### Basics

Before you learn how to determine the moon's phases in your head, you'll need to learn and remember some important moon-related terms, so you can describe each phase.### Moon's Age

No, this doesn't refer to the moon being roughly 4.5 billion years old. It refers to the number of days since the most recent new moon, and ranges from 0 to about 30 days.Over at cycletourist.com, there's a Moon age calculator that uses the date and your time zone to give the exact moon age. Try a few different dates on that calculator, so you can get an idea of the different ages.

### Northern vs. Southern Hemisphere

The way the moon proceeds through its phases appears differently, depending on whether you're in the Northern or Southern hemisphere. Here's how the moon's cycle appears in the Northern Hemisphere:To make this progression more memorable, we'll just focus on stages 3, 5, and 8 from the above diagram:

These phases look like the word “DOC”. The D is shaped like the half-moon, the O is shaped like the full moon, and the C is shaped like the crescent.

In the Southern hemisphere, however, the moon starts its phases from the left side, and proceeds until it disappears on the right, like this:

Looking at only stages 2, 5, and 7 from this picture, we get:

You've probably already guessed that this progression can be remembered using the word COD.

To remember that DOC applies to the Northern hemisphere, use the fact that north is

*up*on most maps, and just think of Bugs Bunny's classic saying, "What's

*up*, Doc?" For COD, recall that a codpiece is designed to be worn in a man's “Southern hemisphere.”

### Events vs. Phases

In the 8 stages shown the above pictures, 4 are really only short one-day*events*, while the rest are part of

*phases*, each of which are about a week long. I'll start with the events, since most people learn about those first.

### Events: New Moon vs. Full Moon

The full moon is simple enough to remember. It's when the moon is as full, or as brightly lit as its going to get in its current cycle, as shown in stage 5 of either of the above graphics.Some people confuse the concept of a new moon with a full moon. If you think of

*NEW MOON*as being the same as

*NO MOON*, that shouldn't be a problem. Image 1 in either of the above graphics shows a new moon.

### Events: First Quarter vs. Third Quarter Moon

The term quarter moon often seems confusing, especially since they refer to what most people think of as a half-moon.*Quarter*, in this context, refers to the point in the cycle, not the shape itself.

Think of quarters in a moon cycle like quarters of a year. After a new year, just like a new moon, comes the first quarter of the year. The third quarter of a year is the last to happen before you head into the next new year, so a third quarter moon is the last you see before the next new moon.

In the above graphics, stage 3 is the first quarter moon, and stage 7 is the third quarter moon. Since the side that's lit in either quarter varies with the hemisphere from which it's being viewed, the DOC and COD mnemonics are very important.

### Phases: Waxing vs. Waning

*Waxing*is the phase in which the moon is getting brighter from night to night. Stages 2, 3, and 4 all show a moon in a waxing phase. If you think of either ear-wax buildup, or waxy yellow buildup on a linoleum floor, this will help you remember that waxing refers to the moon building up, or getting brighter.

*Waning*, on the other hand, refers to the moon getting darker and darker, as in stages 6, 7, and 8 above. To help you remember this, you can think of Lil Wayne, and use Lil to help you recall that waning means that the light on the moon is getting “littler.” Alternatively, you might think of Wayne Newton as “Wayne

*Newmoon*,” to remind you that waning moons are getting closer and closer to the new moon.

### Phases: Crescent vs. Gibbous

The*crescent*phase is easy to recall, especially if you're familiar with crescent rolls or crescent wrenches. When use to refer to the moon, crescent simply means that less than half the moon is illuminated, as in stages 8, 1, and 2 above.

*Gibbous*, on the other hand, is an unfamiliar word to many. It refers to the phase when half or more of the moon is illuminated, as in stages 3, 4, 5, 6, and 7 above. Imagine you're trying to negotiate a deal with someone, and they keep saying, “

*Gibbous*more than half or no deal!”

### Quick Notes Before Moving On

Waxing or waning, when combined with crescent or gibbous, identifies the exact phase of the moon at any given time. The waxing crescent phase runs from the new moon to the first quarter moon. Next comes the waxing gibbous moon, which runs from the first quarter moon to the full moon.After that, there's the waning gibbous, from the full moon to the third quarter moon, and the waning crescent, from the third quarter moon back to the new moon.

