### Highest Powers of 3 and $$\left(1+\sqrt{2}\right)^n$$

Let $$\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$$, where $$a$$ and $$b$$ are integers. What is the greatest common divisor of $$b$$ and $$81$$?
Source: 2011-2012 SDML High School 2a, problem 15.

Let $$(1+\sqrt{2})^n = a_n + b_n \sqrt{2}$$. I've thought about this some more, and there's a nice way to describe the highest power of $$3$$ that divides $$b_n$$. This is probably outside of the scope of the intended solution, however.

First note that $$(1-\sqrt{2})^n = a_n - b_n \sqrt{2}$$, and so from $$(1+\sqrt{2})(1-\sqrt{2})=-1$$ we get $$(1+\sqrt{2})^n (1-\sqrt{2})^n = {(-1)}^n$$. This gives ${a_n}^2 - 2 {b_n}^2 = {(-1)}^n.$ Now define the highest power of a prime $$p$$ that divides $$n$$ to be $$\operatorname{\nu}_p(n)$$.
From cubing and using the above result it's straightforward to prove that if $$\operatorname{\nu}_3(b_n) = k > 0$$ then $$\operatorname{\nu}_3(b_{3n}) = k+1$$.
Note $$(1+\sqrt{2})^4 = 17 + 12\sqrt{2} \equiv -1+3\sqrt{2} \pmod{3^2}$$. Cubing and using the first formula as before, we can in fact show that $(1+\sqrt{2})^{4\cdot 3^n} \equiv -1 + 3^{n+1}\sqrt{2} \pmod{3^{n+2}},$ and squaring we also have $(1+\sqrt{2})^{8\cdot 3^n} \equiv 1 + 3^{n+1}\sqrt{2} \pmod{3^{n+2}}.$ Now assume $$\operatorname{\nu}_3(b_m) = k, \operatorname{\nu}_3(b_n) = l$$ and $$k\neq l$$. From the top formula if $$3 | b_i$$ then $$3 \not{|} a_i$$, and it follows that $\operatorname{\nu}_3(b_{m+n}) = \min(k,l).$Putting this all together, write $$n = 4\cdot m +k$$, where $$0\leq k <4$$. If $$k\neq 0$$, then $$\operatorname{\nu}_3(b_{n}) = 0$$. If $$k=0$$, let the base-3 expansion of $$m$$ be $$a_i \cdot 3^i + \ldots + a_0$$. Then $\operatorname{\nu}_3(b_{n}) = \min_{a_j \neq 0} j+1 .$
For $$n=2012$$, we have $$2012 = 4\cdot 503 = 4\cdot(2\cdot 3^5 + 3^2 + 2\cdot 3 + 2)$$ and so $$\operatorname{\nu}_3(b_{2012})=1$$. We don't actually need to compute the entire base-3 expansion for 503, of course; we only need to observe that it's not divisible by 3.

For $$n=2016$$, we have $$2016 = 4\cdot 504 = 4\cdot(2\cdot 3^5 + 2\cdot 3^2)$$ and so $$\operatorname{\nu}_3(b_{2016})=3$$.

### 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…

### Mixed Models in R - Bigger, Faster, Stronger

When you start doing more advanced sports analytics you'll eventually starting working with what are known as hierarchical, nested or mixed effects models. These are models that contain both fixed and random effects. There are multiple ways of defining fixed vs random random effects, but one way I find particularly useful is that random effects are being "predicted" rather than "estimated", and this in turn involves some "shrinkage" towards the mean.

Here's some R code for NCAA ice hockey power rankings using a nested Poisson model (which can be found in my hockey GitHub repository):
model <- gs ~ year+field+d_div+o_div+game_length+(1|offense)+(1|defense)+(1|game_id) fit <- glmer(model, data=g, verbose=TRUE, family=poisson(link=log) ) The fixed effects are year, field (home/away/neutral), d_div (NCAA division of the defense), o_div (NCAA division of the offense) and game_length (number of overtime periods); off…

### Gambling to Optimize Expected Median Bankroll

Gambling to optimize your expected bankroll mean is extremely risky, as you wager your entire bankroll for any favorable gamble, making ruin almost inevitable. But what if, instead, we gambled not to maximize the expected bankroll mean, but the expected bankroll median?

Let the probability of winning a favorable bet be $$p$$, and the net odds be $$b$$. That is, if we wager $$1$$ unit and win, we get back $$b$$ units (in addition to our wager). Assume our betting strategy is to wager some fraction $$f$$ of our bankroll, hence $$0 \leq f \leq 1$$. By our assumption, our betting strategy is invariant with respect to the actual size of our bankroll, and so if we were to repeat this gamble $$n$$ times with the same $$p$$ and $$b$$, the strategy wouldn't change. It follows we may assume an initial bankroll of size $$1$$.

Let $$q = 1-p$$. Now, after $$n$$  such gambles our bankroll would have a binomial distribution with probability mass function \[ \Pr(k,n,p) = \binom{n}{k} p^k q^{n-k…