# What is the distribution of destiny numbers

Today my wife surprised me with the fact, that our destiny numbers (see e.g. http://numerologylife.com/destiny_number) are the same, namely we both have a destiny number of two.

But I suspect that this might not be so surprising, since the destiny numbers are calculated based on birth dates, and this might led to a skewed distribution of destiny numbers towards lower numbers.

So my question is:

Given that a destiny number is calculated as iterated digit sum of someones birth date, e.g.

the 11th of March, 1985 := 1+1+3+1+9+8+5 = 28 = 2+8 = 10 = 1+0 = 1

what is the distribution of this numbers (let's say after 2000 years starting from year 0)? So how high is the probability to have a destiny number of 1, 2, 3, and so on?

Bonus points are given, if someone can show, how this distribution changes over time and what the probability is that two persons which are N years apart, have the same destiny number.

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The distribution should be essentially even, since the repeated digit sum is essentially equivalent to computing the remainder of the number when dividing by $9$. (With zero remainder being a $9$ sum.) But some trick might occur due to the frequency of the month/days, I suppose. Given enough years, that will even out. – Thomas Andrews Feb 8 '14 at 15:02

The destiny number is essential the remainder when the sum of the day, month and year are divided by $9$, except that when we see $0$ we say $9$.

If we just looked at the sum of day and month to begin with, we would get the following almost balanced distribution across a year:

1   39
2   41
3   42
4   41 (42 in a leap year)
5   42
6   40
7   41
8   39
9   40


Including the year is simple. For example $2014$ divided by $9$ (or equivalently its digit sum) gives a remainder of $7$, so we need to push the cycle seven steps down, or two steps up, to give, since it is a non-leap year:

1   42
2   41
3   42
4   40
5   41
6   39
7   40
8   39
9   41


Ignoring leap years, the pattern would be completely balanced over $9$ years. If leap years happened every four years (as from $1904$ to $2096$) then the pattern would be completely balanced over $36$ years. With the Gregorian calendar the pattern would be completely balanced over $3600$ years. But even with shorter periods the pattern would be close to balanced. In reality, the distribution of destiny numbers will be more balanced than the distribution of birthdays, since any seasonal component is smoothed out over time.

So it is not unreasonable to approximate by saying that each destiny number has close to a probability of $\frac19$ of appearing or that two random people have close to a $\frac19$ chance of sharing the same destiny number. If you have more information about them, such as sharing a birthday, then you can say more: if they were born exactly $N$ years apart then they will share a destiny number if $N$ is a multiple of $9$ and otherwise not.

If you have four people, the chance that at least two of them share a destiny number is about $54\%$; this is a simplified version of the birthday problem, showing that coincidences are common.

Whether this is meaningful is a question for numerology rather than mathematics; in other words "no".

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