Published on Sunday, December 27, 2009 in

Vacation beach chairsI'll be on vacation through the end of the year. Don't worry, Grey Matters will return with new posts on January 3rd, 2010!

I did arrange for some help with the Timed Quizzes post, so updates on that post will continue as normal.

In the meantime, look around the site via the menu at the top, or explore the blog entries, which go back to March of 2005, via the menu on the side. You may find something on the site you never knew about!


Merry Chris-maths!

Published on Thursday, December 24, 2009 in , ,

12 Days of Christmas TreeTwo years ago, just before Christmas, I examined the mathematics of The 12 Days of Christmas.

This year, singingbanana, whom you may remember from the 15 Puzzle Challenges post earlier this month, gives us a great new way to visualize triangular numbers in a very Christmasy fashion:

Of course, if you like your Christmas geekery taken to extremes, check out the Singing House of Sioux Falls, SD:

I wish you all a merry Christmas, and a happy new year!


Tamariz Memorized Stack (Mnemonica) Trainer

Published on Sunday, December 20, 2009 in , , , , , ,

Playing CardsGreetings, memorized deck fans! Today, we have a new treat for you.

Over at Furniture´s Card Magic Corner, Furniture has posted a new memorized deck tool called the Tamariz Memorized Stack (Mnemonica) Trainer.

It's a spreadsheet file with 3 tabs, and can be viewed online in Google Docs. The 1st tab simply holds the deck order, the 2nd tab challenges you to recall a card by its position and vice-versa. The 3rd tab is a bonus, and will help quiz you on how to cut to a particular card, in order to deliver it to a given position.

Besides using it online, you can also download it from Google Docs, and use it offline in programs like Excel or Numbers.

For those of you who use the Aronson stack, MemDeck329 has already prepared a new version for you, which is available as a zip file here. If you use a different stack, the spreadsheet is easily altered to fit your stack in the 1st tab.

If you find this useful, you'll probably also enjoy these other free online memorized deck tools.


Arthur Benjamin's formula for changing math education

Published on Thursday, December 17, 2009 in , , , ,

I recently sprained my arm, so I'm going to keep today's column short to minimize typing.

One of Grey Matter's favorite performers, Arthur Benjamin, returns for this post in a short, yet interesting, talk about to improve math education.


Mysteries of Money, Pizza and Rockets

Published on Sunday, December 13, 2009 in , , , ,

Pizza TheoremSome problem remain unsolved because they're simply so challenging. Other mysteries remain unsolved simply because they haven't bugged enough people that someone finally sits down and solves the problem. Today, we'll take a look at some interesting mysteries that have finally been solved.

Want some money? I've got a game that will guarantee you money! I have 2 envelopes that both contain money, but 1 of the envelopes contains twice as much money as the other one. You only get to keep the contents of 1 envelope, but you do get to make a choice. Just to make in more interesting, I'll let you choose either envelope, check to see how much money is inside, and then decide if you want to switch or not. If you don't switch, you keep the money in that first chosen enevelope. If you do switch, you keep the money in that 2nd envelope.

At first, this sounds like a simple variation of the Monty Hall Dilemma, but there's an important difference. In the Monty Hall Dilemma, you know the hidden contents (2 goats, 1 car). In this two-envelope challenge, you start with no knowledge of the exact contents, and then gain a partial knowledge.

The long run of this challenge has always been how to maximize your return. You could just always switch, and you'd get the larger envelope half of the time. The same thing would happen if you never switched. Yet the calculation of expected return suggests you should always switch. Earlier this year, a group of researchers from Australia actually worked out a surprisingly simple strategy for maximizing your returns in this two-envelope paradox.

Either way, you've got your money, let's go get some pizza with it! On Grey Matters, I've actually discussed pizza quite a bit, from the Pizza Theorem to using geometry in a pizza parlor to save money. Here's another challenge: How do you fairly divide a pizza?

If pizza places always divided their pizzas into exactly equal slices and always cut exactly through the center, this would be a relatively simple problem. However, whenever you get a pizza in the real world, some pieces are larger, some are smaller, and the cut isn't always through the exact center. As Yogi Berra once said, "In theory, there's no difference between theory and practice. In practice, there is."

Under these conditions, how do you divide up the pizza fairly? It would be especially handy if there were no calculations, so that the pizza doesn't get cold while you work out the answer.

