Rules

Somebody has mixed up the calendar! Can you help straighten it out?

Just like the 15 puzzle, the object of this puzzle is to return all the pieces to their correct order. Instead of 15 pieces to move around, however, the puzzle below has 41 pieces. In addition, the calendar aspect adds an extra dimension to the challenge!

Note the year and month above the calendar puzzle itself. Your challenge is to arrange the pieces so that they form the correct calendar for that month and year. The following rules apply:

• The blue square marked with a “1” will always be located somewhere in the top row. This should make sense. You don't start any calendar on the 2nd week.

• The blue numbered tiles are arranged on their corresponding days in the month. If the 1st fell on a Tuesday in the given month, then the blue “1” needs to be placed in the Tuesday column, and so on.

• The blue numbered tiles must be arranged in numerical order, reading from left to right, then top to bottom.

• The white lettered tiles are used to fill in the remaining spaces not used by the dates, and must be arrange in alphabetical order, reading from left to right, then top to bottom.

For example, if you're given a 30-day month that begins on Tuesday, the goal would be to arrange the puzzle like this:

Note, in this example, that A and B are used to fill the first Sunday and Monday respectively, since the month begins on a Tuesday. After the month ends, the remaining lettered pieces, C through K, are used to fill the remaining spaces.

The Year Range selection menus can be used to choose the range of years for your next puzzle (the default range is 2000 to 2099), and the New Puzzle button will generate a new month and year, as well as shuffle the pieces.

For help in solving the puzzle, try the next section.

Post your record times and moves, along with the year and month you solved, in the comments!

Solving

The two challenges that were combined to make the sliding calendar puzzle are taught separately here in the Grey Matters Mental Gym.

First, learn how to solve the classic versions of the 15 puzzle in this tutorial.

After that, you need to learn how to determine the day of the week for any given date.

To apply them to this particular puzzle, consider the year and month you're given, along with the first of the month. Once you figure out on which day of the week the first day falls, you also know which lettered tiles (if any) come before it.

Once you get the first lettered and numbered tiles in place, it's mostly like solving the original 15 puzzle.

When you get to the last two rows, the tricky part becomes working out which letter goes below which other piece. If you take into account where the letters in the top row (again, if any) left off, and how many days are in the month, this isn't too difficult.

Yes, you could cheat and look up the given month calendar online, but it's far more impressive to do it without looking.

Introduction

This tutorial is meant as a sequel to my unit circle tutorial. Much of this tutorial assumes you have already been through that one, so if haven't already done so, please go through the previous tutorial now.

The unit circle is meant as a sort of idealized circle, from which all other measurements can be scaled up. Radians, for example, made it easy to determine how far a wheel was traveling, given its rotation in radians. All that was required was a single multiplication.

As a matter of fact, all the measurements on the unit circle work this way. Sine and cosine can be applied to real world measurements with a single multiplication to scale them up or down.

That's why, in the unit circle, all the trigonometric functions do a sort of double duty. They're ratios, as you've been taught, but in the idealized world of the unit circle, they can also be treated as absolutes, since they result in useful coordinates. For example, sine and cosine are both ratios that relate in different ways to the length of the hypotenuse, but also give exact coordinates to graph.

In this tutorial, we'll go beyond sine and cosine, and explore the other trigonometric functions of tangent, cotangent, secant, and cosecant. The goal of this tutorial is to explain them in a way that can be easily understood, comprehended, and remembered.

To explain these, we're going to scale the triangle up beyond the bounds of the unit circle. In the previous tutorial, the hypotenuse always had a length of 1 because it was representing the radius of the unit circle.

The hypotenuse of a right triangle is, by definition, the longest of the three sides, so this meant that, with any given angle, the other two side must always fall on or inside the unit circle itself, because they have to be 1 (in the case of 90 degree or π/2 angles) or less.

What happens, though, if we scale up a right triangle in the unit circle so that the width or height is 1 instead of the hypotenuse? In this tutorial, we'll try that out, and examine the useful measurements that result.

To start off simply, however, we'll focus on the one thing that doesn't change as the right triangle is scaled up - its slope.

Slope

Question: What's the slope of a 30 degree angle? Many would answer something like, “What kind of question is that? It's 30 degrees!“

I'd give that answer half credit. It does recognize that, regardless of the lengths of the sides of the triangle, the slope does remain the same.

However, slope is usually stated as decimal number, so that it's easy to multiply by. Here's a quick refresher course in slope, excerpted from the series The Mechanical Universe:


“Change in elevation” over “change in horizontal distance” is usually stated in a way that's catchier and easier to remember:



For example, take a 45 degree angle in the unit circle. When the hypotenuse was 1 (the radius of the unit circle), what was the rise and what was the run? If you remember the hand trick from the video, it shouldn't be too hard to recall:



So, for a 45-degree angle the slope is 1. In other words, for every 1 unit you move horizontally, you're going to move 1 unit vertically, as well. This makes sense for a 45-degree right triangle, since the two shorter sides are the same length.

What about that 30-degree angle I asked about earlier? Lets work through that problem:



You can see a more exact answer at Wolfram|Alpha. It can also be stated as the square root of 3 over 3.

As you'll see it again and again, you should find a way to remember the square root of 3.

The number is less exact than in our 45-degree answer, but the meaning remains the same. With a 30-degree angle, for every 1 unit you run (move horizontally), you're going to rise (move vertically) 0.57735... units. That's more than half a unit, but less than a slope of 0.60.

We've been talking quite a bit about slope, but not much about the unit circle here. This is a very important tangent however, as you'll learn in the next section.

Tangent

Ever been told by someone that you're going off on a tangent? Ultimately, the complaint is that you're going off on a line that will only take the discussion farther and farther from the main point.

It's almost exactly the same definition in math. One mathematical definition of tangent is a line that touches a circle (a unit circle, for our purposes) at only one point. Starting at that one point, and then traveling along the tangent would only take you farther and farther away from that main point.



As you can see, a tangent is easy to draw. Draw a radius line at any angle, and at the point where it touches the circle, draw a line perpendicular (at a 90° angle) to the radius, and that's the tangent.

To keep things simple in this tutorial, we're only going to consider the horizontal and vertical tangents. In this particular section, we'll only focus on the vertical tangent.

The formula to graph a vertical tangent is easy, it's x=1. When y=0, x=1. When y=5, x=1. When y=4,287, x=1. You get the idea. Not surprisingly, the only point at which it intersects the unit circle is

In the previous tutorial, we always created the length of the hypotenuse (the longest side) to be 1. Imagine that, instead of the hypotenuse having length 1, we had the base have a length of 1.

