Moon Phase For Any Date

Published on Thursday, January 03, 2013 in , , ,


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 MORE than 0, so we think of it as PLUS 8. 13 was 6 LESS than 19, so we think of 13 as MINUS 6.

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.


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:
• 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.


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.

Why Modulo 30? 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.

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.

Why Focus on Groups of 19 Years? 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.

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.

Why Multiply the Years by 11? 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.

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 78 days short of a tropical year. This is approximated in the estimation formula through multiplying by 11.

Why Do Years Range From -9 to 9, Instead of 0 to 18? Over the 19-year Metonic cycle, that small 18 day difference can start to add up. 18 years × 18 day = 188 day = 218 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 18-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.


Mental Division: Decimal Accuracy

Published on Sunday, July 29, 2012 in ,


While there are many mental math sites and videos freely available on the internet, very few seem to focus on division.

That's likely because division rarely comes out nice and neat, like addition, subtraction, and multiplication do. You have to deal with “leftovers” in the forms of remainders, fractions, or decimals.

Fractions are often seen as the most human-friendly way of expressing left-over numbers. Consequently, being able to give exact decimals, especially for unusual fractions such as 5/7 or 13/15, seems like an amazing feat to most people.

This tutorial will focus solely on division problems whose results are less than 1. For example, you'll learn how to deal with 5 ÷ 7 (or the equivalent fraction 5/7), but not how to deal with, say, 16 ÷ 7 (or 16/7).

Fortunately, there are numerous patterns of which we can take advantage, that will help make the task much easier than many people would suspect. You can start learning about these patterns in the next section.

Fractions to Memorize

When dealing with smaller divisors, such as those from 2 to 11, it's quite useful to have the decimal equivalents of all such problems less than 1 memorized.

You probably already know all the answers for dividing by 2 through 4 already:


1/3=.333... (the dots are used to refer to the endless repetition of digits)


Dividing by 5 is very easy. You just double the numerator, and put a decimal point in front of it:


You also already know how to divide most of the numbers by 6, as well:


All you have to know is two more 6ths:


I'll come back to the 7th, but for now, I'm going to cover the 8ths, since they're not much more difficult than 4ths. The trick here is to multiply the numerator by 125, and stick the decimal point in front of it:


The last one almost everyone is familiar with is how to express 10ths:


New Patterns

To be able to memorize every fraction/division problem up to dividing by 11, then, all most people need to learn is how to handle 7ths, 9ths, and 11ths.

9ths are especially easy, as you simply start with the decimal point, and repeat the numerator endlessly:


If you know your multiples of 9 up to 10, then you can handle 11ths. Simply multiply the numerator by 9, and express that as a 2-digit number (9 × 1 = “09”). Repeat these 2 digits endlessly, and you've got your 11ths:


Finally, there's the 7ths. The 7ths have a very unusual pattern, starting with 1 ÷ 7:


Notice that same sequence of numbers, 142857, repeats over and over again. To make things even easier, every 7th has this same repeating pattern. The only thing that changes is which number comes immediately after the decimal point. Here are the 7ths:


To determine which number comes first, simply multiply the numerator by 14, and note the digit in the 10s place. That will be the digit that comes first. For example, with 3/7ths, you'd do 3 × 14 = 42, and since 4 is the digit in the tens place, you'd start the pattern with 4, giving you .42857142857... and so on.

Once you're comfortable with these basic memorized patterns, you're ready to move full speed ahead to the next section, where you'll learn to divide by numbers near 100.

Dividing by 100

Dividing a 2-digit number by 100 is easy, of course. Take the numerator, place the decimal point in front of it, and you're done. 46/100? .46! 67/100? .67!

Dividing by 99

Surprisingly, dividing a 2-digit number by 99 is almost as easy as dividing by 100. The only difference is that the numerator repeats without end. 46/99? .464646...! 67/99? It's .676767...!

Think of 99 as being like 100, but since the 9s repeat, so does the numerator.

