Ponder This Challenge - October 2013 - B&W coloring of a chess board

Ponder This Challenge:

Color the squares of an 8x8 check-board in two colors (black and white) and calculate the percentage of black squares in every one of the 8 rows, 8 columns and 30 diagonals.

Our challenge this month is to find such coloring which gives the same percentage for every row; different one for every column and as many different numbers in the diagonals as possible.

Supply your answer as 8 lines of 8 B/W characters.

For example, the following is a solution for a 2x2 case of the problem:
BW
BW

The column percentage is 0%, 100%; all rows are 50% and the diagonals has 0%,50% and 100% (each appears twice).

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

  The number of black squares in the columns are eight distinct numbers in the range between 0 and 8.
  The total number of black squares on the board should be divisible by 8, so clearly the missing number is 4.
  There are 23 different possible percentages that can be realized on the diagonals:
  0,1/8,1/7,1/6,1/5,1/4,2/7,1/3,2/5,3/7,3/8,1/2,5/8,4/7,3/5,2/3,5/7,3/4,4/5,5/6,6/7,7/8, 1.
  Since there are only two 8-long diagonals, we cannot get the four possibilities (1/8, 3/8, 5/8 and 7/8). Also we only have four 7-long diagonals, so we cannot get the six possibilities ( 1/7, 2/7, 3/7, 4/7, 5/7, 6/7). Thus the maximum is 19.
  One can get it quite easily by playing a little with row and column permutations.

  We chose to show Mark Pervovskiy's solution since it was the nicest looking one:
  WWWWBBBB�
  WWWWBBBB�
  WWWWBBBB�
  WWWWBBBB�
  WWWWBBBB�
  WWBWBWBB�
  WBBWBWWB�
  BBBWBWWW

  Thanks to Motty Porat for sending us ten wonderful solutions:

  PP P P      O O OO  N NN  N    DDD D       E E E E    RRRR
    PP  PP   O  OO O    NN  NN     D DDD    EEEE       RR  RR
    PP  PP   O  0O O     NN NN       DDDD    EEE   E   RR RR
  P PP  P    O  O OO     NN NN       DDDD    EE E  E   RRRR
  PPPP         OOO O     NNN N     D DD D   EEEE       RR R
  PPPP       O  OO O     N NNN     D DD D   EEEE        RR R  R
  PPPP       O  OO O     NN NN     D DD D    EEE   E   RRR R
  PPPP        O OOO     NNN N    D D DD      E E E E   RR  RR

  T  T TT      H  HH H      IIII       S SSS
  TTTT        HH  HH       III   I    SSSS
  TTTT         HH H  H       II II    SSSS
  TTTT         HH H  H       I III     SSSS
  TTTT        HH  HH         I III     SSS S
  TT T  T     HH  HH         I III      SSSS
  T  TT T     HH  HH         I III      SS SS
   T TTT      H   HHH        II  II    SSS S

