### Interview Doubler

Problem:

Find the smallest positive number that doubles when you move the last digit to the front.

Solution:

The answer is $$105263157894736842$$, and the solution is similar to Twisty Temperature. Let this number be $x = x_{n}\cdot 10^{n-1} + ... + x_{1}\cdot 10^1 + x_{0}$ with $$0 < x_{n} < 10$$, then after moving the last digit to the front we get $y = x_{0}\cdot 10^{n-1} + (x-x_{0})/10.$ We also have that $$y = 2 x$$, so our equation becomes $19\cdot x = x_0 \cdot (10^{n}-1).$ Now 10 is a primitive root modulo 19, and so it follows that $$x$$ is integral if and only if $$n$$ is of the form $$18\cdot m$$. Also note that we need $$2 \le x_0 \le 9$$ since we require $$0< x_{n} < 10$$. When $$m=1$$ and $$x_0 = 2$$ we get $$x =105263157894736842$$; when $$m=2$$ and $$x_0 = 2$$ we get $2 (10^{36}-1)/19 = 105263157894736842105263157894736842.$ That this number indeed doubles when you move the last digit to the front can be verified by WolframAlpha.

### 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