shade and count

February 10, 2012

Each cell in the 8×8 square shown here is colored black or white with the following rules:

  1. each row has exactly same number (4) of white and black squares.
  2. each column has exactly the same number (4) of white and black squares.

Show that the sum of numbers in the 32 white squared cells is equal to the sum of numbers in the 32 black squared cells.

UPDATE: Hint in comment 1.

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2 Responses to “shade and count”

  1. Ramesh Says:

    I also thought that having equal number of white and black colored cells in each row and column meant that there will always be pairs of rows/columns that are flipped versions of each other. Janardan Sapre came up with this nice counter example:

    0|0|0|0|1|1|1|1
    0|1|1|0|0|1|1|0
    1|0|1|0|0|1|0|1
    0|0|1|1|1|0|0|1
    1|1|0|1|1|0|0|0
    1|1|0|1|0|1|0|0
    0|0|1|1|0|0|1|1
    1|1|0|0|1|0|1|0


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