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)
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