The line dividing the dark part of the moon from the light part is known as the

*terminator*, which can make from some amusing sci-fi movie references.

Once you're comfortable with all the terms, you're ready to start learning the formula for determining the moon phase in your head, which begins in the next tab.

### About The Formula

The moon phase calculation formula has much in common with the*Day of the Week For Any Date*method taught here in the Mental Gym, including the developer of the formula, John Conway.

Just like the day of the week feat, you're going to translate the year, date (1-31), century, and year into key numbers, and add them up.

In the day of the week feat, you then performed a modulo 7 operation on this number, but for the roughly 30-day moon cycle, you're going to perform modulo 30. For a clear explanation of this concept, read BetterExplained's post,

*Fun With Modular Arithmetic*.

Unlike the calendar formula, the moon phase formula is not exact. The real moon's orbit is quite complex, and exact calculations include corrections for every detail.

The formula you'll learn is simple enough to do in your head, and will be accurate to within about a day or so. You'll learn how to deal with this in the

*Presentation*section of the Practice tab.

### Month Key

The month key is very easy to remember. For March to December, you simply use the standard month number. For example, March's key number is 3, April's key number is 4, and so on up to December's key number, which is 12.January and February aren't hard to recall. January's key number is 3, and February's key number is 4. If you can remember the month numbers, and these numbers for the first two months, you'll be able to remember the year key without trouble.

### Date Key

If you thought the month key was easy, the date key is even easier! All you use is the number of the date, 1 through 31, as the date key.So, your first step is to look at the month and date, and add the two keys together. July 12th? That's 7 + 12 = 19. February 7th? That's 4 + 7 = 11. January 19th? That's 3 + 19 = 22.

Once you have a total of the month and day keys, it's usually a good idea to add 30 to the total before working with the century and year keys. With those, you're going to be doing some subtracting, and adding 30 will help ensure that the running total remains a positive number.

When you're comfortable getting and adding the month and day keys, it's time to learn the year and century keys.

### Century Key

When performing this feat, it's best to use a date that's meaningful, such as a birthday or anniversary. So, all you really need to know are keys for the 1900s and 2000s.If you're given a date in the 1900s, you simply subtract 4 from the running total. If you're given a date in the 2000s, you'll subtract 8 instead. That's it!

### Year Key

Unlike the previous keys, there are a few steps to working out the year key. Don't worry, they can still be done in your head.First, you'll need to know your multiples of 19: 0, 19, 38, 57, 76, and 95.

This is because the first step in working out the year key is finding out how close you are to the nearest multiple of 19, whether that multiple is more or less than the year number.

Let's use 2008 as an example, focusing on the year as simply 8. 8 is between 0 and 19, but obviously closer to 0. To get 8 starting from 0, you'd add 0 + 8, so think only of +8 at this stage.

It's important to note that this was just the first step to finding 2008's key, so +8 is the number we need for the 2nd step, not the key itself.

Before moving on to the 2nd step, though, let's try another example year, such as 2013. Working with 13, we see that it's between 0 and 19, but closer to 19. Starting from 19, you'd get to 13 by working our 19 - 6, so 13's number at this stage should be thought of as -6.

This is what I mean about finding the distance to the closest multiple of 19. 8 was 8

**than 0, so we think of it as**

*MORE***8. 13 was 6**

*PLUS***than 19, so we think of 13 as**

*LESS***6.**

*MINUS*What about, say, 59? That would be +2, because 57 is the nearest multiple of 19, and 57 + 2 = 59. How about 88? That becomes -7. Do you understand why?

This first step seems complicated when you first learn about it, but with practice, it's not hard to determine the number you need quickly and easily. Regardless of the year, notice that you'll always get a number ranging from -9 to +9 as a result.

Now it's time to explain the second step. First, I'll explain the long way of working through this step, then I'll give you a shortcut that gives the same result.

**LONG VERSION:**Whether the number you got in the first step was positive or negative, you're going to work through this step as if it's a positive number, and then restore the proper sign at the end.

Going back to our 2008 example, we got +8 in the first step, and we remember that plus sign. Next, we multiply that by 11, and get 88. Finally, 88 modulo 30 is 28, and we recall that the original number was positive, so we know that the 28 is positive, as well. This is the final number we need, so we would

*add*28 to our running total.