Dividing by 90

The technique for dividing any number from 1 to 89 by 90 takes us back to grade school math. Remember learning long division and getting answers like “3 remainder 7”? That's the type of thinking you'll need here.

To divide a 2-digit number by 90, just divide the numerator by 9, but work out the answer in the “x remainder y” format. Once you know that, x will be the first digit after the decimal point, and y repeats after that.

For example, let's work out 58 ÷ 90. Ask yourself, what is 58 divided by 9? Assuming your know your multiplication tables, you should think of it as 6 remainder 4. That means the decimal equivalent of 58 ÷ 90 is .6444....

Remember this trick for 90 by thinking of the way you divided by 9 when you had 0 knowledge of fractions.

Dividing by 91

To dividing by 91, you'll first need to know how to multiply any number from 1 through 90 by 11. Below is a video to give you a good quick tutorial if you're not already familiar with the technique (If your browser supports Flash, here's another excellent tutorial). There's also a page where you can practice.

To divide by 91, you start by multiplying the numerator by 11, and then subtract 1. This number will likely be 3 digits, but if it's only 2 digits, place a 0 in front of it (76, for example, becomes 076). Put a decimal point in front of this 3 digit number, and you've got the first 3 digits of your answer.

To get the next 3 digits of the answer, subtract each of the digits from the previous step from 9. At this point, the number will repeat with the same 6 digits forever.

As a full example, let's work out 44 ÷ 91. We multiply 44 by 11 to get 484, and subtract 1, so we have 483. The first 3 digits of the answer, then are .483. Next, we work out 9 (always) - 4 (first digit) to get 5. 9 - 8 (second digit) = 1 and 9 - 3 (third digit) = 6, so the next 3 digits are these answers, 5, 1, and 6. Putting that all together, and repeating those same 6 digits, the decimal comes to .483516483516....

If you think of the 6 digits in the answer as two 3-digit numbers that always add up to 999, (Such as 483 and 516 in our previous example) it makes this easier. Since 91 ends in 1 just like 11, it's easy to remember that the technique for dividing by 91 involves multiplying by 11.

Dividing by 98

98 has a particularly surprising pattern. To start, you begin with the numerator, adding a 0 in front if it's a 1-digit number, and keep doubling the number. For example, with 6 ÷ 98, we start with 06 after the decimal point and keep doubling:
06 12 24 48
Checking with Wolfram|Alpha, we see that 6 ÷ 98 does indeed start with .06122448!

There are two more rules you need to know for this technique. First, anytime your doubling sequence gives you a number of 49 or more, you need to add 1 before continuing. Second, when you continue the sequence, double that modified number, but only give the last two digits. Continues the sequence as if these last two digits were just another two digits in the sequence, remembering to follow both of these rules.

To make this clearer, let's work out 16 ÷ 98. We start the series as before:
16 32 64
Stop! That 64 is over 49, so we need to add 1, making it a 65:
16 32 65
From there, we double 65 to get 130, but remember the second rule, that we only include the final two digits, the 30:
16 32 65 30
From here, we continue as if 30 were just another 2-digit number in the sequence. The next number would be 60, which is more than 49, so we add 1:
16 32 65 30 61
Sure enough, 16 ÷ 98 works out to be .1632653061 (and beyond).

Once you get the hang of these rules, you can carry out the decimal equivalent as far as you like. Remember the add 1 rule even when you're starting with a number equal to or more than 49. For 78 ÷ 98, you'd start by adding 1 to 78, and continue from there:
79 59 18 36 73
Do you follow the pattern there? If so, then you can mentally work out that 78 ÷ 98 is .79591836734693877551020408 and even go beyond that if you like!

Since 98 is 2 away from 100, remembering this slightly unusual doubling sequence shouldn't be a problem.

Take some time and master these specific numbers. When you're ready to move on, I'll show you how to use what you already know to automatically handle even more division problems.


As with many things in life, simplifying before working on the problem is the first thing you should do. If you get a problem such as 25 ÷ 35, or a number like 25/35ths, reduce the problem by dividing both parts by the same number, and keep doing this until you make the two numbers as small as possible. When you realize this boils down to 5/7, you can use the 7ths technique to realize that this works out to .7142857....

