On August 21, 2017, there will be a total solar eclipse visible in the United States, which hasn't happened since 2012, and won't happen again until 2024!

It's the perfect time to present an impressive knowledge of the moon, such as being able to estimate the moon phase for any date in 2017! Does learning this feat sound difficult? Surprisingly, it's much easier than you may think.

For 2017 only, the phase of the moon formula is simple: (Month key number + date) mod 30. The result is the age of the moon in days from 0 to 29. I'll explain each part of this formula below.

**Month key number:** January's key number is 3, February's key number is 4, and all other months' keys are their traditional numbers; March is 3, April is 4, May is 5, and so on up to December, which is 12.

**Date:** This is simply the number represented by the particular date in the month. For the 1st, add 1. For the 2nd, add 2. For the 3rd, add 3, and so on.

**mod 30:** If you get a total of 30 or more, simply subtract 30. Otherwise, just leave the number as is.

As an example, let's try the published date of this post, July 28th. July is the 7th month and the date is the 28th, so we work out 7 + 28 = 35. Since that number is greater than 30, we subtract 30 to get 35 - 30 = 5. This result tells us that the moon's age will be 5 days on July 28th, 2017.

But what does it mean to say that a moon is some number of days old? Here's a simple explanation:

**0 days** = New moon (the moon is as dark as it's going to get)

**0 to 7.5 days** = Waxing crescent (Less than half the moon is lit, and it's getting brighter each night)

**7.5 days** = 1st quarter moon (Half the moon is lit, and getting brighter each night)

**7.5 to 15** days = Waxing gibbous (More than half the moon is lit, and getting brighter each night)

**15 days** = Full moon (The moon is as bright as it's going to get, and will start getting darker each night)

**15 to 22.5** days = Waning gibbous (More than half the moon is lit, and it's getting darker each night)

**22.5 days** = 3rd quarter moon (Half the moon is lit, and it's getting darker each night)

**22.5 to 29 days** = Waning crescent (Less than half the moon is lit, and it's getting darker each night)

So, our 5-day-old moon would be a waxing crescent (Less than half the moon is lit, and it's getting brighter each night). Sure enough, Wolfram|Alpha confirms this estimate! Moon Giant confirms this estimate, as well.

There are a couple of finer points to note. First, this simple method happens to work only in 2017. The method won't be this simple again until 2036 and then 2055. If you want to learn to calculate the moon phase for any date in the 1900s, you can learn the full feat over in the Grey Matters Mental Gym.

Second, remember that this is an estimate. The actual error margin is plus or minus one day. So, getting an estimate of 5 days means that the moon is somewhere from 4 to 6 days old. When the overlap includes a first quarter moon, a full moon, or a 3rd quarter moon, I usually describe this as, "While it might not be technically accurate, most people would look up and describe it as a 1st quarter moon" (or full moon, or 3rd quarter moon).

This method is a simplification of John Conway's original moon phase estimation formula from *Winning Ways for Your Mathematical Plays*, vol. 4. Practice it and have some fun amazing your friends and family during the coming solar eclipse!

## Solar Eclipse Mental Feat!

Published on Friday, July 28, 2017 in fun, math, memory, mental math, self improvement

## Review: Perfectly Possible

Published on Sunday, July 09, 2017 in books, fun, magic, magic squares, math, mental math, reviews, self improvement

Many regular Grey Matters readers will be familiar with Michael Daniels' *Mind Magician* site, where he teaches numerous math and memory feats, such as calculating cube roots in your head instantly. He's recently written a new ebook on the 4-by-4 magic square, titled *Perfectly Possible*. I found it to be well worth the time and money invested, and wanted to share my thoughts with Grey Matters readers.

This is going to be a difficult review, as I can't give too much away, but I also want to share with you the quality of this method. I'll start with the qualities promoted by Michael Daniels himself:

- Completely impromptu. No set-up, gimmicks, or cribs.

- New, improved method - minimal memory and the simplest of calculations.

- Suitable for close-up or stage performances.

- Produces elegant magic squares.

- Can be immediately repeated for different totals.

- Includes a browser application that helps you to learn and practice (Internet connection not required).