What happens with 2013? Applying the first step to 13 gave us -6 (remember?), so we'll work through this process as just 6, remembering the minus sign for later. 6 × 11 = 66, and 66 modulo 30 is 6. Restoring the minus sign from earlier, we get -6. This simply means that we'd

*subtract*6 from our running total.

**SHORTCUT:**Let's run all the numbers from -9 to +9 through the above process, and see if we can find some patterns:

The first thing to notice here is that neither the sign nor the ones digit (the rightmost digit) change. It's also easy to see that the multiples of 3 don't change at all.

If you think of the multiples of 3 as being 2-digit numbers (±03, ±06, and ±09), then you can also see that the tens digit of each number is the ones digit modulo 3. For example, 7 modulo 3 is 1, so we just put a 1 in front of a 7 to make 17.

Just like in the long version, when you do the modulo 3, you need to temporarily drop the sign, then put it back afterwards, for this formula to work out properly.

Let's work through the shortcut method with a year ending in 53, as an example. In the first step, 53 becomes -4 (you should still understand why). We temporarily drop the minus sign, and perform 4 mod 3 to get 1, so we put a 1 in there to make -14. This means we need to subtract 14 from our running total.

The more you practice and perform this feat, the more quickly these numbers will come to mind. Sooner than you expect, the numbers will come from your memory quicker than from calculation.

Now that you understand how to properly figure the year, date, century, and year keys, it's time to put them all together and practice, which we'll do in the next tab!

### Getting Results

Now that you understand how to find each key, it's time to put them all together and get the final result!Simply add each of the key numbers together. If the number is greater than 30, take the answer modulo 30.

You'll wind up with a number ranging from 0 to 29, and this number represents the moon's age in days. The chart below shows how the age corresponds to the four major moon events:

To see how this works, let's work through an example date, April 1, 1994. April gives 4 and the 1st gives 1, so that's a total of 5. As I mentioned earlier, I usually like to add 30 at this point to prevent working with negative numbers, so 5 + 30 = 35.

The key for the 1900s is -4, so 35 - 4 = 31. Running 94 through the year key process gives us -11 (Because 95 - 1 = 94, and -1 * 11 = -11, remember?), so 31 - 11 = 20. In this case, 20 is below 30, so we don't have to work out modulo 30.

Our estimate, then, is that the moon's age is

*about*20 days old (±1 day). That suggests we're about 5 days past the full moon, so the moon would be getting darker each night, but a majority of the moon's face would still be illuminated. In technical terms, the moon was in a waning gibbous phase.

If you enter April 1, 1994 into Wolfram|Alpha and scroll down to the bottom, you can verify that the moon was indeed in a waning gibbous phase on that date!

As a matter of fact, Wolfram|Alpha provides a great way to test your ability to work through this formula. Get a random date in the 20th or 21st century by clicking here, then use that date to work through the moon phase formula in your head. When you've got an answer, click the date in the

*Result*pod, and scroll to the bottom to see the actual phase of the moon on that date.

### Presentation

Due to the margin of error of this formula, you can't pin down the specific events (new moon, 1st quarter moon, full moon, and 3rd quarter moon) exactly. Since the various phases are each about a week long, and overlap, you can use this to your advantage.I often use the difference in technical terms used by astronomers (and on Wolfram|Alpha) in contrast with an informal description of the moon, including how average people would refer to the moon. This is usually enough to be considered accurate and verifiable with an outside source.

Don't forget the importance of having an outside source to verify your claim on the moon. If you can't have someone look up the date online, at least carry an almanac or other reference in which your audience can verify the moon phase.

If you make sure people verify your claim with Wolfram|Alpha, you can also usually state your estimate of the moon's age openly. Wolfram|Alpha, at least at this writing, doesn't include this information.

