2011/12 AHSMC Part 1 Problem 14

Problem 14 of 16

Of the $49$ squares of a $7 \times 7$ square sheet of paper, two are to be coloured black while the others remain white. Two colourings are called distinct if one cannot be obtained from the other by rotating the sheet of paper about its centre. The number of distinct colourings is

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Solution

There are $\binom{49}{2} = 1176$ colourings. The number of symmetrical colourings with respect to the middle square is $\frac{49-1}{2} = 24$ . These colourings are counted twice. All the other colourings are counted four times. The desired number is $\frac{24}{2} + \frac{1176-24}{4} = 300$ . The answer is (D).

University of Alberta

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