An observation I’ve made on mathematics is that much of it seems to be able to be traced back to some bored mathematician coming up with a seemingly arbitrarily rule system for fun and following it through to conclusion. In so doing they discover interesting and often seemingly useless principles along the way. This fine bottle of wine then ages in the cellar until a problem comes along that the 100yr old forgotten diversion is just the ticket needed to solve it and suddenly it becomes a cornerstone of modern mathematics.
I too sometimes make up whimsical problems to solve or a series of rules and just follow them to their logical conclusion to see what happens. Typically I find contradiction or I re-derive something obvious, but sometimes I come up with something maybe a little more interesting. Here is something I was toying with; really not much more than a combinatorics problem than something new, but I thought I would share.
The concept is that there are only two entities Male (M) and Female (F) and that a formula consists of multiples of these entities combined in common arithmetic ways. The formulas are then solved by resolving them into the final progeny possible under a simplified set of reproductive rules. Let’s give the example of the simple formula M+F. This formula will resolve 50% of the time to M and 50% to F as the M + F combine to have one offspring of one sex or the other before dying in a sense; The comma delineating the potential outcomes.
M+F = M , F
This rule is Commutative and Associative. Now lets look at the next level of complexity 2M + F:
2M + F = M+M+F = M + (M+F) = M+ (M , F) = M+M , M+F = 2M , (M , F)
or represented visually as such:
As is indicated by the depth into the visual tree of the stable solutions in green there is a 50% chance of 2M, 25% of M and 25% of F. Adding a bit of new notation, here is how I prefer to notate the solution. The numerator is the number of occurrences of the solution in the solution set and the denominator relates the probability which is notated by 2 raised to the power of the depth (eg 25% = 1/4 = 1/2^2)
See below how this notation denotes the probability:
Here is a larger example to better see how combining multiple occurrences reflect in the numerators:
I am going to make just a quick aside to touch on multiplication and division because while I have rules to support it, it feels a little arbitrary and the fun stuff I think is the resolution of addition and I want to get into that in more detail. Here are those rules without a lot of background:
M * M = M² = F
F * F= F² = M
F * M = FM = 1
All multiplication must be performed before probability is resolved; it cannot be performed after.
Ok, back to resolving addition into probability solutions! Working through a number of solutions by hand and then writing code to solve these problems for me, I spotted a pattern when I started to look at the progression of the probabilities:
There are some hints of the Fibonacci sequence in here which as you look closer you recognize the following pattern appearing:
This is intimately related to the Pascal Triangle where you can see these patterns by looking at the diagonals through the triangle:
Now this makes sense because the Pascal Triangle also reflects the Binomial Theorem and related combinatorics which I can see the parallels to. In fact let’s take a look at how you can use the Pascal Triangle to solve a problem:
The solution to 3M + 4F is:
Now look how if you travel 3(M) down one side the triangle and intersect with 4(F) down the other side of the triangle the circled numbers describe the numerators and whose coordinates are related to the denominators are the probabilities in the solution!
Now we can use this fact to get us to a generalized solution for xM + yF. This is because the Pascal Triangle number is equal to the Binomial Coefficient for its coordinates but these coordinates are rows and colums, not diagonals as you will see. Let me expound on that a bit before I continue…
Notice how the coefficients of these polynomials correspond exactly to the triangle
So you can define the Binomial Coefficient for instance of xy^2 of (X+Y)^3 which is the third term of the first polynomial above by finding the value in the Pascal triangle at coordinate 3,2. Note this is a zero-based coordinate system.
This is typically written as:
Incidentally the association to combinatorics here is that this is precisely how you arrive at the ways in which you can choose K values from set of N. For instance this is how many 5 card poker hands can be dealt from a deck of 52 cards.
Ok, back to generalizing a formula for all of this. Let’s start by turning the Pascal triangle on its side so that the sides running down the triangle that we want to use for our coordinate system are now the axis and the tip of the Pascal triangle sits at the origin. Highlighted is the solution to 4M+6F for instance:
Now lets look at the Pascal Row/Column coordinates we need to map to in order to resolve using the Binomial Coefficient equation described earlier:
Through just observation and some basic algebra I was able to map the M/F axis coordinates to the Pascal coordinates in the box and get the denominator of the probability:
M Pascal Coordinate (and binomial coefficient) is :
..and F Pascal Coordinate (and binomial coefficient) is :
Similarly solving for the denominator of the probability:
Then putting it all together to reflect the complete generalized solution you can arrive at the following:
Let’s check it with this problem from earlier:
And that checks!
So there you go, a meaningless rule set carried through its course with some interesting observations along the way. What is it good for? Beats me, but I had fun scrawling stacks of crazy looking formulas all over the place for the last week to noodle this through, discover the Pascal connection and derive a generalized formula!
