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Table Partition Trial (Posted on 2016-07-24) Difficulty: 3 of 5
Can a 75x75 table (consisting of 75x75 identical square grids) be partitioned into dominoes (1x2 rectangles) and crosses (five square figures consisting of a square and its four neighbors.)

Provide adequate reasoning for your answer.

Source: St. Petersburg City Mathematical Olympiad (Russia)

No Solution Yet Submitted by K Sengupta    
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Solution D1 argument | Comment 1 of 3
A pretty standard parity argument:

Color the table in chessboard fashion with the corners black and alternating with white.  There will be one more black than white.  Call this excess (+1).

Dominoes have equal numbers of black and white (0).
Crosses are either center white with 4 black (+3) or center black with 4 white (-3).

Clearly there is no way any number of dominoes and crosses can bring the count to (+1)

  Posted by Jer on 2016-07-24 12:10:20
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