What does "elegant magic squares" mean? One problem with many magic square approaches is that the number can appear unbalanced, such as when 12 of the numbers are less than 15, and the other 4 are over 30. This can give your audience clues about the method. With the

*Perfectly Possible*method, you don't have to worry about that. You're guaranteed a balanced magic square. Elegant also means that you're guaranteed at least 36 different ways in which some combination of 4 squares gives the magic total. Under the right circumstances, this method can yield as many as 52 different combinations!

As with any magic square, the ability to repeat the square immediately with different totals is, of course, essential. Even more impressive, though, is that if 2 people give you the

*same*total, you can still generate a different magic square! Naturally, the same total requires the numbers used to be in the same general range, but this method will allow you to put different numbers in each of the squares with very little difficulty.

That quality is really what makes

*Perfectly Possible*stand out. Unlike the rigid approaches behind most magic squares, the ability to take multiple approaches gives the performer more freedom while disguising the method very effectively. When a change is as constrained as the magic square, finding an approach like this that offers you remarkable degrees of freedom like this is incredible!

If you're interested in creating magic squares, I can't recommend Michael Daniels'

*Perfectly Possible*ebook enough. It's available for $6 on its own, or $8 in combination with

*Mostly Perfect*, its predecessor. If you're seriously consider this as a performance piece, I would also recommend the Unknown Mentalist's

*Why A Magic Square Should Not Be A Magic Square*ebook. It teaches many very effective original presentations that disguise the principle, and will help preserve the mystery by showing you how to prevent audiences from simply searching for "magic square" on the internet during or after your performance.

## Squaring Numbers from 100-199

Published on Sunday, July 02, 2017 in fun, math, mental math, self improvement

Over in the Mental Gym, I have a tutorial on squaring numbers, starting with simpler techniques for multiples of 10 and 5, and working up to squaring numbers as large as 125.

Naturally, I always like to see how much farther I can go, especially when I can still keep things relatively simple. With the technique I'll be teaching in this post, if you're comfortable with squaring numbers up to 125, you're ready to move on to squaring numbers up to 199 (Well, actually 200, since that's not difficult to square).

**Rightmost Two Digits**

We're going to generate the answer from right to left in this technique, working with no more than 2 digits at a time. To get the rightmost 2 digits of the answer, simply square the rightmost 2 digits of the given number. For example, if you're given the number 112 to square in your head, you'd square the rightmost 2-digits of that number, 12, to get 144. You write down the rightmost 2 digits, 44 in this example, and remember the remaining digits to the left, such as the 1 in this example (underscores are used to hold places for numbers not yet written):

**Middle Two Digits**

For the middle 2 digits, simply double the rightmost 2 digits of the given number, then add the number you remembered, if any. Write down the 2 rightmost digits of this answer to the left of the digits previously written, and remember any digits to the left of that. Returning to our 112 example, we look at the rightmost 2 digits, 12, and double that to get 24. Adding the number we remembered from the previous step, 1, we get a total of 25. The rightmost 2 digits of this answer are 25, and there's no digits to the left of those to remember:

**Leftmost Digit(s)**

For the leftmost digits, take any amount you have remembered at this point, and simply add 1 to it. Write down that total to the left of all the digits you've previously written, and you're done! In our 112 example, we didn't remember anything for this stage, so we just write down 1, resulting in:

You can check for yourself that 112^{2}=12,544.

**Tips**

**Single-digit numbers:** When working on either the rightmost 2 digits or the middle 2 digits, you may wind up working with a single-digit answer. These steps *always* require working with 2 digits, so for single-digit answers, just place a 0 to the left of it to make it a 2-digit number.

For example, when squaring 103, you start by squaring 3 to get 9, with nothing to remember:

Next, you'd double 3 to get 6, again with nothing to remember:

Since you don't remember anything from the previous steps, just add a 1 to the left of this answer:

This tells us that 103^{2}=10,609.

**Remembering multiple digits:** Just so you have an example of working with larger numbers, let's try squaring 178. Start by squaring 78, which is 6,084. Write down the 84, and remember the 60:

Next, double 78 to get 156, and add the 60 you remembered from the previous step, giving a total of 216. Write down the 16, and remember the 2:

Finally, add 1 to the number you're remembering and write that down to the left of the previous digits. In this case, we add 1 and 2 to get 3:

The result of 178^{2} is 31,684.

