Tag Archives: puzzles

We Use Math, Integermania, and Best-of-Seven

Welcome to this week’s Math Munch!

“When will I use math?” Have you ever asked this question? Well, then you are in for a treat, because the good people of We Use Math have some answers for you! This site was created by the Math Department at Brigham Young University to help share information about career paths that are opened up by studying mathematics. Here’s their introductory video:

The We Use Math site shares write-ups about a wide range of career opportunities that involve doing mathematics. I was glad to learn more about less-familiar mathy careers like technical writing and cost estimation. Also, my brother has studied some operations management in college, so it was great to read the overview of that line of work. In addition, the We Use Math site has pages about recent math discoveries and about unsolved math problems. Check them out!

Next up is one of my long-time favorite websites: Integermania!

Perhaps you’ve heard of the four 4’s problem before. Using four 4’s and some arithmetic operations, can you make the numbers from 1 to 20? Or even higher? Some numbers are easy to make, like 16. It’s 4+4+4+4. Some are sneakier, like 1. One way it can be created is (4+4)/(4+4). But what about 7? Or 19? This is a very common type of problem in mathematics—which math objects of a certain type can be built with limited tools?

Steven J. Wilson

Integermania is a website where people from around the world have submitted number creations made of four small numbers and operations. It’s run by Steven J. Wilson, a math professor at Johnson County Community College in Kansas. (Steven has even more great math resources at his website Milefoot.com)

There are many challenges at Integermania: four 4’s, the first four prime numbers, the first four odds, and even the digits of Ramanujan’s famous taxicab number (1729).

Here are some number creations made of the first four prime numbers.
Can you make some of your own?

One of my favorite aspects of Integermania is the way it rates number creations by “exquisiteness level“. If a number creation is made using only simple operations—like addition or multiplication—then it’s regarded as more exquisite than if it uses operations like square roots or percentages. I also love how Integermania provides an opportunity for anyone to make their mark in the big world of mathematical research—it’s like scrawling a mathematical “I wuz here!” After years of visiting the site, I just submitted for the first time some number creations of my own. I’ll let you know how it goes, and I’d love to hear about it if you decide to submit, too.

Here are recaps of all the World Series since 1903 from MLB.com

Now coming to the plate: my final link of the week! Monday was the first day of the new Major League Baseball season. I want to share with you a New York Times article from last December. It’s called Keeping Score: Over in Four About a Fifth of the Time. The article digs into the outcomes of all of the World Series championships—not so much who won as how they won. It takes four victories to win a seven-game series, and there are 35 different ways that a best-of-seven series can play out, put in terms of wins and losses for the overall winner. For instance, a clean sweep would go WWWW, while another sequence would be WWLLWW. The article examines which of these win-loss sequences have been the most common in the World Series.

(Can you figure out why there are 35 possible win-loss sequences in a seven-game series? What about for a best-of-five series? And what if we tried to model the outcome of a series by assuming each team has a fixed chance of winning each game?)

A clip of the stats that are displayed in the Times article. Click through to see it all.

I was curious to know if the same results held true in other competitions. Are certain win-loss sequences rare across different sports? Are “sweeps” the most common outcome? After sifting through Wikipedia for a while, I was able to compile the statistics about win-loss sequences for hockey’s Stanley Cup Finals. This has been a best-of-seven series since 1939, and it has been played 73 times since then. (It didn’t happen in 2005 because of a lockout.) You can see the results of my research in this document. Two takeaways: sweeps are also the most common result in hockey, but baseball more frequently requires the full seven games to determine a winner.

It could be a fun project to look at other best-of-seven series, like the MLB’s League Championship Series or basketball’s NBA Finals. If you pull that data together, let us know in the comments!

Batter up, and bon appetit!

******

UPDATE (4/4/13): My first set of five number creations was accepted and are now posted on the Ramanujan challenge page. Here are the three small ones! Can you find a more exquisite way of writing 47 than I did?

Posted by Justin Lanier. Categories: Math Munch. Tags: arithmetic, baseball, careers, combinatorics, data, infographic, open problem, percentages, puzzles, recreational mathematics, sports, video. 9 Comments

Mar. ’13

Sam Loyd, Weight Problems, and Exercises

Welcome to this week’s Math Munch!

Chess master, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

Chess composer, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

First up, remember Sam Loyd? (We’ve featured him twice before.) He was an american chess player and recreational mathematician who lived from 1841-1911. He was also a chess composer, someone who writes endgame strategies and chess puzzles. In fact, he wrote all sorts of puzzles, which his son published in a book called Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks, and Conundrums. (That link will take you to a scan of all 385 pages!) By the way, those 5000 puzzles are only about half of the ones he wrote in his lifetime. It’s no wonder Martin Gardner called him “America’s greatest puzzler.” An interesting anecdote: Sam Loyd claimed until his death to have invented the 15 puzzle, but in fact he did not. The actual inventor was Noyes Chapman, the Postmaster of Canastota, NY.

I wanted to show you some of Sam’s “Puzzling Scales” problems. Why don’t you stop reading now and just solve them both?

These different weights balance because of the torque they apply

There are lots of puzzles like this, based on different weights balancing with each other. A friend sent me this page of weight puzzles based on the idea of torque. The farther out an object is placed, the more torque it applies to the balance, so it’s possible for a 1 pound weight to balance a 2 pound weight if you set them at the right distances. The distance and wight multiply to give the torque applied.

