Consider all the powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and so on…
The unit digits follow the progression 1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6… Nothing too amazing, a nice cyclic relationship, and except for the priming 1, all evenly distributed. But consider the leading digit. In the limit, what are the distribution of the leading digits? I computed a table:
| Leading Digit | Percentage |
|---|---|
| 1 | 0.3010299956639812 |
| 2 | 0.17609125905568124 |
| 3 | 0.12493873660829996 |
| 4 | 0.096910013008056461 |
| 5 | 0.079181246047624776 |
| 6 | 0.066946789630613179 |
| 7 | 0.057991946977686726 |
| 8 | 0.051152522447381332 |
| 9 | 0.045757490560675129 |
The puzzle is to verify and to explain this distribution. Neat stuff.
Some patents are just too much fun, and the pat2pdf script allows you to look them up and get a look at them. Today’s fun patent is for the 