## Five divided by forty-nine

At a family dinner, my brother told me that he had stumbled upon a curious number. Divide five by forty-nine and take a look at the digits: $$5 / 49 = 0.10204081632653061…$$ Do you see a pattern in the decimal places? Yes! It’s powers of two: 1, 02, 04, 08, 16, 32, 65, … Huh, 65? Yes! The next number, 128, is a three-digit number and so the first digit “overlaps” the last digit of 64, making it 65. And this continues for the first digits of 256 (28 + 2 = 30), 512, the first two digits of 1024, etc. Then …