Solvers

• Lorenz Reichel (10/02/2013 05:04 AM EDT)
• Tatiana Stepanova (10/02/2013 04:52 AM EDT)
• Andrew Gacek (10/02/2013 10:43 AM EDT)
• Don Dodson & David Dodson (10/02/2013 11:17 AM EDT)
• Andreas Razen & Radu-Alexandru Todor (10/02/2013 11:44 AM EDT)
• Jason Patskoski (10/02/2013 11:54 AM EDT)
• Joseph DeVincentis (10/02/2013 01:53 PM EDT)
• Dan Dima (10/02/2013 02:12 PM EDT)
• Alan Murray (10/02/2013 03:12 PM EDT)
• Alex Wagner (10/02/2013 03:29 PM EDT)
• Joe Clark (10/02/2013 04:14 PM EDT)
• Alex Fleischer (10/02/2013 04:34 PM EDT)
• Cynthia Beauchemin (10/02/2013 08:45 PM EDT)
• Kan Shen (10/02/2013 09:23 PM EDT)
• Leandro Araújo (10/02/2013 09:33 PM EDT)
• Antoine Comeau (10/03/2013 12:34 AM EDT)
• Liubing Yu (10/03/2013 01:37 AM EDT)
• Dheeraj Reddy (10/03/2013 01:57 AM EDT)
• Kurt Hectic (10/03/2013 05:22 AM EDT)
• James Dow Allen (10/03/2013 06:35 AM EDT)
• Tim Nordenfur (10/03/2013 10:16 AM EDT)
• Michael Wilms (10/03/2013 01:34 PM EDT)
• Adam Kozdron (10/03/2013 02:22 PM EDT)
• Aleksei Galak (10/03/2013 04:13 PM EDT)
• I. J. Kennedy (10/03/2013 05:36 PM EDT)
• Dieter Beckerle (10/04/2013 02:46 AM EDT)
• Todd Will (10/04/2013 11:57 AM EDT)
• Tom Harley (10/04/2013 12:18 PM EDT)
• Philipp Seemann (10/04/2013 12:50 PM EDT)
• Mark Pervovskiy (10/04/2013 02:24 PM EDT)
• Michael Gurvich & Chintan Maggu (10/04/2013 10:52 PM EDT)
• Florian Fischer (10/05/2013 02:06 PM EDT)
• Shirish C (10/05/2013 07:46 PM EDT)
• José Eduardo Gaboardi de Carvalho (10/05/2013 08:36 PM EDT)
• Dezhi Zhao (10/05/2013 11:01 PM EDT)
• Dustin Shively (10/05/2013 11:56 PM EDT)
• Rob Sessions (10/06/2013 12:09 AM EDT)
• Harald Bögeholz (10/06/2013 06:19 AM EDT)
• Nicholas Ngiam & Liu Ren Wei (10/06/2013 11:31 AM EDT)
• Mark Kattenbelt (10/06/2013 02:25 PM EDT)
• Eugene Lee (10/06/2013 02:30 PM EDT)
• Hakan Kjellerstrand (10/07/2013 02:26 AM EDT)
• Hakan Summakoglu (10/07/2013 01:50 PM EDT)
• Chuck Carroll (10/07/2013 03:08 PM EDT)
• Adib Surani (10/07/2013 06:16 PM EDT)
• Joseph Berman (10/07/2013 07:42 PM EDT)
• Yiqian Lu (10/07/2013 08:32 PM EDT)
• Alexei Kopylov (10/07/2013 08:42 PM EDT)
• Adir HaCohen (10/07/2013 11:49 PM EDT)
• Rogerio Ponce da Silva (10/07/2013 06:18 PM EDT)
• Amos Guler (10/08/2013 07:07 AM EDT)
• Matthew Charlap (10/08/2013 07:54 AM EDT)
• David Friedman (10/08/2013 11:34 AM EDT)
• Robin Ryder (10/08/2013 12:32 PM EDT)
• Aharon Malachi (10/08/2013 12:53 PM EDT)
• Hubert Kahlert (10/08/2013 01:53 PM EDT)
• Balakrishnan Varadarajan (10/08/2013 03:10 PM EDT)
• Olivier Mercier (10/08/2013 02:58 PM EDT)
• Julien Briton (10/09/2013 03:24 AM EDT)
• Yasodhar (Pattu) Patnaik (10/08/2013 09:32 PM EDT)
