### A Strange Recursive Relation, Automatic

Hofstadter mentions the following recursive relation in his great book "Gödel, Escher, Bach": \begin{align} g(0) &= 0;\\ g(n) &= n-g(g(n-1)). \end{align} I claim that $$g(n) = \left\lfloor \phi\cdot (n+1) \right\rfloor$$, where $$\phi = \frac{-1+\sqrt{5}}{2}$$, and I'll show this using a technique that makes proving many identities of this type nearly automatic.

Let $$\phi\cdot n = \left\lfloor \phi\cdot n \right\rfloor + e$$, where $$0 < e < 1$$ as $$\phi$$ is irrational, nor can $$e = 1-\phi$$, and note that $$\phi$$ satisfies $${\phi}^2 + \phi - 1 = 0$$. Some algebra gives \begin{align} n-\left\lfloor \left( \left\lfloor \phi\cdot n \right\rfloor + 1 \right) \cdot \phi \right\rfloor &= n-\left\lfloor \left( n\cdot \phi - e + 1 \right) \cdot \phi \right\rfloor \\ &= n-\left\lfloor n\cdot {\phi}^2 - e\cdot \phi + \phi \right\rfloor \\ &= n-\left\lfloor n\cdot \left(1-\phi\right) - e\cdot \phi + \phi \right\rfloor \\ &= n-n-\left\lfloor -n\cdot \phi - e\cdot \phi + \phi \right\rfloor \\ &= -\left\lfloor -n\cdot \phi - e\cdot \phi + \phi \right\rfloor \\ &= -\left\lfloor -n \cdot \phi + e - e - e\cdot \phi + \phi \right\rfloor \\ &= \left\lfloor \phi\cdot n \right\rfloor -\left\lfloor - e - e\cdot \phi + \phi \right\rfloor. \end{align}
Now if \begin{align} 0 < e < 1-\phi &\implies 0 < - e - e\cdot \phi + \phi < \phi;\\ 1-\phi < e < 1 &\implies -1 < - e - e\cdot \phi + \phi < 0. \end{align}
This implies $n-\left\lfloor \left( \left\lfloor \phi\cdot n \right\rfloor + 1 \right) \cdot \phi \right\rfloor = \left\lfloor \phi\cdot (n+1) \right\rfloor .$ Since $$\left\lfloor \phi\cdot (0+1) \right\rfloor = 0$$, we're done.

The point of the algebra was to move all terms involving $$n$$ out, and then checking to see how the remaining term varied with $$e$$. A simple idea, but very useful.

### A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

### Notes on Setting up a Titan V under Ubuntu 17.04

I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under Ubuntu 17.04.

Old install:
Ubuntu 16.04
EVGA GeForce GTX Titan SuperClocked 6GB
2TB Seagate NAS HDD

New install:
Ubuntu 17.04
Titan V 12GB
/ partition on a 250GB Samsung 840 Pro SSD (had an extra around)
/home partition on a new 1TB Crucial MX500 SSD
New WD Blue 4TB HDD