Here's a picture of the situation:



Here's a question for you: If we tried scaling up a 45° right triangle in this manner, what would the coordinates be where the hypotenuse of this new larger triangle intersects the tangent line?

Let's think about this. Because the tangent is defined as x=1, the x coordinate where the hypotenuse intersects the tangent will also obviously be x=1. So, the coordinates we have so far are (1, something). That's half the work already done!

In the previous section, we worked out that the slope of a 45° angle was 1, at that meant that for every 1 unit you moved horizontally, you moved 1 unit vertically. As it happens, a base width of 1 means we are moving over 1 unit horizontally, so we should obviously move up 1 unit vertically!

So, the coordinates of a 45° angle where it intersects the tangent line would simply by (1,1)!

Did that seem easier than it should have? Let's try it with our 30° angle example, too. What was the slope of that angle? It was 0.57735... and so on. Let's take a look at that angle plotted by Wolfram|Alpha.

That means, for every 1 unit we travel horizontally, we travel 0.57735... units vertically. Yep, the coordinates where the hypotenuse meets the tangent line is (1, 0.57735...).

See the pattern? Whenever a right triangle with a given angle has a base length of 1 in the unit circle, the coordinates where the hypotenuse will intersect the vertical tangent line is (1, slope of that angle)!

As with sine and cosine in the unit circle, the slope is doing double duty. It's both the slope itself, and the y coordinate where the hypotenuse intersects the tangent at x=1. In fact, we could just call this number the tangent.

Remember SOHCAHTOA? That helps remind us, among other things that the formula for the tangent is the opposite side's length over the adjacent side's length. Also, remember that those lengths on the unit circle are worked out by figuring sine (the height) and the cosine (the width)

Let's take a closer look at the formulas for slope and tangent:



Yep, the tangent and the slope are always the same, which is why it works out as it does! This should also help you better understand the brief reference to figuring out tangent on your fingers from the previous tutorial.

So far, we've only dealt with scaling triangles up to meet a vertical tangent. What happens if we scale them up to meet a horizontal tangent?

Horizontal Tangent

With a horizontal tangent, we're simply dealing with what happens when y=1, instead of x=1.

Remember how we determined the tangent (the same as the slope) of a 45° angle was 1, so the coordinates where the hypotenuse met the vertical tangent was (1,1)? For a 45°, this works out nicely, since it intersects the horizontal tangent in the same place.

Let's go back to our 30° example, and get a better idea of what changes. The slope is 0.57735..., so let's view what happens when we try this out:



Here's another look at the same situation, via Wolfram|Alpha (albeit slightly distorted). Even without knowing the exact coordinates, we can see that the hypotenuse, in the 30° case, is MUCH longer than where it intersected the vertical tangent.

So, what are the coordinates? We start in a manner similar to before, with the coordinates (something, 1), because the horizontal tangent formula is y=1 (instead of x=1 for the vertical tangent).

In the case of the vertical tangent, we just multiplied 1 (the x coordinate) times the slope to get the y coordinate. Since we're still talking about the same 30° angle, the slope is still the same 0.57735... we used before. We'll just have to come at it from the other way:



You can actually see several important things here. First, we found the coordinates for where our 30° angle intersects the horizontal tangent at (1.73205...,1). Second, while the tangent relationship of y = x × 0.57735... still holds, it would seem easier just to state the relationship the other way around, as in x = y × 1.73205..., especially when that lets you multiply by 1.

Finally, note that you can always find where the hypotenuse intersects the horizontal tangent by dividing 1 by the tangent of the same angle. That's why this number is given the name cotangent.

If you think about it, there are several ways to find the cotangent:



As a brief review, we covered sine and cosine in the previous tutorial, and now we've covered tangent and cotangent. If you think of the horizontal tangent as being a cotangent line, then this is easier to understand.

We keep talking about the slope of the hypotenuse and the coordinates where the hypotenuse intersects the tangent and cotangent lines, but what happens to the length of the hypotenuse as triangles are scaled up to meet the tangent and cotangent lines? That's discussed in the next section.

Hypotenuse Length

We're going to switch from focusing on the slope of the hypotenuse to its length.

This almost means a return to our old friend from the right-triangle, the Pythagorean Theorem: a² + b² = c².

When the hypotenuse is 1 unit long, the squares of the other two sides must add up to 1. For example, in our 30° angle, where the cosine (width) is 0.8660... and the sine (height) is 0.5, we get:



However, the hypotenuse must get longer to meet the tangent lines as we've seen. Let's start as before, scaling up to meet the tangent line (that's the vertical tangent). We've already seen that the coordinates of a 30° angle gives us coordinates of (1, 0.57735...). As in the above example, we note that the coordinates are also the lengths of the two smaller sides. That being the case, let's figure out how long the hypotenuse is when it meets the vertical tangent:



So, we see that the hypotenuse is now 1.1547... units long. Since the original hypotenuse was only 1 unit long, this is a sort of scaling factor for the hypotenuse.

Take another look, though. When we scaled the width of a 30° right triangle from 0.866... to 1, that's a factor of 1.1547... times, as well. The height went from 0.5 units to 0.57735... units, which is also a scaling factor of 1.1547... times!

So, this one factor, when scaling a right triangle up to meet the tangent line tells us how to scale the lengths of all the sides in order to do so! Since we're scaling up from a hypotenuse of 1, this factor also gives us the exact length of the hypotenuse after being scaled up to the tangent line.

In trigonometry, this factor is called the secant. Since we scale the width of the triangle (the cosine, in the unit triangle) up to 1, it shouldn't be surprising that we can find the secant in this manner:



The hypotenuse length was 1 before scaling it up, so you could also work it out by dividing the hypotenuse by the width:

Meeting the Cotangent Line

Let's not forget the length of hypotenuse when scaled up to meet the cotangent line (that's the horizontal tangent line).

Our example 30° right triangle, when scaled up to meet the cotangent line, we found the coordinates (and side lengths) of (1.73205...,1). Let's run the numbers just as before, and find the hypotenuse length:



Oh! The hypotenuse in this case is exactly 2 units long. We took the height (the sine) from 0.5 to 1, which is a factor of exactly 2 units, so we shouldn't be surprised when the hypotenuse (and the width, for that matter) scales up by a factor of 2 units.

