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.
Sounds like a job for Benford’s Law:
http://plus.maths.org/issue9/features/benford/index-gifd.html