Published on 2004-10-01 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
|
<th>
Percentage
</th>
</tr>
<tr>
<td>
1
</td>
<td>
0.3010299956639812
</td>
</tr>
<tr>
<td>
2
</td>
<td>
0.17609125905568124
</td>
</tr>
<tr>
<td>
3
</td>
<td>
0.12493873660829996
</td>
</tr>
<tr>
<td>
4
</td>
<td>
0.096910013008056461
</td>
</tr>
<tr>
<td>
5
</td>
<td>
0.079181246047624776
</td>
</tr>
<tr>
<td>
6
</td>
<td>
0.066946789630613179
</td>
</tr>
<tr>
<td>
7
</td>
<td>
0.057991946977686726
</td>
</tr>
<tr>
<td>
8
</td>
<td>
0.051152522447381332
</td>
</tr>
<tr>
<td>
9
</td>
<td>
0.045757490560675129
</td>
</tr>
The puzzle is to verify and to explain this distribution. Neat stuff.