Since this scaling factor deals with scaling the height up to the cotangent line, this factor is naturally called the cosecant. The formulas for cosecant are as follows:



At this point, you should understand all 6 trigonometric functions: sine, cosine, tangent, cotangent, secant and cosecant. Once understood, however, they can be easily confused. In our final section, I'll offer some mnemonics to help keep them all straight.

Mnemonics

When you have the explanations and formulas in front of you, it's much easier to keep these things straight. How is it possible to keep all this straight in your head?

Let's break things up several ways. First, by the pairs in which you learned them.

Sine and cosine can be thought of as the simple lengths of the sides in the unit circle, when the hypotenuse is 1. Think of signs that tell you the length of a road.

Tangent and cotangent deal with the unit coordinates when the triangles are scaled up to meet the tangent and the cotangent lines respectively. This is its own mnemonic: tangents deal with tangents.

Secant and cosecant are the scaling factors to the tangent and cotangent lines respectively. Think secant means “secaling” factor.

Once you get the functions themselves straight, it can seem tricky to remember which of each pair deals with horizontal information and which ones deal with vertical information. However, there's an almost built-in mnemonic: look for the “o”.

Sine is the vertical length of the triangle. Tangent deals with coordinates when scaled up to the vertical tangent line. Secant deals with scaling factors up to the vertical tangent.

On the other hand, cosine is the horizontal length of the triangle. Cotangent deals with coordinates when scaled up to the horizontal tangent line. Cosecant deals with scaling factors up to the horizontal tangent.

Do you see what I mean by “looking for the o”? The words horizontal, cosine, cotangent, and cosecant all have an o in them, so they all go together.

Similarly, none of the words vertical, sine, tangent, and secant feature an o in them, so they all go together.

Looking for the letter o is an appropriate mnemonic for a circle, don't you think?

Another challenge is remembering the respective formulas. As mentioned before, SOHCAHTOA is the classic way to remember this, but that only gives 3 formulas.

Here's a lesser known yet more amusing mnemonic that gives all 6 formulas. Start by writing down the dividing lines, and writing “OOH AAH” above them:



“OOH AAH” is, no doubt, the sound you made upon first learning about unit circles. On the bottom, you're going to write “OOH AAH” again, but this time from right to left (as in “HAA HOO”):



From left to right, these are the formulas for the three vertical functions, sine (O/H), tangent (O/A), secant (H/A), followed by the three horizontal functions in the same order, cosine (A/H), cotangent (A/O), and cosecant (H/O).

You've also noticed that you can get some of these ratios simply by taking the inverse (dividing 1 by another number) of other factors. Tangent and cotangent are easy:



The ones beginning with t's are easy, so they can be thought of together. The ones featuring c's and s's are a little trickier:



When focusing on secant or cosecant, ignore the first s or c you see, and look for the second s or c. In the case of secant, you'd look at it as seCant. This C lets you know that it's the inverse of the function that begins with C - cosine!

With cosecant, you'd look at it as coSecant, and remember that this S it telling you that it's the inverse of the other S - sine!

Once you have that relationship down, here's a video that will take your knowledge of the trigonometric functions to the next level. It plays quickly, so you should pause it to read and understand everything it says:



Any further questions, class?

Re-Introduction

In this feat, someone gives you a date, and you quickly state the day of the week on which it fell. This new approach is updated for the 21st century, and employs new tips and tricks that help make this feat simpler to learn and quicker to perform.

Approach:

The Day of the Week For Any Date feat combines both memory and mental math. A relatively simple mastery of both, though, will create a response far out of proportion to the required work.

Before I describe the basics of the approach, I'd like to help you get a good idea of your goal, as well as what's possible, by seeing this feat performed by various people in the following videos:

• Dr. Arthur Benjamin performing at TED
• 8-year old Florian Osmani on German TV (in German)

Here's the basic principles, broken down into simple steps:

1) Day and Months Number Conversion: To work out the days of the week mentally, we need to convert them into numbers. We'll also need to convert the months into numbers, to adjust for their effects. These are taught in an easy-to-remember manner.

2) Addition of 3 Numbers: Without using a calculator, can you tell me what 6 + 6 + 31 is? That's about as difficult as the basic formula gets. If you're comfortable doing that, you won't have a problem working through the formula.

3) Subtracting Multiples of 7: Let's say you're asked about the 27th of a month. Regardless of the month or year, we can state with certainty that the 27th of a month will fall on the 6th (since it's 3 weeks, or 21 days, earlier). Since adding 6 is simpler than adding 27, and will give the same result, why not use 6? If you learn to subtract multiples of 7, this makes the arithmetic so easy that you won't have to worry about addition problems any tougher than 6 + 6 + 6!

4) Year Number Conversion: After becoming comfortable with all of the above when given dates in the years 2000 to 2003, you'll learn how to remember and adjust the leap years in the 21st century to key dates. After learning those, you'll learn a simple way to adjust for any year from 2000 to 2099, and even adjust for other centuries!

Running through all these principles, there will be an emphasis on recognizing and taking advantage of patterns. The quicker you can recognize a pattern, the quicker will be your calculation.

We'll start with the codes for the days of the week, since that is our goal. All the formulas and patterns you'll learn later will result in a number from 0 to 6. This number is turned into a weekday as follows:

Day of Week	Number	Mnemonic
Sunday	0	SUNday=NONEday
Monday	1	MONday=ONEday
Tuesday	2	TWOSday
Wednesday	3	Three fingers look like a W
Thursday	4	FOURSday
Friday	5	FIVEday
Saturday	6	SIXturday

To help get you comfortable with converting numbers into days, take the Weekday Codes quiz here. Once you can get a perfect score in a short time, then continue with this tutorial.

Next, you need to learn the codes for the months. Like the weekdays, they range from 0 to 6:

Month	Number	Mnemonic
January	6	WINTER has 6 letters
February	2	February is 2nd month
March	2	March 2 the beat.
April	5	APRIL has 5 letters (& FOOLS!)
May	0	MAY-0
June	3	June BUG (BUG has 3 letters)
July	5	FIVERworks
August	1	A-1 Steak Sauce at picnic
September	4	FALL has 4 letters
October	6	SIX or treat!
November	2	2 legs on 2rkey
December	4	LAST (or XMAS) has 4 letters

Take the Month Codes quiz here to help reinforce the mnemonics.

Note: It's important to note that, in leap years, January reduces by one to 5, and February reduces by one to 1. The other years don't change in leap years. Leap years will be discussed in more detail later.