Here's a general idea of what to say for any given estimate:

**29, 0, or 1 days:**

*“On that date, the moon was in a crescent phase, as the 3rd quarter moon happened about a week ago. In fact, so little of the moon is lit, most people would just refer to it as a new moon.”*

**2 through 6 days:**

*“The new moon was just about (2, 3, 4, 5, or 6) days before that date, so it's getting brighter every night, but less than half the moon is lit on the right side. It's known as the waxing crescent phase.”*

**7 or 8 days:**

*“It's been about a week since the new moon, so the moon is in a waxing phase - in other words, getting brighter each night. The average person might even call it a half-moon or 1st-quarter moon.”*

**9 through 13 days:**

*“The 1st quarter moon was about (1 to 2, 2 to 3, 3 to 4, 4 to 5, or 5 to 6) days before that date, so the moon is getting brighter each night, and more than half is lit. This is called the waxing gibbous phase.”*

**14, 15, or 16 days:**

*“It's been just about a week since the 1st quarter moon, and so much of the moon is lit that most people will say things like, 'All the crazies are out, it's a full moon!'”*

**17 through 21 days:**

*“Just about (1 to 2, 2 to 3, 3 to 4, or 4 to 5) days before that was the full moon, so most of the moon is still illuminated, but it's getting darker each night. The technical term for this is the waning gibbous phase.”*

**22 or 23 days:**

*“On that date, it's been about a week since the full moon, so around half of the left side of the moon is illuminated. Many would refer to this as a half-moon or 3rd quarter moon.”*

**24 through 28 days:**

*“Around (1 to 2, 2 to 3, 3 to 4, 4 to 5, or 5 to 6) days before that date was the 3rd quarter moon, so the moon is getting darker each night, and less than half the moon is illuminated. Astronomer call this the waning crescent phase.”*

The above statements were written with the assumption that the speaker was talking about the Northern hemisphere. For the Southern hemisphere, just switch all the references to left and right.

Once you've practiced both the formula itself and the presentation of your moon estimate, you're ready to perform this feat for someone else.

There are many more ways to use your newfound knowledge of the moon, and you can learn about those in the

*Tips/Tools*tab.

Additionally, you can learn more about the formula itself in the

*Why?*tab.

### Tips and Tools

• This routine works well with most versions of the*Day of the Week For Any Date*feat. I teach one version here in the Mental Gym, as well as a quicker and simpler version called

*Day One*, for sale at Lybrary.com.

• Be alert to key number combinations that total 0, to cut your calculation time:

- Years ending in a multiple of 19 (00, 19, 38, 57, 76, 95): 0 (year) = 0
- The 4th of any month in the 1900s: +4 (date) -4 (1900s) = 0
- The 8th of any month in the 2000s: +8 (date) -8 (2000s) = 0
- Any February or April in the 1900s: +4 (month) -4 (1900s) = 0
- Any August in the 2000s: +8 (month) -8 (2000s) = 0
- Use the Moon Century/Year Combo Key Generator widget below to find years that require no adjustment, such as 2017.

• Instead of trying to describe the moon, you might try having images of 10 to 12 different phases of the moon on your mobile device. Since you have less than half of the 30 different images, it's reasonable that the photo you're showing isn't an exact match to the moon. There's even a moon-shaped toy called

*Moon In My Room*that can display 12 different phases of the moon, which is perfect for this presentation.

• If you want to work with other centuries, I've developed a Wolfram|Alpha widget that will work out the century keys for anywhere from the 1800s to the 3000s. Simply select the century from the menu, then click

*Submit*.

The widget will return 2 numbers,

*r*and

*s*. You can use either one of these as the key for your chosen century, but one if often simpler to use than the other. For example, selecting the 2900s will return

*r*= 6 and

*s*= -24. This means that you can use either +6 or -24 as your key for the 2900s. In this example, adding 6 is the easier way to go.

• Should you decide to focus on a single year, you can use the widget below to get a single key that compensates for the century and the year at once. If you can memorize several keys, you can use the widget to lookup each year and that range.

Use the pull-down menus to select any year from 1800 to 3000, and then click the

*Submit*button. Like the above widget, you're given two choices,

*r*and

*s*, and you can decide which one is easier to use. You can ignore the variable

*y*, as it's simply the last two digits of the year you input.

• If you have

*Day One*, and you're familiar with the Major/Peg System for memorizing numbers, it's not hard to develop a

*Day One*-type approach to the Moon Phase feat.

