A good puzzle has the counterintuitive quality of being very relaxing because of the thought you have to put into them. Here's a few good sites to lose yourself in some classic puzzles.
• AMAZING Productions - Just started this month, this is a blog focused on mazes. Each post is a picture of a maze, and clicking on it will enlarge the picture. These are very high-quality original mazes, and should be printed out and given the time they deserve.
• Desert Labyrinth - Do you prefer your mazes in 3D? If so, check out Desert Labyrinth, a simple yet well-rendered 3D maze game that's different each time you play it! The interface is simple enough that it helps keep the focus on solving the maze, not figuring out how to play. You'll probably want to click on the compass to use the overhead view to help. You can even learn about how the game was built here.
• Puzzler's Paradise - If you enjoy logic puzzles that provide you with clues like, Mary was 2 years older than the person who owned the dog you'll enjoy this site. You can play online with the interactive grids, or print them out and work them by hand! New puzzles are posted often.
• Jigzone - There's nothing quite like a good old-fashioned jigsaw puzzle. There are several sites out there, but Jigzone has two important qualities over its competitors: 1) The interface is better designed and replicates the feel of the experience, and 2) It works even on devices that don't display Flash. As a matter of fact, the experience is even better on a touchscreen mobile device, since you're grabbing the pieces with your fingers directly!
• Puzzles.com - Looking for a wider variety of puzzles? Then this is the place! From original new puzzles to classics from older puzzle books (including many from Martin Gardner), puzzle lovers should definitely bookmark this site and return regularly.
• Chess and Poker.com - If you enjoy the Mental Gym, you'll definitely enjoy this site. Besides strategy guides for its name sakes, it features guides for everything from Tic-Tac-Toe to Dominoes to Carcassonne! Many classic puzzles are also taught here, including Rubik's Cube, Sudoku, and the 15 Puzzle.
• Jaap's Puzzle Page - This one has been a Grey Matters favorite for a long time. It teaches how to solve a wide variety of commercially available puzzles. In many cases, you can play those same puzzles in online versions, and even share them on your own website! If you want to understand the math behind these puzzles, the site usually gives high-quality mathematical explanations, as well.
Do you have any favorite puzzles sites you'd like to share? Let's hear about them in the comments!
A good puzzle has the counterintuitive quality of being very relaxing because of the thought you have to put into them. Here's a few good sites to lose yourself in some classic puzzles.
(Note: This is a repost, with some link updating and minor rewriting, from this same week 3 years ago. I repost it because it has become relevant again.)
If you do math at all at the gas pump, it's probably either related to how many gallons you can get for a given amount of money, or how much money will be required to get a needed amount of gas. If you're willing to do a bit of math and planning before you go get your gas, you can actually work a surprising amount of real savings into the equation, as well.
How do you save on gas? The obvious first answer is to find the cheapest gas you can. My grandfather's method for this was to drive around looking station by station, but that only works well when you're sure you can find gas lower than 35 cents/gallon. Unsurprisingly, the internet is here to help! Sites such as fueleconomy.gov, FuelMeUp, and GasBuddy make short work of finding the lowest gas prices in your area.
Unless you find the cheapest gas in your immediate area, another question begins to raise its head at this point. Sure, if you go a little farther to that station with the cheap gas you can save some money, but if you factor in the gas you'll burn going the extra distance, and the added gas you'll require, are you really saving money? With the current level of gas prices, this isn't a trivial question.
Fortunately, Kimberly Crandell, better known as Science Mom, tackled the question of whether nearby expensive gas or cheaper gas across town was cheaper in July 2007.
As I've explained, there is some math involved, but there are only five different factors involved: The number of gallons needed, the gas mileage of the car, the cost of the closer (more expensive) gas, the cost of the farther (cheaper) gas, and the miles out of the way for the cheaper gas (Google Maps, Yahoo! Maps, or MapQuest will come in handy here). In the article, you learn the formulas to process this, and how to solve for the savings you'll get, as well as the break even points for cost per gallon, total gas gallons, and distance.
Understanding and working through the formulas is one thing, but how about if you would just like to get your answer and go? Once again, the internet is here to help. My favorite tool for this step is Instacalc, which I first mentioned in August 2007.
I've created an instacalc version of Kimberly Crandell's equations where all you have to do is plug in the five factors (remembering that the two prices requested are both price per gallon).