Once you're comfortable with both the month and weekday codes, you're ready to start calculating your first dates. In the next tab, you'll learn how to use these codes to work out dates for the years 2000 to 2003.

Formula

Ready for the formula? Here it is: Month Code + Date + Year Code = Day of Week Code. It's a lot simpler than many people think, but there are some fine points to learn.

We haven't covered year codes yet, so I'm just going to teach you four simple ones with which to start out:

• 2000 = 0
• 2001 = 1
• 2002 = 2
• 2003 = 3

Those shouldn't be too hard to remember, should they? We'll learn more year codes later in this tutorial, but for now, we'll just focus on these years.

Let's start with a simple example. Let's figure out May 1, 2000. The code for May is what? The mnemonic is MAY-0, so May is a 0. The date itself is the first, so we use 1. The year code for 2000 is 0, so our problem is 0 + 1 + 0 = 1.

Which weekday has a code of 1? ONEday is MONday, so Monday is the day of the week on which May 1, 2000 fell. You can verify that at this site. Congratulations, you've just calculated your first date!

Let's try a date that's a little more challenging. Our next date is October 4, 2000. October is 6 (remember “SIX or treat”?), and 2000 is still 0, so that gives us 6 + 4 + 0 = 10. And the weekday that goes with 10 is...

Wait a minute, the weekday codes only range from 0 to 6! What do we do with a 10, or any result higher than 6 for that matter?

If you look on a calendar, the 10th of any month will always fall on the same day of the week as the 3rd, because the 3rd is 7 days earlier. So, with the result, or any number in the formula, we can subtract 7, or any multiples of 7 to reduce the answer. You may want to refresh yourself on the multiples of 7 with the help of Schoolhouse Rock here.

In our October 4, 2000 example, we got 10 as a result, so we can reduce that by 7. 10 - 7 = 3, so our result boils down to 3. Which day of the week is 3? Since 3 fingers looks like the letter W, that's Wednesday. Once again, you can check that here.

This reduction of the multiples of 7 can make the problem itself easier, as well. Let's try figuring out the day of the week for Halloween 2001, or October 31, 2001. October is 6, and 2001 is a 1, so the problem works out to be 6 + 31 + 1 = 38. The closest multiple of 7 to 38 is 35, so we do 38 - 35 = 3, which gives us another Wednesday.

That approach works, but it could be made simpler. When you hear that the date is the 31st, you can reduce that right away by working out that 31 - 28 (the closest multiple of 7 to 31) = 3, and doing 6 + 3 + 1 = 10. True, you would still need to reduce that 10 to 3 again to get Wednesday, but you'd need to subtract multiples of 7 either way.

Note that, by bringing the dates down by multiples of 7, you're making the problem you have to add much simpler. If 6 + 31 + 1 and 6 + 3 + 1 will both give you the same results, wouldn't you prefer to make it easier on yourself?

The scary technical term for subtracting multiples in this manner is modulo arithmetic, which is explained quite clearly at BetterExplained.com.

Let's try this with Valentine's Day in 2003. We start by making sure of the date, February 14, 2003. February is a 2 (remember the mnemonic?), and 2003 is a 3. The problem then becomes 2 + 14 + 3. However, if you spotted that 14 was already a multiple of 7, you should realize that you can drop it out completely! 14, or any multiple of 7, is the same as 0, so you can ignore them. For February 14, 2003, all you really need to add is 2 + 3 = 5. 5 is a FIVEday, or rather a Friday, so that's our answer!

Let's try one last problem before we go. What about February 2, 2000 (Groundhog Day)? February is 2, and 2000 is 0, so the problem we get is 2 + 2 + 0 = 4. 4 is a Thursday (remember FOURSday?), so February 2, 2000 should be a Thursday. Once again, we verify that information here and...OOPS! That site says February 2, 2000 is a Wednesday! What went wrong?

I briefly mentioned this at the end of the previous tab, but it needs to be repeated now. Whenever you're working in January or February dates in a leap year, you need to reduce the month code by 1 to compensate. Effectively the extra day, February 29, hasn't happened yet, so we're subtracting one to adjust for that fact. January becomes 5 (6 - 1) and February becomes 1 (2 - 1).

Since February 2, 2000 is a February date in a leap year, the code for February needs to be 1, not 2. Let's try the equation again, with that in mind. February in a leap year is 1, and 2000 is 0, so the equation is 1 + 2 + 0 = 3, which you should know by now is a Wednesday. As we saw when we originally made the error, Wednesday is indeed the correct day of the week.

Practice the Dates: 2000 to 2003 quiz here, making sure to keep an eye out for January and February dates in 2000, and adjusting your calculations accordingly. Practice these dates until you can calculate them with little trouble, and don't forget to subtract multiples of 7 to make your work easier!

Once you're comfortable with dates from 2000 to 2003, we're going to teach you how to better handle leap years in the next section.

Year Codes: Why?

You'll note that the year codes for 2000-2003 progressed in a nice, simple, 0-1-2-3 order. This is because a normal year consists of exactly 52 weeks (52 × 7 = 364) plus 1 day, to make 365 days. So, from one 365-day year to the next 365-day year, a given date in a given month will fall one day later.

However, a 366-day leap year means that everything must jump ahead 2 days. This also means that the year codes for leap years will jump ahead 2 instead of 1. You might expect 2004 to be a 4, but because it's a leap year, it jumps ahead to 5.

To get you comfortable with the strange nature of leap years, I'm going to start by teaching you the first 7 leap years.

First 7 Leap Years

Here are the year codes for the first 7 leap years, along with handy mnemonics by which to remember them:

Leap Year	Year Code	Mnemonic
2000	0	2000 is mostly 0s
2004	5	Count: 4...5...
2008	3	Right half of 8 looks like 3
2012	1	12 ÷ 12 = 1 (See below)
2016	6	16 ends in 6
2020	4	2 + 0 + 2 + 0 = 4
2024	2	24 ÷ 12 = 2 (See below)

With 2012 and 2024, you'll note that all you have to do is divide their last 2 digits by 12 to get their year code. This pattern keeps working all the way through 2096, which will give you a few extra leap year codes quite easily, assuming you know your 12 multiples:

• 2012 = 1
• 2024 = 2
• 2036 = 3
• 2048 = 4
• 2060 = 5
• 2072 = 6
• 2084 = 7 = 0 (Remember to drop any multiples of 7!)
• 2096 = 8 = 1 (Remember to drop any multiples of 7!)