• Umut Uludag (10/09/2013 08:56 AM EDT)
• Fletcher Dostie (10/09/2013 12:26 PM EDT)
• David Greer (10/09/2013 12:56 PM EDT)
• Vivek Viswanathan (10/09/2013 01:10 PM EDT)
• Andrei Prudius & Satish Vedantam (10/09/2013 04:13 PM EDT)
• Clive Tong (10/10/2013 01:08 AM EDT)
• Shouky Dan & Tamir Ganor (10/10/2013 04:13 AM EDT)
• Steffen Suerbier (10/10/2013 10:01 AM EDT)
• Mathias Schenker (10/10/2013 11:01 AM EDT)
• Serafim Chekalkin (10/10/2013 11:46 AM EDT)
• Ivan Bendyna (10/10/2013 03:07 PM EDT)
• Mukund R (10/10/2013 11:31 PM EDT)
• Maris Van Sprang (10/11/2013 02:03 AM EDT)
• Guangda Huzhang (10/11/2013 03:27 AM EDT)
• Pål Hermunn Johansen (10/11/2013 05:13 AM EDT)
• Berend Jan van der Zwaag (10/11/2013 09:43 AM EDT)
• Daniel Bitin (10/12/2013 03:39 PM EDT)
• Andrea Andenna (10/12/2013 05:28 PM EDT)
• Anatoli Plotnikov (10/13/2013 02:42 AM EDT)
• Klaus Nagel (10/13/2013 03:13 AM EDT)
• Armin Krauss (10/13/2013 09:39 AM EDT)
• Ken Meehan (10/13/2013 11:44 AM EDT)
• Marcel Caria (10/15/2013 09:18 AM EDT)
• Evert van Dijken (10/15/2013 09:45 AM EDT)
• Peter Gerritson (10/15/2013 10:22 AM EDT)
• Igor Novak (10/15/2013 11:16 AM EDT)
• Tao Liu (10/15/2013 01:57 PM EDT)
• Alok Menghrajani (10/15/2013 04:57 PM EDT)
• Christian Pape (10/16/2013 05:58 AM EDT)
• John Tromp (10/16/2013 04:19 PM EDT)
• Frank Methling (10/17/2013 11:01 AM EDT)
• William Heller (10/19/2013 02:59 PM EDT)
• Dennie Tyshchenko (10/20/2013 03:17 PM EDT)
• Motty Porat (10/20/2013 04:24 PM EDT)
• Adrian Orzepowski (10/21/2013 11:47 PM EDT)
• Dan Cervone (10/22/2013 11:30 AM EDT)
• Derek Kisman (10/22/2013 04:34 PM EDT)
• Hugues Juille (10/23/2013 03:40 PM EDT)
• Thomas Egense (10/25/2013 12:21 AM EDT)
• Scott Abbey (10/25/2013 01:23 AM EDT)
• Gilad Weiss (10/25/2013 10:16 AM EDT)
• Antonio García Astudillo (10/25/2013 11:49 AM EDT)
• Minxi Jiang (10/26/2013 09:42 AM EDT)
• Rob Pratt (10/26/2013 11:07 AM EDT)
• David Chang (10/26/2013 11:34 PM EDT)
• Alessandro Silva (10/27/2013 11:27 AM EDT)
• Boris Rakus (10/28/2013 06:18 AM EDT)
• Yaniv Meoded (10/28/2013 08:15 AM EDT)
• Stéphane Higueret (10/28/2013 11:27 AM EDT)
• Greg Bubnis (10/29/2013 01:20 AM EDT)
• Tilmann Steinberg (10/29/2013 11:17 AM EDT)
• Alkeos Tsokos (10/29/2013 12:45 PM EDT)
• David Vainapel (10/29/2013 03:26 PM EDT)
• Maithem Munshed (10/29/2013 10:53 PM EDT)
• Laszlo Sandor (10/30/2013 07:30 AM EDT)
• Simon Xu (10/30/2013 08:03 AM EDT)
• SeongWoo (10/30/2013 05:49 PM EDT)
• Michael Lim (10/30/2013 08:45 PM EDT)
• Milos Cubrilo & Vukasin Vucevic (10/31/2013 02:17 AM EDT)
• Gale Greenlee (10/31/2013 09:43 AM EDT)
• Eugene Perederey (11/01/2013 06:56 AM EDT)
• Katherine Oh (11/02/2013 11:14 AM EDT)