Math Puzzle of the Day…

September 30, 2004 | General | By: Mark VandeWettering

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.

Comments

Comment from Theo
Time 10/1/2004 at 8:08 am

Sounds like a job for Benford’s Law:

http://plus.maths.org/issue9/features/benford/index-gifd.html