Puzzle
4 minute read

Ponder This Challenge - January 2011 - Letters and beans game

Ponder This Challenge:

Teacher: Let me teach you the order of letters.
Student: That's boring.
Teacher: So we'll do it using a game.
Student: That's better.
Teacher: You see the pile of 45 beans on the table?
Student: Yes.
Teacher: Choose a letter from the phrase "IBM RULES" and take beans while counting from 'A' to the letter you've selected.
Student: OK. I choose 'B', so counting 'A', 'B', I'll take two beans.
Teacher: Right! Now it's my turn, and I choose 'L' Counting 'A', 'B',...,'L', I'll take 12 beans.
Student: Got it. Now it's my turn. I also choose 'L', taking 12 beans.
Teacher: Now it's my turn again. I choose 'R' and counting 'A','B',...,'R', I'll take 18 beans.
Student: Aha! Now it's my turn, but I can't play since there's only one bean left.
Teacher: So I win.
Student: OK. Let's play again, but this time we'll start with 71 beans.
Teacher: Fine. You start.
Student: If I play 'B', I can guarantee winning.
Teacher: Good work! You are learning fast. Let's play with another phrase: "XXXXXX XXXX".
Student: OK. Write the number 204193/178481 + 2**-40 at base 8 (octal) and I'd like you to start if the N-th digit after the octal point is 1, otherwise I'll start. For example, the 19th octal digit is 7, so I'll start if N=19.
Teacher: Wow, that was fast - you solved the game.

Find a possible crossed-out phrase ("XXXXXX XXXX" above) that would make the above dialogue true.

To clarify: "**" above means exponentiation; so 2**-40 is one over 2 to the 40th power.

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

  • The rational fraction 204193/178481in base-8 is 0.(11160440223411004470220), where the 0.(xyz) notation means 0.xyzxyzxyzxyz...

    The 2**-40 changes the 14th digit, so the winning starting numbers are {1,2,3,13} and {1,2,3,13,14}+23*k for k>0.
    From this, we get 13 letters that must not appear (A, B, C, J, K, L, M, U, V, W, X, Y, Z); four that must appear (D, I, N, T); and several constraints on the rest.
    We can build several two-word phrases with six and four letters in the words, respectively, that satisfy all these conditions, such as "Hidden Tree", "Things Done", "Prison Diet", and even possible names for the teacher and student ("Dennis Roth" and "Ginger Todd" - thanks Chuck Carroll for this suggestion). The phrase that we chose, however, as almost all of you guessed, is simply "Ponder This".

    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

  • Puzzlet Chung (01/01/2011 08:16 AM EDT)
  • Dmitry Sotnikov (01/02/2011 08:00 AM EDT)
  • John T. Robinson (01/03/2011 12:26 AM EDT)
  • Tom Sirgedas (01/03/2011 10:48 AM EDT)
  • Nyles Heise (01/04/2011 02:04 AM EDT)
  • Dan Dima (01/04/2011 12:27 PM EDT)
  • Luke Pebody (01/04/2011 03:54 PM EDT)
  • Austin Shapiro (01/04/2011 04:58 PM EDT)
  • Clive Tong (01/04/2011 07:06 PM EDT)
  • Franza Cavalcante Jr (01/04/2011 07:21 PM EDT)
  • Daniel Bitin (01/04/2011 07:23 PM EDT)
  • Letao Wang (01/04/2011 11:54 PM EDT)
  • Prateek Agarwal1 (01/05/2011 02:44 AM EDT)
  • Adam Daire (01/03/2011 04:29 AM EDT)
  • Joseph DeVincentis (01/05/2011 11:52 AM EDT)
  • Nick McGrath (01/05/2011 11:58 AM EDT)
  • Piet Orye (01/05/2011 02:52 PM EDT)
  • Chuck Carroll (01/05/2011 04:15 PM EDT)
  • John Duffield (01/06/2011 12:50 PM EDT)
  • Adam M. Mosquera (01/06/2011 02:46 PM EDT)
  • Mark Mixer (01/07/2011 06:40 AM EDT)
  • Mark Mammel (01/07/2011 04:58 PM EDT)
  • Ashutosh Mahajan (01/07/2011 05:57 PM EDT)
  • John Tromp (01/07/2011 07:31 PM EDT)
  • Álvaro Begué (01/07/2011 07:33 PM EDT)
  • Dmitry Litvinenko (01/08/2011 11:06 AM EDT)
  • Thijmen Krebs (01/08/2011 11:14 AM EDT)
  • Dejan Stakic (01/09/2011 11:49 AM EDT)
  • Kerry M. Soileau (01/11/2011 07:58 PM EDT)
  • Daniel Chong Jyh Tar (01/12/2011 05:36 AM EDT)
  • Tamir Ganor (01/12/2011 10:16 AM EDT)
  • Mathias Schenker (01/12/2011 01:50 PM EDT)
  • Gale Greenlee (01/12/2011 02:18 PM EDT)
  • Alex Wagner (01/12/2011 04:26 PM EDT)
  • Chris Shannon (01/13/2011 09:23 PM EDT)
  • Chris Lewis (01/14/2011 01:52 PM EDT)
  • Andrew Buchanan (01/15/2011 08:09 AM EDT)
  • Amrith Raghavan (01/15/2011 01:57 PM EDT)
  • Sevag Papazian (01/16/2011 07:09 PM EDT)
  • Aviv Nisgav (01/17/2011 01:06 PM EDT)
  • Amir Sarid (01/23/2011 08:40 AM EDT)
  • Vitaly Stakhovsky (01/23/2011 08:09 PM EDT)
  • Mayowa Aregbesola (01/23/2011 09:29 PM EDT)
  • Prathu Bharti Tiwari (01/24/2011 04:46 AM EDT)
  • Vivek Nema (01/24/2011 06:08 AM EDT)
  • Naji Ashry (01/24/2011 09:20 AM EDT)
  • Thomas Rohr (01/25/2011 09:07 AM EDT)
  • Kipp Johnson (01/25/2011 12:54 PM EDT)
  • Adriana Clerici (01/26/2011 09:02 AM EDT)
  • David Friedman (01/28/2011 05:37 PM EDT)
  • David Greer (01/29/2011 12:46 PM EDT)
  • Rajendran Thirupugalsamy (01/29/2011 07:38 PM EDT)
  • Andrew Marut (01/30/2011 06:14 AM EDT)
  • Albert Stadler (01/31/2011 05:16 AM EDT)
  • Nikhil Sahasrabudhe (01/31/2011 12:15 PM EDT)

Related posts