I discovered a cool property of positive integers of the form 10^{n}-1, that is, integers made up of n digits of 9s: they have binary representations that have exactly n digits of trailing 1s. For example, 9,999,999 in decimal is 100110001001011001111111 in binary.

The property is interesting in and of itself, but what is more interesting is the process I went through to discover it. It’s a small-scale example of experimental mathematics: I observed something interesting, experimented to collect more data, developed a hypothesis, and constructed a proof.