Check this out: In base 2 take all (or the first several hundred thousand) the odd numbers and figure out the mirror image of the bit string for each number. Ex: mirror of 11111 is 11111 mirror of 1011101 is 1011101 mirror of 100000001111 is 111100000001 You can't really mirror an even number because the far right "0" gets flipped to the other side where there are infinite zeros. So my method just "trims" the string. (those familiar with string processing in programming languages will pick this up right away). Anyway, you end up with a bunch of pairs of numbers. They are always odd of course. If you graph these numbers against their associated mirror values then you will see a linear line for the regular sequence of natural numbers and a kind of function which sort of has a stepping action. Now, can you predict what will happen graph the sorted list of mirror values? I did this with a spreadsheet and about 30000 odd numbers. It was a weird experiment because the mirrorred values, after being sorted, created a graph that is a little odd. I will post a graph later after you get a chance to think about what is happening.