IBM Research | Ponder This | October 2013 solutions

# Ponder This

## October 2013

<<September October November>>

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:
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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

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