If you've read even a little of this blog, you know I love playing with math.

Naturally, when mobile apps are released that let me play with math on the go, I'm even more enthused! Earlier this month, an app called Geogebra was released, and it's a great way to explore math visually!

Geogebra is what is known as a computer algebra system, or CAS, for short. As you input various mathematical equations, you get a visual representation. The video below gives you a better idea of how Geogebra is used. While it focuses on the original online version, much of the operation is the same in the new versions:



The tablet versions of Geogebra were announced on Sept. 1st, and are available for Android, iOS, and Windows 8 tablets. Take the time to explore the other links on that page, as there's great support for the program in the form of a community, examples, videos, and much more!

If you already use the Wolfram|Alpha app on your tablet, Geogebra makes a nice compliment to it, for those who enjoy exploring math.

While going through Mental Floss' Be More Interesting columns mentioned in my previous post, their post on how to navigate with stars caught my attention.

I've posted on how to calculate the moon phase for any date in your head, so why not learn more about the rest of the night sky?

The advice in Mental Floss' star navigation post is good as far as it goes. Yes, there are really only a few constellations you need to know to find your way around the sky, but the column stops short of practical teaching.

A website called quietbay.net used to feature a great tutorial on finding the important constellations, but that site has vanished from the internet. Fortunately, the Internet Wayback Machine has come to the rescue!

Here is the archived version of quietbay's clear, visual, and interactive constellation tutorial. It only takes about 15-20 minutes for the full tutorial. Being an archived version, there are a few images missing here and there, and only once or twice are those missing images are essential to finding the stars in the tutorial, but overall, it's still quite workable, and will quickly teach you how to located Polaris, Betelgeuse, Orion, the Big Dipper, Cassiopeia, and even Jupiter, if it's in the sky.

You should also note that it's a northern hemisphere-based tutorial, so the constellation Crux isn't included. Unless you're viewing from the southern hemisphere or the northern tropics, you won't be able to see Crux. If you can see it, Crux is one of the easier constellations to locate.

Try out the tutorial, read the Wikipedia article on Crux, and practice with the real night sky, and you'll be amazed how quickly you can get a good, basic knowledge of the night sky!

UPDATE: This site goes as far back as 2003. This approach was turned into a book in 2010, titled Stikky Night Skies. It teaches 6 constellations, 4 stars, a planet, and a galaxy, and only takes about an hour to read. There is a sample tutorial on the book's website, teaching only about Orion and Betelguese.

If you'd like to learn more in this same way, I highly recommend Laurence Holt's Stikky Night Skies!

Image: © Glenn Francis, www.PacificProDigital.com

I don't often post about physical feats, but when I do, I make sure they're interesting!

Mental Floss has teamed up with Dos Equis, and their famous Most Interesting Man In The World campaign, to develop a series of posts, collectively titled, Be More Interesting.

Instead of the usual mental feats I emphasize here on Grey Matters, these are more physical in nature. They're unusual enough, however, that remembering, practicing, and demonstrating the techniques are definite conversation starters.

The great majority of the techniques are described in posts which can be found in Mental Floss' Be More Interesting gallery.

Many of them are survival skills, such as building a wilderness shelter, winning a duel, and surviving without water for extended periods of time.

There are also numerous “bucket list”-type feats, such as surfing a volcano, or piloting a hot air balloon. The remainder tend to be feat you would demonstrate at a parties or other gatherings, such as picking a lock, or dominating at arm wrestling.

Some of the other techniques require more visual instruction, so Mental Floss has chosen 8 of the feats to teach on video. At this writing, only 5 of them have been recorded, and they're shown at the bottom of this post.

If you like these type of feats, and/or even the usual type of mental feats found in Grey Matters, check out Mental Floss' book Be Amazing and O'Reilly's Mind Performance Hacks. You can also learn similar feats online for free, in places such as Reddit's Learn Useless Talents subreddit, and even Scam School features the occasional stunt.

Do you have any favorite stunt you like to demonstrate that makes you more interesting? I'd love to hear about it in the comments!

How to open champagne with a saber:



How to rip a phonebook in half:



How to start a fire without matches:



How to walk on a tightrope:



How to break a board:

September's snippets are ready!

These mathematical snippets are all designed in intrigue your curiosity while being simple and straightforward.

• Although it seems to be an unusual source for math information, Business Insider recently published an article titled 7 Animated GIFs That Will Make You Instantly Understand Trigonometry. It's a great article, because it doesn't claim to be any more or less than what it is - just clear, informative animations. If you like these, imgur.com has a more complete gallery you may enjoy, as well.

• If you've enjoyed my discussions of the Monty Hall problem in 2006, in 2010, and in 2012, you might enjoy the BBC's article and video on the Monty Hall problem, which they know as the Deal Or No Deal problem.

• I bet that, if I let you choose 5 random integers, I'll be able to find 3 of them that sum to a total which is evenly divisible by 3. What are the odds of me being able to win this bet? The Mind Your Decisions blog takes a look at the surprising answer to this riddle.

• NumberPhile's latest video is oddly amusing. Both James Grime and Brady were sent packages simply labeled Warning: Contains Numbers. The surprise in the package was, well, interesting...

Instead of posting on Thursday, I thought I'd wait until today, so we can enjoy some Friday the 13th fun!

Do I believe in the superstitions about Friday the 13th? No. Without them, however, Friday the 13th is just another day, so they do serve at least one good purpose!

