/01/2008 04:58 AM EDT)