Here's a quick refresher course in reducing division problems and fractions:

Being familiar with the quick divisibility tests for the numbers 2, 3, 4, 5, 6, 9, and 10 will help greatly here.


Attempting to reduce the fraction or division problem should always be your first step, but sometimes you just can't reduce the problem to dividing to 2 through 11. This is where knowing how to divide by numbers near 100 comes in handy. See if you can make the problem larger, into one you're already familiar with.

Let's say you want to know the decimal equivalent of 11/25ths. You don't have a technique for handling 25ths, but if you scale both parts of the problem up so it reads as 44/100, then it's easily apparent that the answer is .44.

What about 13/15ths? Multiply both numbers by 6, and you get 78/90ths, which you should easily work out to .86666... using the strategies you've learned.

Similarly, dividing by 13 can be turned into dividing by 91 by multiplying times 7 (because 91= 13 × 7) and dividing by 33 becomes dividing by 99 just by multiplying by 3. You can also divide by 49 by scaling up by 2, which turns it into a problem of dividing by 98.

If you need to divide by 14, and you can't reduce it to 7ths, you can multiply both numbers by 7 to turn it into a problem of dividing by 98. 11/14ths won't reduce to 7ths, but will scale up to 77/98ths, or .7857142857....

Reduce AND Enlarge

Again, reducing should always be the first step, but sometimes enlarging the problem after reducing can help. For example, here's how to handle 24 ÷ 39. Both numbers are divisible by 3, so the same result can be obtained by reducing it to 8/13. We can then scale this problem up to 56/91, and work out that it's .6153846....

Using this approach, you'll be able to handle any problem involving dividing by 13, 14, 15, 18, 20, 25, 30, 45, 49, and 50. Often, you'll be able to handle many other numbers, as well. 26/28ths is the same as 13/14ths, and thus 91/98ths.


Sometimes, however, you're stuck with a division problem or fraction such as 32/43rds. That won't reduce to anything, and scaling the 43rds up to 86ths or 129ths doesn't bring up any familiar divisors.

If you look closely, however, you can see that it's quite close to 33/44, or 3/4, so you can estimate that it's roughly .75. 32/43 works out to roughly .7441860, so you can see that this is a good guess.

Most lessons in estimating fractions, such as this video, focus on rounding to 0, ½, or 1, but if you can round to the nearest 3rd, 4ths and 5ths, your estimated answers will improve greatly.

I hope you've found this tutorial useful and enjoyable!


How To Play and Win Notakto: 3+ Boards

Published on Tuesday, May 01, 2012 in , ,


For those who have come across this post by accident, this is Part 2 of a 2-part post on a tic-tac-toe-like game called Notakto. Part 1 can be found here, and will give you a more complete introduction to the game, as well as lessons on how to play and win on 1 or 2 boards against someone else.

Everything from this point on assumes that you've practiced all the strategies and can win 1- and 2-board Notakto games every time.

You should know and understand the importance of terms like sacrifice, boot trap, and 2X trap. You should also be very familiar with the few rules we've introduced so far:
• When playing on an odd number (1, 3, 5, etc.) of Notakto boards, you can guarantee yourself a win by being Player 1 - the odd-numbered player. When playing on an even number (2, 4, 6, etc.) of Notakto boards, you can guarantee yourself a win by being Player 2 - the even-numbered player.

• Your first move will always be placing your X in the center square of any empty board (empty board refers to any Notakto board with no Xs already on it).

• When the other player marks their X on a board with pre-existing Xs, your next move will be made on that same board. When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

Starting from this point, I'll show you how to generalize what you already know to play and win Notakto on 3 boards, 4 boards, and beyond!

In the next section, you'll start by learning how to play 3 boards.

3 Boards

Let's start with the simplest example of a 3-board Notakto game.

Since there are an odd number of boards, you start the game, and mark an X in the center of any board. In this example, the other player responds by marking an X on the same board, and your reply is to sacrifice that board.