**Interest for 2 time periods:** This technique is especially handy for quickly calculating how many times the principal will grow at interest rates of 99% or less for 2 time periods. The only additional step is to put a decimal point between the ten-thousands and thousands place. For example, since 103^{2}=10,609, that means that principal invested at 3% per year for 2 years will grow to 1.0609 times its original size. Although you're not likely to ever see it happen, principal invested at 78% per year for 2 years would grow to 3.1684 times its original size, because 178^{2} is 31,684.

Practice this, have a little fun with it, and you'll have an impressive new skill!

## Magic, Math and Memory Videos!

Published on Sunday, June 25, 2017 in Knight's Tour, magic, math, memory, memory feats, self improvement, videos

I recently ran across a number of videos I figured would be interesting to regular Grey Matters readers, so I thought I would share them.

We'll start things off with a little math magic, courtesy of Tom London and his appearance on *America's Got Talent* earlier this week:

Yes, I could explain the method, but I don't want to ruin the fun and the mystery. Just enjoy the magic of the prediction for what it is, since that's how it's meant to be enjoyed.

If you want mathematical explanations, however, I highly recommend checking out PBS Digital Studios' *Infinite Series*. These are videos on assorted advanced mathematical topics, yet they're taught in a very accessible way. Back in March, I discussed a puzzle which required the understanding of Markov chains to solve. Compare that to their video *Can a Chess Piece Explain Markov Chains?*, which also happens to employ my favorite chess piece, the knight:

If you enjoy Grey Matters, you may also the work of 4-time USA memory champion Nelson Dellis, who focuses on both mental and physical fitness. He has a series of memory technique videos, as well as interviews with masters of mental skills. Both of these are available on his YouTube channel, as well. As a taste of his skilll, watch his video, *Memorizing 28 names in less than 60 seconds!*:

Curious how he's able to do that? He explains in the next video in the series, *HOW TO // "Memorizing 28 names in less than 60 seconds!"*.

At this point, I'll wrap things up so you can get started on a potentially mind-expanding journey.

## Yet Still More Quick Snippets

Published on Sunday, June 11, 2017 in magic squares, math, snippets, videos

It's now been a few months since Grey Matters was back, so now it's time to bring Quick Snippets back!

This time around, we have plenty of mathy goodness, so it's best to just jump right in!

• Besides the Clay Institute's famous selection of Millennium Problems, which will make you a millionaire if you prove or disprove any one of them, there's a lesser known set, known as John Conway's $1,000 problems. Not long ago, the 5th conjecture, which claims that working through a certain procedure (described in the link an video below) will always end in a prime, was disproven by physicist James Davis. The Numberphile video below details the problem and James Davis' counterexample:

For more about the million-dollar Millennium Problems, watch the BBC's *Horizon* documentary, "A Mathematical Mystery Tour of Unsolved Mathematical Problems."

• Speaking of fun discoveries in recreational mathematics, check out Allan William Johnson's "Magic Square of Squares", discovered back in 1990, and just recently posted over at Futility Closet.

• James Grimes introduces us to a different sort of "Square of Squares", in his latest Numberphile video, "Squared Squares". The challenge here is to make a perfect square shape from a set of smaller square shapes:

• Presh Talwalkar, of Mind Your Decisions, posted an interesting puzzle recently. It's titled, "The Race To 32,768. Game Theory Puzzle". Read the article up to the point where you're challenged to work it out yourself, or watch the video below up to the 2:12 mark, and try and figure it out for yourself. If you get stuck, try going over my *Scam School Teaches the Game of 15* post for inspiration.

• Late last month, mathematical video maker 3Blue1Brown posted a must-watch video on the visualization of all possible Pythagorean triples. Even if you remember everything from you math classes about Pythagorean triples, this video is both eye-popping and an eye opener:

• We'll wrap this set of Quick Snippets up with help from Mathologer. His videos are always interesting, but his latest one is one of those unusual approaches to math that makes you appreciate its beauty. This video is titled, "Gauss's magic shoelace area formula and its calculus companion", and it teaches an simple but unusual method for working out the area of any polygon that doesn't intersect itself. The host even goes on to show how this approach can be adapted in calculus to work out the area contained by curves!

## Tour the U.S.!

Published on Sunday, May 28, 2017 in fun, Knight's Tour, puzzles, self improvement

*Starting in Delaware, you must tour the 48 contiguous United States, visiting each state exactly once. Where will you finish?*

That's a puzzle from Futility Closet. I'll link directly to it later, so as not to get too far ahead. It's really a puzzle within a puzzle, however.