If you prefer, I've also created a metric version of this calculator, for readers in other countries. Whichever version you use, I hope this helps save you some money and that you find it useful!
Update: If you've already started enjoying my previous post's recommendation of William Spaniel's Game Theory 101 videos, you'll enjoy his method of finding cheap gas without perfect information. This method, based on game theory, is equally mathematical, but requires fewer calculations.
The sciences, especially math, are not often associated with fun. To combat that idea, I've gathered together links to many math and science documentaries online that challenge the notion that learning can't be fun.
• Eureka! - No, I'm not referring to the SyFy drama. Eureka! is a series of 5-minute animated shorts, each of which focused on an aspect of basic physics, such as intertia or mass. They're very creative, and great to help you catch up on the basics of physics.
• Game Theory 101 - Game theory is the study of human choices made in situations with defined rules, such as games. At this writing, there are several online courses teaching game theory, but William Spaniel's simple and direct videos make it easier to understand than most of the others. To help you follow along, he's even made a free spreadsheet calculator available!
• Nice Guys Finish First - While I'm on the topic of game theory, here's a BBC documentary on one of the most classic aspects of game theory – the Prisoner's Dilemma. At first, it seems like just a simple theoretical game. However, when you consider that it can model everything from business negotiations to international politics, it becomes much more important.
• Breaking Vegas: The MIT Blackjack Team - If you saw the movie 21, you saw the somewhat dramatized version of the story of the MIT Blackjack Team. This documentary provides a better understanding of exactly how the team came together and eventually fell apart. There's also a British documentary about the MIT team called Making Millions the Easy Way with more details.
• The Nature of Things: Martin Garder - Martin Gardner brought such a fun approach to mathematics for the masses, that it's said he turned a generation of kids into scientists, and a generation of scientists into kids. If you're not familiar with his work, this documentary is an excellent place to start.
• Algebra: In Simplest Terms - Sol Garfunkel hosts this series where Algebra is brought to life with real-world uses. This series is a great reply to the math student in all of us who is always asking, When am I ever going to use this? If you can catch his other series, For All Practical Purposes, before its removed from Google Video on April 29, 2011, I suggest you watch it, as well.
• James Burke's Documentaries - I can't say enough good things about James Burke's documentaries. While the zig-zag approach he took to teach history in his documentaries may seem run-of-the-mill to a generation that grew up with the internet, it was a breakthrough approach in its time. It still helps history come alive and feel more real, and thus more accessible. I've even gone to the trouble of annotating every episode of his first two documentary series with Wikipedia links.
• The Universe: Beyond The Big Bang - If you particularly liked James Burke's Infinitely Reasonable episode, Beyond the Big Bang should be your next stop. The two are very close, but this one spends two hours going into detail. Seeing Einstein's theories presented with a carnival ride analogy is a highlight of this show.
• Cosmos - For many in the 70s, Carl Sagan was the first to challenge their notions about the nature of the universe, as well as humanity's place in it. Thankfully, Hulu brings Sagan's delightful brand of astronomy right to your desktop.
• Scientific American Frontiers - This classic series, hosted by Alan Alda, and named after the magazine that brought you Martin Gardner, always seemed hard to catch on TV, with it's schedule of airing only once or twice a month. Now you can catch episodes on Hulu, or at PBS' own site.
• NOVA - NOVA is probably one of the longest running series anywhere on TV, having started in 1974. It was already in its 16th season when the Simpsons premiered! Many of its episodes are so classic, you may have seen them in school. To this day, it remains one of the best ways to keep up with the sciences.
• Hunting the Hidden Dimension - OK, I've already covered NOVA as a whole, but their documentary about Benoit Mandelbrot and fractals is a standout. It's also a fascinating lesson about how long simple ideas can remain hidden.
• A Brief History of Time - Stephen Hawking's classic work is a wonderful examination of the nature of the universe, especially concerning our understanding of it since the days of Einstein.
• CalTech: The Mechanical Universe - This 52-part series is a true college-level course in sciences, taught with recreations and computer graphics that really aid understanding. It covers everything from atoms to planets in a very accessible way, including some episodes on basic calculus that aren't hard to understand. The youtube edition is missing some episodes, but you can find a more complete set at Google Video until April 29, 2011.
