..well, not so much a puzzle, more something that seems counter-intuitive yet logically bombproof. A prize for anyone who shoots a hole in it. THREE WAYS TO PROVE 0.999(rec) = 1 1. Switching between fractions and decimals, 1/3 = 0.333(rec). Multiply both sides by 3 and you get 1 = 0.999(rec) 2. Take 1 - 0.999(rec) = 0.000(rec). Therefore the distance between 1 and 0.999(rec) is nothing, so they must be the same. 3. Let x = 0.999(rec). Multiply both sides by 10 to get 10x = 9.999(rec). Next, subtract the first equation from the second equation to get 9x = 9. Therefore x = 1. What am I missing (other than something better to do at this time of night) ?