At that point, the game reduces to a 2-board Notakto game with the other player effectively going first. Even before they place their first X on the remaining boards, you should already know that you're going to win.

This is the basis of another general rule you should keep in mind:
Your basic strategy is to sacrifice boards, until you get down to the final remaining 2 boards. These remaining 2 boards will be played just as you would play any standard 2-board Notakto game.

Marking a Different Board

In this next example, you play first once again, but your opponent marks an X on the edge (it could just as easily be a corner or the center) of a different board. You recall this rule, and respond accordingly:
When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.
With 3 boards, all 3 now have an X on them. From there, you play to eliminate the first board you can, and play the rest as a standard 2-board Notakto game.

There are a few more lessons we can learn from this game. First, if your opponent doesn't mark a previously-empty board in the center, the defenses you've learned will usually prevent them from placing an X in the center of that same board until they're forced to. Of course, if they do mark a previously-empty board with an X in the center, you can easily sacrifice it or set up a boot trap, as needed.

Because of the rule that tells you when to be player 1 or player 2, combined with the rule that tells you when to place an X in the center of a previously-empty board, you can always guarantee yourself a minimum number of boards with Xs in the center.

With an even number, such as 4, going second and only marking previously-empty boards after your opponent does the same thing, it's easy to see a pattern of theirs-yours-theirs-yours guarantees you a minimum of 2 boards with Xs in the center (it's also possible ALL of them could have Xs in the center). Extending this to 6, you should easily see that the minimum number of boards with Xs in the center will be 3. Given any even number n, the minimum number of Xs in the center you'll have is n/2.

What about odd numbers? With an odd number of boards, you always start, effectively guaranteeing yourself an extra board with an X in the center. So, for any odd number of boards n, the minimum number of boards with Xs in the center works out to be (n + 1)/2. For 3 boards, that is (3+1)/2 = 4/2 = 2 boards minimum with Xs in the center. For 5 boards, you'll have a minimum of 3 boards with Xs in the center, and so on.

Why are the Xs in the center so important? When learning how to play 2-board Notakto, your defense depended on at least one of the two boards having an X in the center. Even if you didn't end the game on this particular board, it was the board with the X in the center which allowed you to keep control of both boards and win with either one.

Because of this, we're going to amend the rule taught from the simple 3-game demonstration above. The revised rule is below:
Your basic strategy is to sacrifice boards, until you get down to a set of boards, only 1 of which has an X in the center. On the sole remaining board with an X in the center, you will build your boot trap. If there are two or more boards without Xs in the center, play to sacrifice them until you get down to 2 boards (one of which is the board with an X in the center). These remaining 2 boards will be played just as you would play any standard 2-board Notakto game.
It's time once again to practice. Practice 3-board Notakto either online or with the iPad Notakto app until you can win every time.

Once you feel confident playing and winning 3-board Notakto every time, you're ready to apply what you know to win on 4 boards, 5 boards, and beyond! Just make sure you can win every time on a given number of boards, before moving on. You'll probably notice that it takes you less and less time to play perfect games on each level.

Once you can win every time on any level, it's time to stop playing against computers and learn how to present this as a game to real people.

How To Present Notakto

Now that you know how to play and win Notakto every time, it's time to learn how to present this to a real person. My first bit of advice comes from Bob Farmer, who would often include the following warning in his Flim-Flam! column:
Caveat Scamtor: Ethical Hustlers warn the Mark the game is fixed. Money lost is an educational investment. Gambling may be illegal where you live. Information in this column may be wrong, so don't bet the farm until you've verified it's right.
Since Notakto is so similar to tic-tac-toe, that's often the best way to introduce it. Here are a few key points that help when introducing the game:

“Ever play tic-tac-toe? It's fun until you get to the point where both players are good enough that it's always a tie.” - This brings up the topic, and establishes familiarity.

“My friends and I liked the fact you could play it almost anywhere, but hated those tie games, too.” - This takes the familiarity, and gives a reason for the non-standard rules you're about to introduce.