First, practice just recalling the codes with the Leap Year Codes: 2000 to 2024 quiz here.

Once you're comfortable recalling all the codes, practice working out dates for those years with the Leap Year Dates: 2000 to 2024 quiz here. Don't forget that that the month codes for January and February are both reduced by 1 in leap years!

Once you've practiced those years, you're ready to learn how to handle any leap year in the 21st century!

All Leap Years

If you're given a leap year that ends in a multiple of 12, you can already handle those through 2096 quite easily, of course. What about the remaining leap years?

If there were no such thing as leap years, the pattern year codes would simply repeat every 7 years. Because of the effect of leap years every 4 years, however, the pattern of year codes usually repeats every 28 years.

I say “usually” because years ending in 00 are an exception. Years ending in 00 are only leap years if they're divisible by 400. So, 1600, 2000, and 2400 are leap years, while 1800, 1900, and 2100 are not.

Thanks to the 00 exception the calendar only repeats EXACTLY every 400 years. However, when you're dealing with a range of years in which EVERY (without exception) 4th year is a leap year, then you can still rely on the 28-year rule.

This means that you can depend on the 28-year rule for every leap year from 2000-2096! I'll break this down in a simpler manner, so you can see how this is useful.

For the leap years 2028 through 2052, all you have to do is subtract 28 years, and you'll get a year with the same year code! 2028 - 28 is 2000, which you already know has a year code of 0, so 2028's year code is 0. 2032 - 28 = 2004, whose year code is 5, and so on.

While doing 2028 - 28 = 2000 in your head is simple enough, some people find that working out problems like 2040 - 28 or 2052 - 28 during a performance to be a little challenging. There's a way to make it simpler.

If you're worried about subtracting 28 from a number, add 2 and then subtract 30 instead. For example, instead of doing 2040 - 28, work out 2040 + 2 - 30 = 2042 - 30 = 2012. 2012 is a 1 year, so 2040 is a 1 year as well! What year has the same year code as 2052? Add 2 and subtract 30, and you'll get your answer in no time.

Similarly, for the years 2056 to 2080, you subtract 56 to get the year code. The mathematical short cut here, if you feel you need it, is to add 4 and then subtract 60. What's the year code for 2064? 2064 + 4 - 60 = 2068 - 60 = 2008 = a year code of 3! How about 2068? You should get a year code of 1, just like 2012.

How about 2072? Did you start by adding 4? Stop! 72 is a multiple of 12, so we just work out that 2072 is 6 from the pattern of 12s above. Don't forget to take advantage of the 12 pattern when you can! You should ask yourself if a year is divisible by 12 first, before subtracting.

Finally, for the years 2084 to 2096, just subtract 84. For 2084 and 2096, of course, just use the 12 pattern we discussed earlier.

If you need a shortcut for 84 for the remaining years, simply subtract 4, then subtract 80. 2092 - 4 - 80 = 2088 - 80 = 2008 = year code of 3! I'm sure you have the idea by now.

You can practice this process with the Leap Year Codes: 2000 to 2096 quiz here. Again, don't forget to take advantage of the 12 pattern when you can.

Also, don't forget to practice actual dates in these leap years with this Leap Year Dates: 2000 to 2096 quiz.

Once you're comfortable working through both of these quizzes, it's time to learn how to determine the code for every year from 2000-2099 in the next section!

2000-2099

Are you ready to handle any date from 2000 to 2099? You're probably more ready thank you think!

Once you can handle leap years, the remaining years are simple.

When given a non-leap year, you need 2 pieces of information: The year code for the nearest leap year BEFORE the given year, and how far the given year is from that leap year. When you have these two pieces of information, simply add them together (remembering to drop any multiples of 7, as we've discussed before), and you have the year code.

For example, take 2009. The closest leap year is 2008, which has a year code of 3 (remember?), and 2009 is 1 year later. So, we work out 3 + 1 = 4, so 2009's year code is 4!

How about the year code for 2051? The nearest leap year BEFORE that is 2048, and we can use the 12 rule to determine that the year code is 4. Since 2051 is 3 years later, we do 4 + 3 = 7 = 0 (don't forget to drop out multiples of 7!), so 2051 has a year code of 0.

How about a tricky one like 2094? It's 2 years after 2092, which has a year code of 3 (remember how we know that?), so 2 + 3 = 5, so 2094 has a year code of 5.

Since you can't be more than 3 years after a leap year in any date from 2000-2099, this is a relatively simple adjustment.

To get practiced with this approach for determining year codes for any year, use this Year Codes: 2000 to 2099 quiz.

Once you get comfortable with that quiz, move on to the Dates: 2000 to 2099 quiz here.

Being able to determine the day of the week for any date in the 21st century is an impressive feat on its own. Once you're comfortable with doing that, you can move on learning how to handle dates in other centuries in the next section.

Other Centuries

Often you'll get asked about dates in the 20th century, especially if you're discussing someone's birthday. How do you handle those?

For dates from 1900 to 1999, simply work out the similar date for the 2000s, and then add 1. That's it!

For example, Let's say someone tells you they were born on January 20, 1985. Start as if you were working out January 20, 2085. 85 is a 1 year and January is a 6, so 1 + 6 = 7 = 0. Reduce 20 to 6 (cast out multiples of 7!) to get 6, and add 1 for the 20th century to get 7, which drops to 0. That 0 is the code for Sunday, and sure enough, January 20, 1985, was a Sunday.

Once you've worked out the day of the week for a given date in the 21st century, there's a simple pattern to alter the day for other centuries:

• 2300 to 2399 = add 1
• 2200 to 2299 = add 3
• 2100 to 2199 = add 5
• 2000 to 2099 = add 0
• 1900 to 1999 = add 1
• 1800 to 1899 = add 3
• 1700 to 1799 = add 5
• 1600 to 1699 = add 0

There is one VERY important note here for January and February dates in the years ending in 00: If a year ends in 00, it's only a leap year if it's divisible by 400. The years 1600, 2000, and 2400, and so on are leap years, while years like 1900, 1800, and 2100 are not. If you're given a January or February date in a year ending in 00, double check whether it's a leap year before you make the leap year adjustment.

Take a few minutes to study these adjustments, and then practice using them in the Dates: 1600 to 2399 quiz here.

Assuming you've put in the practice, you should be ready to give the day of the week for any date. In the next section, I'll provide a few tips and some background that can help improve your performance.