Use the Moon Century/Year Combo Key Generator widget above to determine keys for each of the years 2000 through 2018, and link from the same year images I use in

*Day One*to the Major/Peg images for each adjustment (I add a spear to the peg image to represent the minus sign, so I know when to subtract). The years 2019 - 2099 can be reduced to these images via modulo 19, similar to the use of modulo 28 in

*Day One*. To adjust for a date the 1900s, simply add 4.

I jokingly refer to this approach as

*Night One*.

• Here are some excellent videos that can provide helpful background knowledge:

- The Universe: The Phases of the Moon
- NASA | Moon Phase & Libration: Moon With Additional Graphics - The moon as it will appear all throughout 2013, with 1 frame representing 1 hour!
- How Far Away is the Moon? (The Scale of the Universe) - If you think that scale is amazing, ask yourself how big Jupiter would be if it were shrunk to the same scale.
- What are years? - Related: What is a Leap Year?

• If you'd like to understand more about each part of the estimation formula itself, you can learn more about it in the

*Why?*tab.

### Why?

While some may prefer to use the formula without understanding how it works, I've added this section for people who want to understand it better.**If you were to draw a line from the center of the Earth through the center of the moon to a fixed star, it would take about 27.321661 days (27 days, 7 hours, 43 minutes, 11.5 seconds, known as a sidereal month) to do so. However, it wouldn't return to the exact same phase, since the Earth has moved roughly 45 million miles in that same amount of time.**

*Why Modulo 30?*To return to the same moon phase takes longer, an average of about 29.530588 days (29 days, 12 hours, 44 minutes, 2.9 seconds, known as a synodic month).

The phase of the moon in this 29.530588-day cycle is approximated through the use of modulo 30, which is easier to work with mentally.

**When estimating moon phases, it's useful to know how long it will be before the moon will be in the same phase on the same date. As it turns out, the answer is surprisingly accurate 19 years and 2 hours! This is known as the Metonic cycle.**

*Why Focus on Groups of 19 Years?*Over a 100-year period, the difference is only about 10.5 hours, so working in groups of 19 works very well in the estimate. The estimation itself adjusts every 100 years, and takes this into account.

**If the moon revolved exactly 12 times around the Earth every tropical year (the technical name for the year as most people know it), we wouldn't need to make any adjustment for years, as the moon would always be in the same phase on the same day.**

*Why Multiply the Years by 11?*If it took the moon, say, 2 days short of a tropical year to revolve exactly 12 times, then we'd have to adjust by adding 2 days for every year. This would be done simply by multiplying by 2.

It turns out that the time required for 12 revolutions of the moon is about 10

^{7}⁄

_{8}days short of a tropical year. This is approximated in the estimation formula through multiplying by 11.

**Over the 19-year Metonic cycle, that small**

*Why Do Years Range From -9 to 9, Instead of 0 to 18?*^{1}⁄

_{8}day difference can start to add up. 18 years ×

^{1}⁄

_{8}day =

^{18}⁄

_{8}day = 2

^{1}⁄

_{8}days!

When originally developing the moon phase estimation formula, John Conway worked out the rather ingenious -9 to +9 approach you learned. What's so ingenious about it?

Go to this Desmos Calculator page, and drag the slider back and forth from 0 to 18 years. The red bar shows the

^{1}⁄

_{8}-day difference building up over that time. The numbers on the y-axis (the vertical axis) represent the total number of days in error.

By comparison the blue bar, representing the -9 to +9 approach, through the first 9 years, and then begins to decrease dramatically as you add more years!

Hopefully, this gives you a better idea of how this reduces the margin of error.

### What is the Margin of Error?

With all the approximate numbers we're using, the resulting accuracy of this estimation is quite amazing.To give you an idea of just how accurate it can be, I've provided the following calendar. You can use the drop-down menus to select any month from January 1900 to December 2099, and the calendar will instantly change.

*Est.*, in each day, represents the estimated age of the moon you get by working through the method in this tutorial.

*Real*is the actual age of the moon, using calculations from

*Practical Astronomy with Your Calculator*, 3rd edition (The newest edition includes spreadsheet calculations), modified from javascript code by Janet L. Stein Carter.

As I've modified them, these calculations reflect the phase of the moon as it appears at noon in Greenwich Mean Time on each given day. The moon's age in degrees, from 0° to 359°, is converted into its age in days in the synodic month, and then rounded to the nearest whole number.

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