“To prevent the ties, we decided that both players should play as X. After all, if you only mark Xs on the board, somebody has to get 3 in a row sooner or later, right?” - This introduces the all-Xs rule in a sensible way.

“After some testing, we found it makes for a longer and more interesting game when the person who makes 3 Xs in a row is the loser instead of the winner.” - This brings up the other major rule change. Note that both new rules are introduces with the real reasons for their existence.

“We've found it's even more fun when played on more than one board at the same time! When we do that, any board with 3 Xs in a row is out of play, and the last person to make 3 Xs in a row on the last available board is the loser.” - This quickly establishes the idea and rules of multiple board play. It is important for later than you do not mention a specific number of boards at this point.

Who Plays First?

At this point, you've already mentioned that the game will be played on multiple boards, but avoided mentioning exactly how many. This is about to give you the advantage in the game.

Ask the other player whether they'd like to go first.

If they decide that they're going first, draw 2 boards (or 4, if you prefer), and invite them to place their X anywhere. If they decide you're going first, draw 3 boards (5 is often too intimidating for a first game), and mark your X in the center square of any board, as usual.

You've subtly guaranteed yourself a win by having their choice of who goes first determine the number of boards!

If you want to state the number of boards before asking whether they go first, mention specifically that you'll be playing on 3 boards (or any odd number), and ask whether they'd like to go first. If they let you play first, play and win the game as normal.

If they decide they're going to play first instead, mention that, since the game is new to them, you should start with a practice game, so they can get the idea. Play the practice game, preferably playing to lose to build their confidence. Remind them once or twice through this game that it's a practice game. Once this “practice game” is over, play a legitimate game, mentioning that, because they went first last time, you get to go first this time.

You can find more tips on presenting Notakto in the next section.


• One more suggestion for the “practice game” technique: You could always have your practice game take place on a single board, explaining that this is to help them get the idea. Then, for the real game, you can move on to 3 boards as promised, in which you go first in that game, since they went first in the practice game.

• After you make each move during the game, you'll notice that usually all available boards will have an odd number (1, 3, or 5) of Xs. Naturally, this means that before you make your move, there will be only one board with an even number of Xs. This can act as a strong signal to tell you which board you need to play.

There are two exceptions to this pattern:

1) At the beginning of the game, when your opponent has just marked an X on a previously-empty board. In this situation, you're going to mark a new board with an X in the center square, of course. That makes this one case where each board played starts with an odd number of Xs before you play, while the same is true after you play (since you're marking a new board).

2) The other exception happens at the end of the game, the one described in the Attacked! section of this tutorial. If you're down to two boards, one of which has an X at the center and the other one doesn't, and your opponent sacrifices the board with the X at the center, then each move you make will leave an even number of Xs on the remaining board, instead.

• If you want to understand more about Notakto, here are a few helpful resources:

- Another puzzle (bgonline.org discussion, where the idea was first posted to the internet)
- Neutral Tic-Tac-Toe (MathOverflow discussion)
- Impartial Tic-Tac-Toe Presentation (PDF)
- The Secrets of Notakto: Winning at X-only Tic-Tac-Toe (PDF)

• This winning strategies don't need to be top secret. If someone genuinely shows an interest in learning how to play Notakto and win every time, feel free to teach them and/or send them to this tutorial. Above all, enjoy it and have fun!


How to Play and Win Notakto

Published on Tuesday, May 01, 2012 in , ,


Math professor and Backgammon expert Bob Koca was playing tic-tac-toe with his 5-year-old nephew, when the nephew whimsically suggested that they should both play X. After mathematically analyzing such a game, Professor Koca realized that this was a deceptive new version of the classic game of Nim.

Thane Plambeck later dubbed this game Notakto (pronounced No-tac-toe). In this tutorial, you'll learn how to play this game so that you can win every time!

Notakto is played on a standard 3-by-3 tic-tac-toe board, and the rules are as follows:

• Both players alternate making an X on the board.

• Players may mark on X on an available space (again, any space not already occupied by an X) any available board during their turn.