Calendar Background

The current calendar system we use is known as the Gregorian calendar, since it was introduced by Pope Gregory XIII. It was first put into use in 1582 by the Catholic countries, so the calculations you've learned aren't really effective for dates before 1600.

In addition, many non-Catholic countries didn't adopt the Gregorian calendar until much later. Britain and its colonies didn't adapt the calendar until 1752. The use of the Gregorian calendar as a worldwide standard, however, didn't happen until the 1920s!

Tips

• I can't emphasize enough the speed advantages of dropping multiples of 7, and becoming comfortable with that process. After you get use to doing this for dates from the 7th to the 31st several times, it almost becomes automatic.

• Carry a perpetual calendar! It's one thing to do this feat and know you're right. When you're doing it for an audience, they'll need some way to verify that you're correct. Originally, this meant carrying around a bulky book of calendars, but many mobile devices today make this much easier.

You'll generally want an app that mainly generates calendars for a wide variety of year, without appointment features, such as QuickCal for the iPhone and iPod Touch. iPad users can use YearViewer, and Android users can use Two Hundred Year Calendar or Day of Week.

• Want to practice on the go? Download these free mp3 files that give a date, then pause, then give the day of the week. The pauses range from 30 seconds down to 3 seconds, so you can challenge yourself as you get better. They're available in both DATE/MONTH/YEAR order (common in the UK, Australia, and Europe) and MONTH/DATE/YEAR (common in the US).

• As you've seen, working out year codes can take longer than just remembering the month, date, and week codes. When performing, the smart thing to do is ask for the year first, work out the year code as needed (including whether a leap adjustment will be needed), and only then ask for the specific date.

That way, not only do you get the year calculation out of the way, but you'll be able to determine the weekday more quickly and it will appear more impressive to your audience.

• When you're comfortable performing the feat this way, but you find you desire to be quicker, there is a more advanced step you can take. You can completely eliminate the calculations of the year code by memorizing the 100 codes needed for the years 2000-2099.

To do this, you'll need to be familiar with the Link System, the Shape Peg System and the Phonetic Peg System (AKA the Major System) (with images for 0 to 99).

Once you've practiced those systems and are comfortable with them, you use the Phonetic Peg System for images to represent the last two digits of the year (0 to 99), and the Shape Peg system for the year code (from 0 to 6). You then use the Link System to mentally link those two images together.

If you decide to go this method, here's a complete chart of the years from 2000 to 2099 with their corresponding year codes. Because the images people use with the above systems are so widely varied, I've avoided suggesting any mnemonics.

Year	Year Code
2000	0
2001	1
2002	2
2003	3
2004	5
2005	6
2006	0
2007	1
2008	3
2009	4
2010	5
2011	6
2012	1
2013	2
2014	3
2015	4
2016	6
2017	0
2018	1
2019	2
2020	4
2021	5
2022	6
2023	0
2024	2
2025	3
2026	4
2027	5
2028	0
2029	1
2030	2
2031	3
2032	5
2033	6
2034	0
2035	1
2036	3
2037	4
2038	5
2039	6
2040	1
2041	2
2042	3
2043	4
2044	6
2045	0
2046	1
2047	2
2048	4
2049	5
2050	6
2051	0
2052	2
2053	3
2054	4
2055	5
2056	0
2057	1
2058	2
2059	3
2060	5
2061	6
2062	0
2063	1
2064	3
2065	4
2066	5
2067	6
2068	1
2069	2
2070	3
2071	4
2072	6
2073	0
2074	1
2075	2
2076	4
2077	5
2078	6
2079	0
2080	2
2081	3
2082	4
2083	5
2084	0
2085	1
2086	2
2087	3
2088	5
2089	6
2090	0
2091	1
2092	3
2093	4
2094	5
2095	6
2096	1
2097	2
2098	3
2099	4

We'll start with the basics by introducting the concept of radians. What exactly are radians?

Radians are an alternative to degrees. Let's make sure that everyone is on the same page, with a quick and fun refresher course in degrees:



Most people are familiar with degrees. They're an absolute necessity when studying things like geography (where on Earth am I?) or astronomy (When will I be able to see a given star from where I am on Earth?). So why do we even need an alternative to degrees?

I'll highlight the problem with a smaller-scale example. Imagine a public park with circular running tracks of varying sizes, and two people, one who is trying to build his speed and stamina for a 100 meter race, and the other person who is timing the runner. They find a track, and the timer stands in the center of the circular track. The runner notes that the circular track is some weird amount, say, 142 meters in circumference, so he needs to know the location of the 100 meter mark.

The timer whips out his calculator, divides 100 by 142, getting 0.704225352. He multiplies this by 360 degrees, since the track is circular, and cheerfully exclaims that all the runner has to do is run 254 degrees (253.521127 degrees, to be more precise) to cover 100 meters on that track. At this point, the runner gives the timer a funny look.

See the problem now? 254 degrees is great when describing how far the timer, standing in the center of the circular track, has to turn to watch the runner. This doesn't help the runner much, as the runner isn't standing in the center. That's the whole thing with degrees; they're great for observers, such as the timer, but not so much for people or things moving around a circumference, such as our runner. This is why there's a need for an alternative to degrees.

Note that the absolute distance of 100 meters isn't of much help, either. On the different sized tracks in the park, that distance will be a different number of laps on each track. So, absolute distance and degrees aren't handy, thus we introduce radians. So what exactly are they?

Since the radius of a circle has a constant relationship with the circumference, that of Pi times twice the radius, how about counting off how far we've traveled around the circumference of a circle in terms of that circle's own “radius units”? It's a good idea, but “radius units” sounds weird (I think it's the repetition of the letter U), so we use the term radians instead!

It boils down to this: 1 radian is simply the length of a circle's radius traveled around that same circle's circumference.


That also gives us the formula to calculate radians: Radians = distance traveled / radius.

Math books like to scare you by writing this same formula as θ = s / r. θ is pronounced “theta”, and simply refers to the answer in radians. s means “arc length”, or, the distance traveled around the circumference. r, the only straightforward shorthand in this formula, means the length of the radius.

Going back briefly to the oberver's point of view, 1 radian translates into about 57.3 degrees (actually 57.29577951... and on and on). For the oberver, this certainly isn't a nice neat number, like 90 degrees, 180 degrees, or 360 degrees. As we've already learned though, the observer's viewpoint is not the point of radians.