These problems come from a massive bank of puzzles over on Erich’s Puzzle Palace. If you like, you can also play this torque game I found.

Place 1 through 5 to balance the weights.

Place 1 through 6 to balance the weights.

I love problems like this, but I started to wonder, “what if the scales don’t balance? Maybe you could make a puzzle out of that.” I did exactly that, creating a series of imbalance puzzles. Your job is to order the shapes by weight. They start out easy, but there are some tricky ones. I especially like #6.

In each case, order the three objects by weight.

I’m also hosting an imbalance puzzle-writing contest. My two favorite puzzlists will win a print of their choosing from my Stars of the Mind’s Sky series of mathematical art. You should try your hand at writing one. Just email it to Lost in Recursion.

Finally – we all love great problems and puzzles, but skill building is an important aspect of mathematics as well, and exercises help us build skill. Exercises are often dull, but I found a website with some exercises I quite like, and I wanted to share them with you. Check out the Coffee Break section over on StudyMaths.co.uk.

Detention Dash

Find the Primes

Odd One Out

Detention Dash, for example, is just a timed multiplication chart, but typing the answers in on my computer really made me feel some of the patterns in the numbers. You should try it. Odd One Out also keeps you on your toes and makes you think about different kinds of numbers. I find them surprisingly fun. I hope you agree.

Bon appetit!

Posted by Paul Salomon. Categories: Math Munch. Tags: balances, exercises, games, puzzles, sam loyd, torque, weights. 8 Comments

Jan. ’13

Ghost Diagrams, Three New Games, and Scrabble Tiles

Welcome to this week’s Math Munch!

A ghost diagram composed of two different tiles.

“An organism is more than the sum of its organs. When the organs are fitted together, the organism becomes something more. This surprising something more we call “spirit” or “ghost”. Ghost Diagrams finds the ghosts implicit in simple sets of tiles.”

So writes Paul Harrison, creator of the amazing Ghost Diagram applet. Paul creates all kinds of free software and has his Ph.D. in Computer Science. I found his Ghost Diagram applet through this huge list of links about generative art.

A ‘111-‘ tile connected to a ‘1aA1’ tile.

Given a collection of tile types, the applet tries to find a way to connect them so that no tile has any loose ends. A tile type is specified through a string of letters, numbers, and dashes. Each of these specifies an edge. You can think of a four-character tile as being a modified square and a six-character tile as being a modified hexagon. Two tiles can connect if they have edges that match. Number edges match with themselves—1 matches with 1—while letter edges match with the same letter with opposite capitalization—a matches with A.

It’s amazing the variety of patterns that can emerge out of a few simple tiles. Here are a couple of ghost diagrams that I created. You can click them to see live versions in the applet. There are many other nice ghost diagrams that Paul has compiled on the site. Also, be sure to check out the random button—it’s a great way to get started on making a pattern of your own. I hope you enjoy tinkering with the ghost diagram applet as much as I have.

And now for some more fun: three new games! When I ran across Loops of Zen, I had ghost diagrams on my mind. I think they have a similar feel to them. The goal in each level of Loops of Zen is to orient the paths and loops so that they connect up without any loose edges. I feel like this game—like good mathematics—requires both a big-picture, intuitive grasp of the playing field and detailed, logical thinking. Put another way, you need both global strategy and local tactics. Also, if you like playing Entanglement, then I bet you’ll like Loops of Zen, too.

Last week we wrote about Flatland. This book and the movies it inspired describe what it might be like if creatures of different dimensionality were to meet each other. The game Z-Rox puts you in the shoes of a Flatlander. Mystery shapes pass through your field of vision a slice at a time, and it’s up to you to identify what they are. It’s a tricky task that requires a good imagination.

Hat tip to Casual Girl Gamer for both of these great mathy games.

Steppin’ Stones

Steppin’ Stones is a fun little spatial puzzle game I recently came across. You should definitely check it out. It also provides a nice segue to our last mathy item for the week, because a Steppin’ Stones board looks a lot like a Scrabble board. Scrabble, of course, is a word game. Aside from the arithmetic of keeping score, there isn’t much mathematics involved in playing it. In addition, the universe of Scrabble—the English dictionary—is not particularly elegant from a math standpoint. However, it’s the amazing truth that even in arenas that don’t seem very mathematical, math can often be applied in useful ways.

From a comic about Prime Scrabble on Spiked Math.

In Re-evaluating the values of the tiles in Scrabble™, the author—who goes by DTC and is a physics graduate student at Cornell—wonders whether the point values assigned to letters in Scrabble are correctly balanced. The basic premise is that the harder a letter is to play, the more it should be worth. DTC does what any good mathematician does—lays out assumptions clearly, reasons from them to make a model, critiques the arguments of others, and of course makes lots of useful calculations. One tool DTC uses is the Monte Carlo method. In the end, DTC finds that the current Scrabble point values are very close to what the model would assign.

I really enjoyed the article, and I hope you will, too. And since Scrabble is a “crossword game”, I think I’ll leave you with a couple of “crossnumber” puzzles. Here are some straightforward ones, while these require a little more thinking.

Have a great week, and bon appetit!

P.S. I can’t resist sharing this video as a bonus: a cellular automaton of rock-paper-scissors! Blue beats green, green beats red, and red beats blue. Hooray for non-transitive swirls!

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, applied math, art, board games, cellular automata, comics, dimensions, games, geometry, puzzles, tessellation, video. 27 Comments