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!

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).
  

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 112²=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 103²=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 178² 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 103²=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 178² is 31,684.

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