Let's start off with a nod to those superstitions, then, including why even the non-superstitious may need to take them seriously:



How long until the next year without a Friday the 13th? Well, if you look at a calendar, the only time a Friday the 13th can occur is when the 1st of the month falls on a Sunday. Let's take a close look at when each month begins.

January 1st can, of course, fall on any day of the week. We'll refer to this uknown day as d. Now, regardless of which day d is, the first of February happens 31 days later. More accurately, it happens 4 weeks and 3 days later. So, whichever day of the week d is, February will begin 3 days of the week later. Working through the individual months in this way, here's what we find for a non-leap year:

January begins on day d.
February begins on day d + 3.
March begins on day d + 3.
April begins on day d + 6.
May begins on day d + 1.
June begins on day d + 4.
July begins on day d + 6.
August begins on day d + 2.
September begins on day d + 5.

Stop at this point and notice an important feature here. We have months than begin on d, d + 1, d + 2, d + 3, d + 4, d + 5, and d + 6. So, by September of a non-leap year, we must have a month that began on a Sunday. That, in turn means at least one of those months must have a Friday the 13th!

Is this also true for leap years? Let's work through a month of leap years in the same way and find out:

January begins on day d.
February begins on day d + 3.
March begins on day d + 4.
April begins on day d.
May begins on day d + 2.
June begins on day d + 5.
July begins on day d.
August begins on day d + 3.
September begins on day d + 6.
October begins on day d + 1.

In a leap year, it takes until October to guarantee that a month will begin with a Sunday, and therefore to guarantee that a month will have a Friday the 13th.

In other words, if you're waiting for a year without a Friday the 13th, it's not going to happen without another major calendar reform.

Now, as many Grey Matters regulars already know, the Gregorian calendar (the one most people currently use) repeats itself exactly every 400 years. So, September 13, 2413 will also be a Friday the 13th!

Out of curiosity, what is the most common day of the week on which the 13th falls? Over at QED Cat, they took the time to examine all 4800 months, and found that the 13th is more likely to fall on a Friday than any other day! Thursday and Saturday are the least common days, occuring only 684 times each out of 4800 months, while Friday occurs 688 times every 4800 months! Granted, this is only a difference of about 14.25% versus about 14.3%, so it's not like Friday occurs wildly out of proportion to the other days.

I wrap this post up with one final thought. 2013 has 2 + 0 13ths on a Friday. The next one will be in December, exactly 13 weeks from today!

No, I didn't misspell algorithms in the title.

Logarithms are percieved to be very difficult, or even mysterious. If you take the time to understand them, however, they're not mysterious or difficult, and they can even make some things easier for you! Let's get right to the basics.

If you're familiar with exponents, such as 10², logarithms are basically just a way of rearranging the exponent problem in a different way. This video will explain that idea in a little more detail:



While that boils down what logarithms are, the only real use they describe in that video is the pH of pool water, and it doesn't delve into that much. Over at BetterExplained.com, they have a great article on Using Logarithms In The Real World that gives you a better grasp of their importance. It's a must read on the topic!

As you can see in the graphic up in the corner, logs are easily calculated on many calculators, and computers. However, Numberphile has an excellent lesson in working with logarithms without aid from electronics:



Besides showing the use of log tables, this video also introduces one of my favorite aspects of logarithms. It takes most arithmetic operations and simplifies them in an interesting way. Multiplication becomes addition, division becomes subtraction, exponents become multiplication, and roots become division. In short, logarithms can almost seem like cheating, as they turn more difficult operations into simpler ones!

Even though you see seemingly endless pages of log tables in the video above, it is possible to work out logs in your head, and with much less memorization than you might think. I be derelict in my duty at Grey Matters if I didn't show you how to do something along these lines!

Nerd Paradise has an excellent lesson on working out base 10 logs in your head, and the comments feature several great additional tips. CuriousMath.com used to have a similarly great lesson, which has been rescued by the Internet Wayback Machine. The comments here are also quite helpful.

A lesson in working on exponential problems through logs has also been rescued. The 3rd example, involving finding 5th roots, is surprisingly simple, once you've mastered the basics.

You don't even need to do any mental calculation to take advantage of the power of logarithms. Go back and read my Benford's Law post. It features a quick lesson from James Grime and a Scam School video that shows you how to take advantage of Benford's Law in a very clever way.

Take some time and learn about logarithms. There's plenty to discover, learn, and enjoy about them!

I've been having a rough week this week, between being offline for most of Sunday, and dealing with a family emergency the rest of this week.

A little magic always cheers me up, so that's the focus of today's post.

Let's start with a quick opener that's a little different from most Grey Matters fare.

It's this week's episode of Scam School, and there's no math or memory involved. It's just pure, classic sleight-of-hand, developed by Marcus Eddie, who is teaching his SPLINTER! routine:



Now that you're awake and got your splinters removed, let's turn to a little more traditional magic. Not only are playing cards involved, but there is a mathematical basis, so it's probably a little more what you're used to on this site.

Our old friend Card Colm has been experimenting with the Gilbreath Principle. His latest results are in his most recent column, Rosette Shuffling Multiple Piles. It turns out that using a special adaption of the Riffle Shuffle, known as the Rosette Shuffle, it is possible to mix 3 piles of cards together, and still get startlingly predictable results!

I do have to keep this short, due to all I'm dealing with this week, but I hope you found these magical tidbits as enjoyable as I did!