• The person who makes a horizontal, vertical, or diagonal line of 3 Xs on the board is the loser.

If you'd like to try out this game for yourself, you can play the first level online here (click reset if you pass the first level to stay). If you have an iPad, you can download and play the Notakto app here.

When referring to general types of squares, I'll use the terms center, corner, and edge to apply to the various squares as follows:

When I need to refer to a square in more specific terms, I'll refer to the various squares with these terms:

I'll also frequently use the terms rotations and reflections.

If you rotate a given pattern of Xs through quarter turn (90°) increments to match another pattern of Xs, then those two patterns are rotations of each other. Two given patterns of Xs that are mirror images of each other, with left and right switched, and/or top and bottom switched, are reflections of each other.

These are important concepts, since patterns that are rotations or reflections will share the same strategy.

Now that you've got a basic understanding of the concepts involved, it's time to learn how to win 1-board Notakto!

Winning 1-board Notakto

Mathematician Timothy Chow originally likened the winning moves to that of a chess knight. If you're not familiar with chess, chess knights move either 1 square vertically and 2 squares horizontally or 1 square horizontally and 2 squares vertically, resulting in an unusual L-shaped move.

If we think of a chess knight on a Notakto board, you'll note that the knight in the corner square below (on your left) can only move into either of the edge squares marked with a dot. The knight located in the edge square (on your right) can only move into either of the corner squares marked with a dot.

Because of the unusual way the knight moves, a knight in the center doesn't have any possible moves.

Here's how to use the knowledge of a knight's move to win at Notakto.

To guarantee yourself a win on 1-board Notakto, you must play first, and start by placing your X in the center.

After each time the other player marks an X in a given square, you'll place your X a knight's move away from that square.

As seen in the knight's move graphic above, there will often be two open squares that qualify. Before placing your X, make sure that you're not inadvertently making 3 in a horizontal, vertical, or diagonal line of 3 Xs, thus losing the game.

Played properly, the resulting game should look something like this:

Why This Works

By starting with your X in the center, that limits every following move to be played on centers and edges, which are squares where you can use the knight's move strategy.

If the other player places an X in a corner at this point, the knight's move will place your X on an edge square. Conversely, If the other player places an X in the edge at this point, the knight's move will place your X on a corner square. This results in a boot-shaped arrangement of Xs at this point:

I refer to this arrangement (and any rotations and reflections of this arrangement) as the boot trap. Take a close look at it. The other player in the boot trap above won't place their X in the lower edge or the upper right corner, because they would instantly lose the game.

That leaves 4 remaining squares that seem harmless enough. However, when they place their X on any one of those squares, and you place your X a knight's move away (again, making sure you don't accidentally complete a horizontal, vertical, or diagonal line of 3 Xs), the board then looks something like this:

Notice that placing ANY X on this board must result in a losing play! From the boot trap, you can always force this situation, which means you'll always be able to win the game.

Practice 1-board Notakto online or with the iPad Notakto app until you can always win.

Once you feel comfortable enough with the 1-board strategy, it's time to learn to win 2-board Notakto!

Playing 2-board Notakto

For playing Notakto with 2 or more boards simultaneously, the rules are similar, but there are a few changes:

• Both players alternate making an X on the board.

• Once an individual board has a horizontal, vertical, or diagonal line of 3 Xs on it, that board becomes unavailable (no more moves may be made on it).

• Players may mark on X on an available space (any space not already occupied by an X on any available board) any available board during their turn.

• The person who makes a horizontal, vertical, or diagonal line of 3 Xs on the last available board is the loser.

While you'll still use the boot trap and the knight's move strategy, there are some new adjustments and new strategies to learn in order to win multi-board Notakto every time.

