Write, side by side, the numeral 1 an even number of times. Subtract from the number thus formed the number obtained by writing, side by side, a series of 2s half the length of the first number. You will always get a perfect square. For instance,
1111 - 22 = 1089 = 33²
Can you say why this is?
11...1 - 22...2 = 11...1 11...1 - 2(11...1)
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2n times n times n times n times n times
= 11...1 00...0 - 11...1
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n times n times n times
= 11...1 x (100...0 - 1)
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n times n times
= 11...1 x 99...9
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n times n times
= 11...1 x 9 x 11...1
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n times n times
= 3² x 11...1²
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n times
= 33...3²
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n times