The answer to this problem was found, not surprisingly, by a group of mathematics students who enjoyed hanging out in their favorite pizzeria. Study the answer well, as many pizzas were left cold in pursuit of this important information.

One mystery that got plenty of attention this week was a mysterious spiral in the sky over Norway. It appeared as a white spiral with a blue streak shooting out of the middle. The mystery was solved when it was discovered that it was a misfiring rocket launched from Russia. The video here shows how the spiral and streak occured, along with some stunning photographs of how it actually appeared.

There are still plenty of unsolved math-related problems, but have you ever noticed that the closer a problem is to our immediate interests, the more quickly it gets solved? Even the backers of the Millennium Prize realize this, which is why they attach a $1 million prize to these unsolved problems.


Scott Kim and the Art of Puzzles

Published on Thursday, December 10, 2009 in , , , ,

Scott Kim's TED TalkScott Kim is an eclectic mix of mathematician, puzzler, artist and too many other labels to fit within a blog post. In this post, we're going to let Scott Kim take us through his view of the art of puzzles.

Most people know Scott Kim from his Inversions (generically known as ambigrams). He's also the organizer of the Gathering For Gardner.

Scott Kim has also found a place in more recent works related to Martin Gardner, since Gardner was one of Kim's inspirations. In Puzzlers' tribute: a feast for the mind, Scott Kim provided the opening “Playful Tribute”, and an article on his work called, “Inversions: Lettering with a Mathematical Twist”. Martin Gardner effectively re-payed the tribute by writing about Kim in The colossal book of mathematics: classic puzzles, paradoxes, and problems in chapter 16, “The Amazing Creations of Scott Kim”.

Over at TED, they just posted video of a 2008 talk given by Scott Kim about his designs and puzzles, which is well worth watching:



15 Puzzle Challenges

Published on Sunday, December 06, 2009 in , , , , , , , , ,

15 PuzzleYouTube user singingbanana, whose videos will interest Grey Matters readers, has just posted some interesting videos concerning the humble 15 puzzle.

If you'd like to follow along with these videos, checkout the cut-the-knot's Fifteen puzzle page (Java required) to try the puzzle out for yourself.

To learn the basics of solving 15 puzzles, see Jaap's 15 puzzle page.

The first video concern's Sam Loyd's classic 15/14 version of the puzzle. If you're using the cut-the-knot version to which I linked above, you can use the Cheat button to start from the 15/14 version to try this challenge out:

As you may have guessed by the ease with which singingbanana offers the $1,000 reward, getting to the standard solution from a position starting with the 15 and 14 switched is impossible.

The question is, why is it impossible? That's answered clearly in the follow-up video, where three challenges using the 15/14 version are added:

Can you meet these challenges? Try them out for yourself before proceeding any further.

Some of you may remember that I taught an approach to a 15 puzzle magic square a few years back, which I later demonstrated on video. However, the one I teach requires that you start from a standard 15 puzzle. If you start from the 15/14 version, as in the final challenge in the above video, you can't do it exactly as I teach.

However, if you understand the principle taught in the video, you should see how it's possible to adapt my magic square approach. Again, try it out for yourself before proceeding any further.

Here's my answer to all three challenges, and you'll see how I adapted the magic square for the 15/14 puzzle:

How did you do?


Yet Still More Quick Snippets

Published on Thursday, December 03, 2009 in , , , , ,

LinksDecember is here, and so are this month's snippets!

• Last month, I talked about Game Theory in Popular Culture, and linked to the final scene of The Dark Knight as one example. Over at the Quantitative Peace blog, there's an excellent full analysis of this scene as it relates to game theory.

• Richard Wiseman, whom you may remember from his Quirkology work, has released a magic app called Telepath (iTunes Link). At first, it may seem like iSensor or iForce, but the fact that your spectayor picks up the iPhone instead of you will puzzle anyone who is wise to those other methods.

• The recently-mentioned Mind Your Decisions blog had a few problems lately, as noted here and here, but they seem to be fixed now. Presh has even found time to author two new game theory posts, both featuring fictional Charlies: Brown, from Peanuts, and Eppes, from Numb3rs.

• Over at Edu-tastic, they posted a list of the Top 50 Bloggers to Help You Study, Focus and Learn Better, and Grey Matters is the first blog mentioned in the Memory and Brain Training category! Grey Matters readers should definitely check out this list, as there are many great blogs you may not have known about before that you'll probably enjoy.