To guarantee yourself a win in 2-board Notakto, the other player must go first. In fact, there is a simple pattern that will help guarantee you a win with any number of boards:
When playing on an odd number (1, 3, 5, etc.) of Notakto boards, you can guarantee yourself a win by being Player 1 - the odd-numbered player. When playing on an even number (2, 4, 6, etc.) of Notakto boards, you can guarantee yourself a win by being Player 2 - the even-numbered player.
There's also a simple rule for remembering where to mark your first X:
Your first move will always be placing your X in the center square of any empty board (empty board refers to any Notakto board with no Xs already on it).
When you go first, all the boards are empty, so of course you can make your first move this way. If you're Player 2, there will always be at least one other empty board after the other player's first move, so you can be assured of still having an empty board on which to mark the center square.

These are great strategies for knowing how to start the game in your favor, but as I mentioned earlier, you'll need to learn some more strategies to assure yourself a win. The first new strategy is learning how to sacrifice a board.


Since only the person who makes a horizontal, vertical, or diagonal line of 3 Xs on the final available board loses, taking other boards out of play by purposely completing a line of 3 Xs can be very helpful. Taking a board out of play in this way is called sacrificing, just like the same concept in chess.

Let's start with the simplest possible example of sacrificing a board. In the example below, the other player is Player 1, and you are Player 2 (because there's an even number of boards, remember?).

Their first move is the center square of one of the boards, and your reply is the center square of the remaining empty board. From this arrangement, no matter where they put their next X, there will always be 2 Xs in a line. Your response is to complete that line with a 3rd X, and sacrifice that board:

Once that sacrifice is made, the game effectively reduces to a 1-board Notakto in which the other person is the second player. As shown in the animation above, you set up your boot trap, and proceed to win the game just as before!

While the above game can and does happen, not every game happens in such a simple and straightforward manner.

Our next example will still show the concept of sacrificing a board, but it will happen later, and after a more complex series of moves.

This game starts, again, with the other player going first. This time, they're going to start in a corner (though it could just as easily be an edge), and you respond just as you should, marking an X in the center square of the remaining board.

This time, however, the other player marks an X on the same board on which you just played your center X. What do you do in this case?

Simple! You respond just as you would in 1-board Notakto, and set up your boot trap.

From here, we'll assume the other player marks an X in line with the X they played first. Naturally, you complete the row of 3 and sacrifice that board. The game then returns to the board on which you've set up your boot trap, and you win in the usual way:

This latter, more complex game actually shows a number of concepts that will be important through the rest of this tutorial:

1) Any strategy you learn in this tutorial can be delayed and still remain effective.

2) This is a another good lesson in determining which board to play:
When the other player marks their X on a board with pre-existing Xs, your next move will be made on that same board. When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.
In the previous tab, we discussed a similar rule that helps determine where you start. This rule, on the other hand, determines on which board you will continue play.

Try practicing 2-board Notakto online or with the iPad Notakto app using this approach, but don't be discouraged if you don't win.

Playing this way will guarantee a win if the computer responds as we've assumed above, but you'll notice that there are some situations that haven't been covered yet. The most common is when they start on a corner or an edge, and the next time they make a move on that board, there's no way to make 3 Xs and sacrifice the board!

In the next section, you'll learn how to use that kind of play to set up another trap for the other player!

The 2X Trap

When the other player marks their first X in a corner or an edge, it's quite possible that their next move could prevent you from sacrificing the board on your next move. It's not uncommon to see a situation such as this (the X in the right board is yours, the 2 on the left belong to the other player):

Another arrangement that could prevent an immediate sacrifice happens when two Xs are marked on edges that aren't directly across from each other (Again, the X in the right board is yours, the 2 on the left belong to the other player):

In both of the above cases, the next move is yours. What is the best move to make?

With both arrangements, the answer is exactly the same! You should place an X to make the pattern shown on the left board:

Why? The next time your opponent places an X anywhere on a left board with that arrangement, they'll wind up with 2 Xs in a line. When they do, you can place the 3rd X and sacrifice that board! Yet again, you can prepare your boot trap on the right board, and win the game as usual.

Because this design forces your opponent to place 2 Xs in a line, I refer to this as the 2X trap. It's easy to visualize, as the 2X trap always has 1 X in a corner, and 2 edge Xs, both of which are a knight's move away from the corner X.