However, in the next tab, we'll introduce the unit circle concept itself, and see how radians make things nice and easy for the person or thing that is moving around it.

Since we now have the scalable concept of radians to work with, we can now develop an entire scalable circle with which to work. We'll start by drawing on graph paper a circle centered at the (0,0) mark that has a 1 unit radius (remember cartesian coordinates?). It looks at it does below, with the coordinates marked where it crosses an axis:


That fact that it's a circle with a one-unit radius gives us the shorthand term for it: unit circle. Let's take a closer look at it what a unit circle can do.

Here's your first question: Since radians measure how far around a circle you've traveled, what's the distance around a full unit circle in radians? Let's work this out.

Our circle has a radius of 1 unit, so the circumference (our total distance traveled, in this case) is 2 times the radius times Pi, so we have 2 times 1 unit, giving 2 units, times Pi, resulting in 2 * Pi, or 2π for short (That π symbol is the Pi symbol, not a small letter n). That's just the total distance traveled, though.

To convert that to radians, we need to divide by the radius, of course. Our radius is 1 unit, so we work out 2π/1, which is simply 2π. In other words, the distance around a full circle is 2π radians in length!

Now you see why 1 radian is such a weird angle in degrees (1 radian = 57.29577951... degrees). When doubled and multiplied by Pi, it has to give a nice even 360 degrees. Again, we're going to stick to radians in this discussion, so just think of 2π radians as a full circle.

From here, it's not hard to see that π radians gives us half of a circle, and π/2 radians gives us a quarter circle. Three-quarters of a circle, then, would give us a distance of 3π/2 radians.


So, what distances are we traveling in radians when we traveling 45 degrees? 45 degrees is the centered observer's way of saying an eighth of a circle. The mover thinks of a circle as being 2π radians, so 2π times 1/8 would be 2π/8, or the same as π/4 radians.

Every 1/8 of a trip around the circle, then, would be in units of π/4 radians:

• 1/8 = π/4 radians
• 2/8 = 2π/4 = π/2 radians
• 3/8 = 3π/4 radians
• 4/8 = 4π/4 = &pi radians
• 5/8 = 5π/4 radians
• 6/8 = 6π/4 = 3π/2 radians
• 7/8 = 7π/4 radians
• 8/8 = 8π/4 = 2π radians

When you see the patterns and understand the process, it's actually not hard to understand.

Since angles of 30 degrees, or 1/12 of the distance around the circle, are also common, they're not hard to work out in radians. 1/12 of 2π radians comes down to π/6 radians. From there, we get:

• 1/12 = π/6 radians
• 2/12 = 2π/6 = π/3 radians
• 3/12 = 3π/6 = π/2 radians
• 4/12 = 4π/6 = 2π/3 radians
• 5/12 = 5π/6 radians
• 6/12 = 6π/6 = π radians
• 7/12 = 7π/6 radians
• 8/12 = 8π/6 = 4π/3 radians
• 9/12 = 9π/6 = 3π/2 radians
• 10/12 = 10π/6 = 5π/3 radians
• 11/12 = 11π/6 radians
• 12/12 = 12π/6 = 2π radians

Here's everything you've learned so far in one diagram, with the 90 degree units marked in black, the 45 degree units marked in red, and the 30 degree units marked in blue:


So, if you can remember that 2π radians takes you all the way around the circle, and that π radians, as well as 6π/6 and 4π/4 radians, takes you halfway around the circle, working out the rest of the numbers isn't difficult at all.

In the next tab, we'll discuss another amazing way the unit circle becomes useful when we focus on figuring out coordinates.

I have some quick refresher questions before we move on. When you've traveled π/2 radians, at what (x,y) coordinates are you? You're at (0,1).

Here's a tougher one: When you've traveled π/4 radians, at what (x,y) coordinates are you? That one's a little trickier. For this one, we're going to have to go back the viewpoint of the observer in the center, and thus back to degrees.

The radius of the unit circle is, of course, always 1 unit. If we construct a right triangle with the radius line as the hypotenuse, we could construct a right triangle for any angle in the circle:


Since π/4 radians from the central observer's point of view is 45 degrees, we'll create a 45 degree right triangle to help work out those coordinates:


Since we're looking for the (x,y) coordinate, the length of the side adjacent to our angle (the one running along the x axis) would give us the x coordinate. The height of the side opposite the angle (the one running straight up to meet the radius/hypotenuse) would give us the y coordinate.

If you remember all those lessons about right triangles, it's at this point where you begin to realize how many tools we can use here.

• Pythagorean Theorem: a² + b² = c²
• Sine of any angle = opposite/hypotenuse
• Cosine of any angle = adjacent/hypotenuse

(Don't forget: SOHCAHTOA)

Since the hypotenuse is 1, and both the sine and cosine divide by the hypotenuse, not only does this make the math nice and easy, but the answers will also be the exact coordinates we need!

The x coordinate is the length of the side adjacent to the angle, so we need to use the cosine formula. By cheating and using a calculator (make sure to set it in degrees, not radians!) we find that cos(45 degrees) = 0.707106781.... To get the y coordinate, we find the length of the side opposite the angle by using sin(45 degrees), which is also 0.707106781....

So, we have our coordinates for a 45 degree, or π/4 radian, angle: (0.707106781...,0.707106781...). Hmmm, that's another one of those numbers that goes on forever. Isn't there a better way to state that number?

Let's run through the Pythagorean Theorem approach, especially know that we know that sides a and b are equal, and see what we can come up with:



“OK,” you say, “so what? We just came up with the same numbers.” Let's back up a bit to to where a squared (and b squared, in the specific case of a 45 degree right triangle) was equal to 1/2. Instead of using .5, let's see where working with that 1/2 as a fraction can take us:



Hmmm...writing one over the square root of 2 is certainly much easier than that long string of numbers. Indeed, when you're tested on unit circles on many standardized tests, they'll usually ask you to write the π/4 radian or 45-degree coordinate in exactly that way.

Before moving on, though, I'd like to make that fraction a bit neater by making the bottom a whole number. We do that by multiplying both the top and bottom by the square root of two, and expressing the coordinates as fractions:



Not only is this a cleaner way to write the coordinates, it will make this and the other coordinates we work out much easier to remember, as you'll see in later sections.

The fractions may look funny, but all they're really saying is, “If you take the square root of 2, and divide it in half, that's a shortcut to working out this coordinate.” Or, put the opposite way, “Hey, if you take this coordinate, double it, and then square it, you get a nice, simple number, in this case - 2!”

