Ticket to Ride is a “cross-country train adventure” board game I bought and had the chance to play recently. You score points by claiming train routes connecting U.S. and Canadian cities and then using your train routes to complete specific journeys you choose at the beginning of the game. There’s also an iPad version of Ticket to Ride.
Of course, I look at this map and can easily think of it as a weighted graph. This graph has no Euler circuit and no Euler path. Montreal, New York, Charleston, Miami, and many others have an odd degree. The graph does have a Hamilton circuit, which I actually discovered fairly quickly even though the problem is NP.
In algebraic chess notation, the rows of a chessboard are numbered 1 through 8 beginning at the bottom row and the columns are lettered a through h going from left to right.
Given the coordinates of two chessboard tiles (say e7 and h3), how can you tell if the two tiles share the same color without looking at a chessboard?
Think of the letters a, c, e, and g as odd and the letters b, d, f, and h as even. Then the coordinates for every tile can be categorized as (even, even); (odd, odd); (even, odd); or (odd, even).
(even, even) and (odd, odd) tiles will always share the same color. The same is true for (even, odd) and (odd, even) tiles for the opposite color.
For example, e7 is (odd, odd) while h3 is (even, odd). These two tiles must be of different colors.
Notes
I would love to find a more elegant way to phrase this.
On the last day before winter break, a colleague of mine introduced me to the puzzle game Lunar Lockout. The game is played on a 5 by 5 board using six pieces: one red “spacepod” and five “helper-bots”. Here’s the setup for the game’s first puzzle (out of 40). Because this is a beginner puzzle, the last helper-bot is not involved.
Here are the rules to the game, taken from the rules card.
The Object To maneuver the Red Spacepod, assisted by the Helper-Bots, until it lands on the Emergency Entry Port at the center of the Mothership’s Landing Grid.
Spacepods and Helper-Bots travel between the grid lines, and can only move forward, reverse and side-to-side. No diagonal movement is allowed.
You can only move a Pod or Bot on the grid if its movement can be blocked by a Bot or a Pod. Without this block, the Pod or Bot will jet off the landing Grid into space—forever! (Use your imagination!)
A move is complete when you land on the grid square next to the blocking Pod or Bot (it acts like your brakes).
The Spacepods must jet over the Emergency Entry Port if you have not set up the correct blocks.
The solution to the first puzzle is straight-forward:
It may not surprise you that there are competitions (internationally organized by the World Cube Association) to see who can solve a cube the fastest. And because there are competitions for that, there are also competitions to see how fast one can solve a cube with only one hand, with one’s feet, or while one is blindfolded. (This is a full list of sanctioned event types).
A competitor is timed on five different solves of a cube (the cube is scrambled in a different manner each time) and then the slowest and fastest times are discarded. The average of the middle three fastest solves is then used to rank competitors and determine a winner. This makes sense. Sometimes a competitor may not have fully warmed up. Perhaps a certain cube configuration just takes significantly more (or less) moves to solve than others. Any of a host of random hidden factors could interfere with performance. The trimmed mean of solution times gives a more accurate picture of the ability of a cube competitor.
For a similar reason, teachers usually have a policy of dropping the lowest quiz or test grade for a term. However, we would never even think of discarding a student’s highest grade, even if that grade is atypical of a student’s overall performance (and I would never advocate such a thing either). But it did get me to think more about the purpose of using numerical averages as term grades.
By most metrics, either the median or the trimmed mean is a better measure of “typical”-ness than the mean. But we use the mean instead perhaps because:
Our sense of fairness requires us to use a measure that incorporates all performance outcomes in a term of study.
The mean’s tendency to be heavily influenced by outliers is sometimes an advantage: students know that they can greatly affect their overall grade if they “knock one out of the park”.
In a similar vein, the median’s resistance to outliers may actually discourage students from studying, especially towards the end of a term when even a 0% mark would not greatly affect the final grade.
Disguised as an entertaining description of juggling and the Site Swap notation, the presentation is really about science and mathematics, how they work, what they mean, and why they’re important.
The arrival of David Dinkins, the first and still only African American mayor of New York City, was a pleasant surprise at the beginning of the night. He was there to introduce Mr. Wright, but what does Mayor Dinkins have to do with a math presentation? Unbeknownst to me, Mayor Dinkins was a math major at Howard University and graduated magna cum laude.
Mr. Wright’s talk was incredibly entertaining. I’ve rarely seen a speaker that was as able as Mr. Wright to engage an audience with such a wide range of ages as well as mathematical backgrounds. And the talk touched on everything from mathematics to science to philosophy and had a great deal of inspiration and humor as well.
I’ve included two short video clips of the talk that I captured on my phone. They don’t do it justice as the presentation was much more profound and Mr. Wright’s juggling abilities much more impressive than what is shown here.
Now I have a strong urge to learn how to juggle myself!
Through the New York City Math Circle, I am taking a course for teachers called More Enrichment. Last night was the first class. I saw of a lot of great solutions, but what has me more excited are some great problems that remain unsolved. Here’s one that the presenter Gil Kessler filed under “Colorful Problems”.
Consider an 8 by 8 chess board:
and 31 dominoes:
If we try to tile the chess board with our 31 dominoes, we will leave two squares uncovered. Furthermore, of the two uncovered squares, one will be grey and one will be black.
The question is: if we arbitrarily remove two squares of different colors from the board, can we still tile the remaining squares with our 31 dominoes?
In class, it was mentioned that the answer is “yes”. Now to figure out the proof!
I used to think solving a Rubik’s Cube involved a lot of visual and spatial acumen, but that was before I started learning how to solve one about a week ago. It turns out that solving a Rubik’s Cube is actually more like solving a quadratic equation: you have a set of very simple moves and then a set of memorized algorithms (and in Rubik’s Cube parlance, they are actually called “algorithms”). You execute a few simple moves to place the cube in a desired position and then go into an algorithm to make sure each piece is oriented and aligned correctly. Repeat until solved.
Once I start an algorithm (which can range from four to twelve moves), I actually look away from the cube since I find it easier to just think about the correct sequence of turns!
These data come from 1,000,000 trials in Python. There’s a nonzero chance of eight, nine, ten, eleven, or twelve sets appearing among 12 random cards. The frequency of 12 sets though is less than 1 in 1,000,000, which makes the twelve set puzzle I posted before exceptionally rare.
What’s interesting to me: the distribution looks binomial.
I did not come up with this game, but I can’t quite remember where I first heard about it either. This implementation is mine though. You can also click on the image to see the spinner.
In the game of Skunk!, players spin a wheel that is marked with $100, $500, and Skunk! spaces. Landing on a space with a dollar value adds the corresponding amount to a temporary pot. Repeatedly spinning accumulates more and more money in this pot. Before every spin, the player has a choice to make: take the money in the pot (and end the game) or spin again. However, if a player lands on Skunk!, the pot is emptied and he or she goes home with nothing!
Games like this are my favorite way to introduce a host of concepts in probability theory: expected value, variance, and the geometric probability distribution. The crucial question: at what point should you take the money and run?
Notes
You can see the HTML, CSS, and JavaScript that control everything by viewing the source on this page. In class, I have paper money to keep track of the pot. It’s actually quite fun to hit the big SPIN! button on a SmartBoard.