As anyone who has tried the Knight's Tour knows, moving around with the limitation of landing in each space only once can be quite a challenge.

The original Futility Closet puzzle can be solved either by thinking about that puzzle on its own with a little analysis. However, I think it might be a little more fun to try and solve the puzzle by trying various ways to get around the 48 contiguous United States, visiting each state exactly once.

To that end, I've written this as a puzzle you can play below. It's an interactive map, so you can scroll (by clicking and dragging outside of the U.S.) and zoom (using the + and - in the upper left corner of the map) as needed, which helps when you're trying access smaller states.

You start by clicking on any state. That state will turn blue, representing your current location. Some states will also appear in green, and these are states which border your current state, and to which you haven't already traveled.

To move to a new state, simply click on any one of the green states. Clicking on any other states won't have any effect. Once you click a green state, that state will become blue (again, denoting your current location), and a new set of bordering states will become green. Also, the state which you just left becomes red, which denotes that you can never move to that state again.

Quick side note: The way I've programmed it, you can't move directly from Arizona to Colorado or directly from New Mexico to Utah, as I didn't consider meeting at a single point to be a bordering state.

If you want to work on the original Futility Closet puzzle, zoom in on Delaware, and click on it to start. Otherwise, start on any state. Try and see if you can get to all 48 states just once each time. You'll get an alert if you've either solved the puzzle, or become trapped. In either case, the map will go blank and you can try again.

Even if you don't get all 48 states, try and beat your highest number of states each time. As you play, perhaps you'll even get a realization that will help you solve the original Futility Closet puzzle.

The more you try this, the better you'll get, and you'll probably be able to solve the 48-state U.S. tour puzzle and the Futility Closet puzzle before too long. At this point, you can challenge your friends, and show your skills at solving the tour. What kind of experience did you have solving the puzzle? I'd love to hear about it in the comments below!

## The Collective Coin Coincidence

Published on Sunday, May 21, 2017 in fun, magic, math, money, psychology, Scam School, videos

This week, Diamond Jim Tyler demonstrates a new take on an old trick. Regular Grey Matters readers won't be surprised to learn that I like it because it's based on math, and it's very counterintuitive. We'll start with the new video, and then take a closer look at the trick.

This week's Scam School episode is called *The Collective Coin Coincidence*, and features Diamond Jim Tyler giving not only a good performance, but also a good lesson in improving a routine properly:

Brian mentions that this was an update from a previous Scam School episode. What he doesn't mention is that you have to travel all the way back to 2009 to find it! The original version was called *The Coin Trick That Fooled Einstein*, and Brian performed it for U.S. Ski Team Olympic gold medalist Jonny Moseley. It's worth taking a look to see how the new version compares with the original.

Brian and Jim kind of rush through the math shortly after the 4:00 mark, but let's take a close look at the math step-by-step:

Start - The other person has an unknown amount of coins. As with any unknown in algebra, we'll assign a variable to it. To represent coins, change or cents, we'll use: c

1 - When you're saying you have as many coins (or cents) as they do, you're saying you have: c

2 - When you're saying you have 3 more coins than they do, the algebraic way to say that is: c + 3

3 - When you're saying you have enough left over to make their number of coins (c) equal 36, that amount is represented by 36 - c, so the total becomes: c + 3 + 36 - c

Take a close look at that final formula. The first c and the last c cancel out, leaving us with 3 + 36 which is 39. If you go through these same steps with the amount of coins (in cents, as it will make everything easier) as opposed to the number of coins, it works out the same way. This is what Diamond Jim Tyler means when he explains that all he's saying is that he has $4.25 (funnily enough, he says that just after the 4:25 mark).

As long as we're considering improvements, I have another unusual use for this routine. If you go back to my *Scam School Meets Grey Matters...Still Yet Again!* post, I feature the Purloined Objects/*How to Catch a Thief!* episode of Scam School, which I contributed to the show. It's not a bad routine as taught, but my post includes a tip which originated with magician Stewart James. This tip uses the Coin Coincidence/Trick That Fooled Einstein principle to take the Purloined Objects into the miracle class! I won't tip it here, so as not to ruin your joy of discovery.