Answers to the puzzle in Sunday's post:
Besides 28 ÷ 7 = 13 and 25 ÷ 5 = 14, there are 20 other sets of numbers that can be substituted in the comedy routine:
12 ÷ 2 = 15
14 ÷ 2 = 25
16 ÷ 2 = 35
18 ÷ 2 = 45
15 ÷ 3 = 14
18 ÷ 3 = 24
24 ÷ 3 = 17
27 ÷ 3 = 27
16 ÷ 4 = 13
24 ÷ 4 = 15
28 ÷ 4 = 25
36 ÷ 4 = 18
15 ÷ 5 = 12
35 ÷ 5 = 16
45 ÷ 5 = 18
18 ÷ 6 = 12
36 ÷ 6 = 15
48 ÷ 6 = 17
49 ÷ 7 = 16
48 ÷ 8 = 15
April's editions of snippets is dedicated to math and videos, so let's get started right away!
• Many people originally knew Danica McKellar as Winnie from The Wonder Years, and now know her as the author of several books designed to make math less intimidating for girls. Ever wonder how that change came about? In her brand new blog, she answers that question in her Why Math? post. Between that, her current roles, and her baby boy Draco, she's been keeping pretty busy:
• One of the newest items in the Grey Matters Store is my original 2011 Dice Calendar iPad, iPod Touch, and iPhone cases. The dice part is easy to understand, but the calendar part is hidden as part of the design. Here's the explanation:
• Scam School recently delivered two podcasts that dealt with math somewhat indirectly. By that, I mean that the focus is on numbers and patterns instead of equations. Try and see if you can work out the answers for yourself before getting the answers:
The first of these two podcasts is called Petals Around The Rose, which I originally brought up in their forums:
The other one concerns a trio of number puzzles:
• During their 1941 movie, In The Navy, Abbott and Costello present a comedy routine in which Costello keeps showing that 28 ÷ 7 = 13:
10 years later, Marjorie Main and Percy Kilbride would perform a similar routine in Ma and Pa Kettle Back on the Farm. Only this time, the proof was that 25 ÷ 5 = 14:
This routine originated in the 1921 musical Shuffle Along, written by Flournoy Miller and Aubrey Lyles. This routine was still going around in 1972, when Flip Wilson performed the former version with Michael Jackson! Olivette Miller Darby, who was Miller's daughter, saw the routine and sued Flip Wilson for unauthorized use of the material.
Even though the math itself isn't exactly sound, here's some food for thought: Assume that there are always a single 1-digit number and two 2-digit numbers involved, what other sets of numbers could be used in such a routine? I'll give the answer in Thursday's post.
(NOTE: Check out the other posts in The Secrets of Nim series.)
Do you read Grey Matters regularly? If so, you'll have a great advantage in figuring out Star Nim. Even if not, you can have some fun playing and figuring out the game right in this post!
First, let's get the rules of Star Nim down. It's a standard Nim game. As defined in the first Nim post, that simply means that the last person to remove an object is the winner.
Star Nim is played on a board like the one below (in fact, that's where you'll be playing it!), which starts with a penny in each circle (8 total). As with all versions of Nim, players alternate taking turns. On each turn, a player may remove 1 penny, or 2 pennies connected by a straight line segment. Neither player may remove 0 objects on their turn.
In the version below, the computer will always go first, and you'll be the 2nd player. This is because the 2nd player can always win with the right strategy, and the purpose of playing this game is to figure out that winning strategy.
To start, click the Start New Game button, and pennies will appear in the circles, except for any pennies the computer has taken as its first move. Click on any penny to remove it, and then the computer will ask if you wish to remove a 2nd penny. If you wish to removed 2nd penny, click OK, and if you don't wish to, click Cancel.
The computer will then make its move, removing either 1 penny or 2 pennies connected by a line, and then it's your turn again. If you click on a penny and no other pennies are along lines connected to that one, the computer will not ask you if you wish to make another move.
Remember that the object of the game is to be the last person to remove a penny, and that you're playing over and over again in order to figure out what the winning strategy is for the 2nd player. Don't read beyond the game until you've at least tried this game several times.
Did you figure it out? This isn't one of the more challenging variants of Nim, so it usually doesn't take too many times to figure out the winning strategy.
A good knowledge of the other versions of Nim is a good help, of course. If you recognize the board from the Penny Star Puzzle variation of the last post's Knight Shift puzzle, you should also realize that the design plays an important part.
Let's start with the last part first. As discussed in Knight Shift post, the pattern of the board can be opened up into a simple circle. If you didn't figure out the strategy already, try playing on the board above while simultaneously making the same moves on a circular version of the board. You'll quickly see why the winning strategy works, as well.