We'll work out the coordinates and their corresponding fractions for the 30- and 60-degree angles in the next section, as well as their multiples.

So, now we not only know the coordinates for a 45 degree, or π/4 radian, point on a circle, we know that the cosine will give us the x coordinate and that the sine will give us the y coordinate on the unit circle. We also know that expressing these coordinates in fractions is much simpler than writing out the endless irrational decimals.

Let's work out the coordinates for the point at 30 degrees (π/6 radians):



Hey! That .5 is nice. That's easy to express as a fraction. But what about that other number? Well, everything else has been expressed over 2, so let's see what happens when we express that number over 2:



Ummm...yuck. Well, square roots seem to be popular in these fractions. Is 1.73205081... the square root of anything? Yep! It turns out that it's the square root of 3! So, the coordinates could be expressed this way:



Next, let's work out the coordinates for 60 degrees (π/3 radians):



Whoah! We get the same numbers as the 30 degree angle, only switched! That's not a coincidence. Remember that the internal angles of triangles always add up to 180 degrees. With a right triangle (a triangle with a 90 degree angle in it) that has a 30 degree angle in it, we find that 180 - 90 - 30 = 60 degrees, the remaining angle. Effectively, the 60 degree coordinates are just the 30 degree coordinates viewed from the other side of the triangle.

Let's review all the information, including the coordinates, that we know so far:


If you think of 1 as also being the square root of 1, you could write 1/2 this way:



So, at all the points where the circle crosses an axis, the coordinates involve 0, +1, or -1 - Simple! And at the oft-used 30° (π/6 radians), 45° (π/4 radians), and 60° (π/6 radians), all the coordinates involve these interesting fractions:



See that? Everything in the fractions is a square root of something over 2! Even better, there's a simple 1, 2, 3 progression! Even with all the complex math involved, it's all boiling down to 0, 1, 2, and 3.

That's easy enough to remember, but how do you remember which fractions are set at which coordinates, especially since it's not hard to make a mistake and switch around the coordinates for both 30° and 60°? Here's a video that will teach you quickly, using that 1, 2, 3 progression:



What about all the remaining major angles, such as 120°, 135°, and so on?

Well, you could go through and work out the coordinates laboriously as we did above, working out the sines and cosines, and so on. However, there's an easier way. We're measuring the 135° (3π/4 radians) angle from the 0° angle. When measured from the 180° (π radians) angle, it works out to be our old friend, the 45° (π/4 radians) angle.

One difference, though, is that we're now on the negative side of the x axis, so that the x coordinate will be negative. The means the coordinate for the 135° (3π/4 radians) angle will be:



Similarly, you can work out the 150° angle as a 30° angle, and so on. They work out to the coordinates we've already determined, but you need to make sure that you adjust the signs (+ or -) for the appropriate section.

Are you ready for this? Here's the entire unit circle worked out:


And here's how to draw it that so you remember all that data:



I like the approach used in this video of counting the π/3, π/4, and π/6 sections separately, so as to keep them mentally separate.

In the next tab, I'll review and break down the patterns for easier understanding. You'll also learn how to do some quick math on your fingers to get the coordinates.

It may seem like a lot that you've learned so far, but it boils down to a few basic things:

• Degrees deal with the observer's point of view. Radians deal with the mover's point of view.
• 1 radian = 1 radius length around the circle's circumference.
• Radians work very well with Pi, to the extent that π radians = 180 degrees, and 2π radians = 360 degrees.
• If you think of π radians as being 180 degrees, and you know your multiples of 30 and 45, the remaining degree angles aren't difficult to work out in radians.
• All the coordinate points where the circle cross an axis involve a 0, a -1, or a +1. Knowing coordinates makes it easy to work out which is which.
• All the coordinate points of π/6 (30°), π/4 (45°), and π/3 (60°) radians involve fractions easily remembered with the 1, 2, 3 progression taught in the video in the previous tab.
• The other major coordinate points can be worked out quite easily from those first 3 sets, as long as you adjust the signs properly.
• Cosine will always give the x coordinate on the unit circle, and sine will always give the y coordinate on the unit circle.

That last fact is especially interesting. Thanks to the simple progressions involved, you can actually work out the cosine and sine on your fingers with a simple trick! If you're familiar with tangents and cotangents, this video teaches how to handle those on your fingers, as well.



Betterexplained.com has a wonderful article you should read at this point, called Intuitive Guide to Angles, Degrees and Radians.

It really drives home the power of radians. Take this example from the article:

Let’s try a real example: you have a bus with wheels of radius 2 meters (it’s a monster truck bus). I’ll say how fast the wheels are turning and you say how fast the bus is moving. Ready?

“The wheels are turning 2000 degrees per second”. You’d think:

Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second. Circumference = 2 * pi * r, so it’s moving, um, 2 * 3.14 * 5 and 5/9ths… where’s my calculator…

“The wheels are turning 6 radians per second”. You’d think:

Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.

Wow! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover.

The reverse is easy too. Suppose you’re cruising 90 feet per second on the highway (60 miles per hour) on your 24″ inch rims (radius 1 foot). How fast are the wheels turning?

Well, 90 feet per second / 1 foot radius = 90 radians per second.

That was easy. I suspect rappers sing about 24″ rims for this very reason.

I'm not sure that's the reason they sing about 24″ rims, but I'll go along with it while it lasts.

Think of the power you have here. Sure, the unit circle seems complex at first, but the power and patterns it presents when remembered and understood make many things simple.

Scaling up to real-world circles and back down to the unit of circles, as well as determining their motion, becomes a simple matter of multiplication. You can work out sines, cosines, tangents, and cotangents on your fingers.

Converting from radians to degrees and back, with a little practice, isn't difficult.

Which brings to mind the question of practice. The simplest practice I can offer is to print out several copies of this PDF, and repeatedly fill it out until you're good enough to do so in under 5 minutes, just as it says.

Sporcle offer this ingenious quiz for practicing radians and degrees. With help from the videos from the previous tab, this shouldn't be too tough.

The most complete quiz on the unit circle that I've found, however, is Math Fanatic's Unit Circle and Trigonometry Quiz. The menu lets you select exactly what aspects you want to practice, and the quiz itself lets you practice each section for as long as you need or want. Since it covers most of trigonometry, and not just the unit circle, there will be some quizzes not covered in my unit circle tutorial.

I hope you find this useful, and I also hope it helped you better understand the unit circle.