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Showing posts from October, 2015

Solving IMO 1989 #6 using Probability and Expectation

An Enormous Number of Kilograms

For years the kilogram has been defined with respect to a platinum and iridium cylinder, but this is now no longer the case. Here's a puzzle about kilograms that's easy to state and understand, but the answer is very, very surprising.

I've always had a fascination with really large numbers. First 100 when I was really little, and as I got older and more sophisticated numbers like a googol and the smallest number that satisfies the conditions of the Archimedes cattle problem.

When I was an undergraduate I interviewed for a summer internship with an insurance company as an actuarial student. They gave me the following puzzle - what's the smallest number that when you move the last digit to the front it multiplies by 2? I calculated for a little while, then said "This can't be right, my answer has 18 digits!". It turns out that the smallest solution does, indeed, have 18 digits.

We can see this by letting our \((n+1)\)-digit number \( x = 10 m + a\), where \…

Solving a Math Puzzle using Physics

The following math problem, which appeared on a Scottish maths paper, has been making the internet rounds.


The first two parts require students to interpret the meaning of the components of the formula \(T(x) = 5 \sqrt{36+x^2} + 4(20-x) \), and the final "challenge" component involves finding the minimum of \( T(x) \) over \( 0 \leq x \leq 20 \). Usually this would require a differentiation, but if you know Snell's law you can write down the solution almost immediately. People normally think of Snell's law in the context of light and optics, but it's really a statement about least time across media permitting different velocities.


One way to phrase Snell's law is that least travel time is achieved when \[ \frac{\sin{\theta_1}}{\sin{\theta_2}} = \frac{v_1}{v_2},\] where \( \theta_1, \theta_2\) are the angles to the normal and \(v_1, v_2\) are the travel velocities in the two media.

In our puzzle the crocodile has an implied travel velocity of 1/5 in the water …

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…

Elo's Rating System as a Forgetful Logistic Model

Elo's rating system became famous from its use in chess, but it and variations are now used in sports like the NFL to eSports like League of Legends. It also was infamously used on various "Hot or Not" type websites, as shown in this scene from the movie "Social Network":


Of course, there's a mistake in the formula in the movie!
What is the Elo rating system? As originally proposed, it presumes that if two players A and B have ratings \(R_A\) and \(R_B\), then the expected score of player A is \[\frac{1}{1+10^{\frac{R_B-R_A}{400}}}.\] Furthermore, if A has a current rating of \(R_A\) and plays some more games, then the updated rating \({R_A}'\) is given by \({R_A}' = R_A + K(S_A-E_A)\), where \(K\) is an adjustment factor, \(S_A\) is the number of points scored by A and \(E_A\) was the expected number of points scored by A based on the rating \(R_A\).
Now, the expected score formula given above has the same form as a logistic regression model. What'…