#math trivia #82 solution

Burt Kaliski Jr. (@modulomathy)
3/22/12 7:50 AM
#math trivia for #March22: #82 is one of several numbers whose digits sum to 10. How many others are there this year? How many overall?

In the range 1 to 366, there are 35 numbers whose digits sum to 10:

— 9 numbers between 1 and 99: 19, 28, 37, …, 91
— 10 between 100 and 199: 109, 118, 127, …, 190
— 9 between 200 and 299: 208, 217, 226, …, 280
— 7 between 300 and 366: 307, 316, 325, …, 361

The overall set is infinite, given that an arbitrary number of zero digits can be included, e.g., 19, 109, 1009, 10009, …, and many other forms.

If zeros are not allowed, then there are just 511 possibilities. (Put another way, everything in the infinite set falls into one of these 511 classes, of the zeros are ignored, based on the ordering of the non-zero digits remaining.). To see this, consider the target sum of 10 as consisting of 10 segments. Divide the segments into a series of subsegments by cutting at one or more of the nine boundaries between the segments. Let the digits of the number be the lengths of the successive subsegments, which will then add up to 10. There are 2^9-1 = 511 ways to cut up the segment depending on which boundaries are cut, and each way will give a different series of digits. (The one way that’s not allowed is to have no cuts at all, which would result in just a single, “10” digit. In all the other cases, the digits are all between 1 and 9.)