Below is a full game animated, in which the 2X trap is used. This particular game involves a rotation of the 2X trap depicted above, so you can get used to seeing it in another arrangment. As with all 2-board Notakto games, the other player goes first.

As with the earlier example games, the 2X trap could be played even if you had developed the boot trap further on the left board. Delayed use of tactics, as I've said before, can still be effective.

Go practice 2-board Notakto either online or with the iPad Notakto app with your knowledge of the 2X trap, and you should find that you're winning more 2-board games than before.

However, your 2-board game still won't be perfect. Sometimes, your opponent can be sneaky and sacrifice your carefully-prepared boot trap! You'll learn how to handle that situation in the next section.

When They Sacrifice Your Boot Trap

Up to this point, every strategy discussed has involved sacrificing all but the last board (if any), and then winning the final board via the boot trap and the knight's move strategy that has been taught.

It isn't difficult to conceive of sacrificing a board, however, and your opponent can do it just as easily as you can. Below is a snapshot of a 2-board Notakto game where the following has occurred:

1) The other player moved first, and placed their X on an edge square (left board). You responded by placing your X in the center of the other (right) board.

2) They marked their next x on an edge square on the right board, and you responded preparing your boot trap.

3) They then marked an X on an edge square on the left board in such a way that you couldn't immediately sacrifice the board. You respond by setting up the 2X trap.

4) They then surprise you by sacrificing the right board, ruining your boot trap.

It's now your move. What do you do?

In this situation, there's only one effective response. You must place your X in the corner directly opposite the other corner X in the 2X trap:

Why does that work? While there are 5 remaining spaces open, the other player cannot place their X in the center or either of the open corner squares, because they'd lose the game. Their only possible response is to place their next X in one of the remaining edge squares, and your response will be to place your X in the other remaining edge square.

After they mark their edge square, and you mark your edge square, it's their move again, and the board now looks like this:

Anywhere they place their X on the left board, they must lose (Remember, the right board is out of play).

When They Sacrifice Early

Sometimes, when the other player sacrifices your boot trap early, the other board only has a single X on a corner or an edge (If the other board had a single X in the center, then you built your boot trap too early and on the wrong board), like this:

The response for this situation is easy, but different enough that it warrants its own section.

The strategy for this situation is simple. Your next move is to mirror the placement of the other player's Xs. What do I mean by mirroring the placement?

If they placed an X on the upper edge, you place your X on the lower edge (and vice-versa). If they placed an X on the right edge, you place your X on the left edge (and vice-versa).

Similarly with the corners, if they placed an X on the upper left corner, you place your X on the lower right corner (and vice-versa). If they placed an X on the upper right corner, you place your X on the lower left corner (and vice-versa).

In the case shown above, there an X in the lower left corner of the right board, and it's your move. The mirroring strategy tells you to place your X in the upper right corner, and keep mirroring their plays as discussed above.

If they mark a mixture of corner and edge squares, and you mirror those appropriately the boards will eventually look like this after your last move:

Does the pattern on the right board look familiar? It's the pattern of 6 Xs with an open diagonal (albeit a rotation of that earlier pattern) that ended the game discussed above.

Alternatively, if the other player keeps marking Xs only in corners, and you keep mirroring the appropriate corners in response, you'll come to this pattern much more quickly after your last move:

Even though there's only 4 Xs in this pattern, the other player must lose. It's their turn, and marking an X in the center or any edge means completing a horizontal, vertical, or diagonal row of 3 Xs, thus costing them the game.

With everything you've learned here, you're now equipped with enough knowledge to win every 1-board and 2-board Notakto game everytime.

To make sure you know how to use this knowledge, it's time again to practice 1-board and 2-board Notakto either online or with the iPad Notakto app.

Keep practicing until you can win the 1-board and 2-board games every time! Try and play without looking back at these strategies, but don't be afraid to look back at them when you need to do so.

Not only have you learned all the strategies you need to win every 1- and 2-board Notakto game, you've also learned all the strategies you need to play on any number of boards! In the next post, I'll take what you've already learned and show you how to generalize those strategies so you can apply them to any number of boards.