If you can generalize the solution, you can also play on any size board.
Obviously, the first secret to winning this game is to let the other person go first. After the other person has moved, you need to make sure that your move leaves the board with two equally-sized groups of pennies remaining on the board.
With an even number of spots on the board, as we have above, this simply means taking the same number of coins as the first player. If the board has an odd number of spots, such as the 9-spot board in chapter 12 of Martin Gardner's book, Time Travel and Other Mathematical Bewilderments, this strategy requires you to take the opposite number of pennies as the first player. You'd take one penny if the first player took two, and vice-versa.
After that, the strategy is much like that of Multi-Pile Nim when you're left with two equal piles. Simply take the same number of pennies every time as the other player did on the previous turn, making sure to take only the pennies directly across from the ones that were taken. Once you understand this technique, the star's symmetry almost becomes a guide for your next properly play.
How long did it take you to figure it out? Did you try playing Star Nim with anyone else? I'd love to hear about it in the comments!
April 10th is both the anniversary of Henry Ernest Dudeney's birth, as well as the anniversary of Sam Loyd's death. Henry Ernest Dudeney was to puzzles in England what Sam Loyd was to puzzles in America.
In honor of those anniversaries, I'd like to take a close look at a particular puzzle from Henry Dudeney's second book, Amusements in Mathematics (available online at Project Gutenberg for free), which he called the Four Frogs.
Henry Dudeney's presentation of this classic puzzle, dating back to 1512, included the following description and fanciful illustration:
341.—THE FOUR FROGS.
In the illustration we have eight toadstools, with white frogs on 1 and 3 and black frogs on 6 and 8. The puzzle is to move one frog at a time, in any order, along one of the straight lines from toadstool to toadstool, until they have exchanged places, the white frogs being left on 6 and 8 and the black ones on 1 and 3. If you use four counters on a simple diagram, you will find this quite easy, but it is a little more puzzling to do it in only seven plays, any number of successive moves by one frog counting as one play. Of course, more than one frog cannot be on a toadstool at the same time.
The original version of this puzzle employed a chessboard, two black knights, and two white knights. The way the frogs move along the lines in the above puzzle is based on the way a chess knight moves on a chess board: one square vertically and two squares horizontally, or one square horizontally and two squares vertically (as shown here).
It seems like a simple enough puzzle, but it's surprisingly challenging. Try it yourself below. That's not just an image, it's a working version of the puzzle. The object in the version below is to switch the positions of the black knights with those of the white knights (when completed, the board should look like this).
To move a knight, simply click on the knight you wish to move, and it will be highlighted in red. Next, click on the destination square, and the knight will move to that square. when you win, an alert window will tell you that you've done it, and how many moves you took. Try and do it in as few moves as possible:
So, what's your record for fewest moves? For that matter, what is the fewest number of moves, and how that we prove that?
At this point, many people would probably throw up their hands in frustration, as analyzing the problem and minimizing the number of moves needed doesn't sound like much fun.
Part of the reason I enjoy mathematical puzzles, and indeed mathematics itself, is the sheer joy of seeing a seemingly complex idea turned into a simple one. This is one of those cases.
Here is the original answer given by Henry Dudeney, along with his original illustrations:
The fewest possible moves, counting every move separately, are sixteen. But the puzzle may be solved in seven plays, as follows, if any number of successive moves by one frog count as a single play. All the moves contained within a bracket are a single play; the numbers refer to the toadstools: (1—5), (3—7, 7—1), (8—4, 4—3, 3—7), (6—2, 2—8, 8—4, 4—3), (5—6, 6—2, 2—8), (1—5, 5—6), (7—1).That's simple and elegant. Take the way in which the knights (or toads) can move, and break them down into a circle. Once you see that, it's quite simple to see AND understand the answer, isn't it?
This is the familiar old puzzle by Guarini, propounded in 1512, and I give it here in order to explain my "buttons and string" method of solving this class of moving-counter problem.
Diagram A shows the old way of presenting Guarini's puzzle, the point being to make the white knights change places with the black ones. In "The Four Frogs" presentation of the idea the possible directions of the moves are indicated by lines, to obviate the necessity of the reader's understanding the nature of the knight's move in chess. But it will at once be seen that the two problems are identical. The central square can, of course, be ignored, since no knight can ever enter it. Now, regard the toadstools as buttons and the connecting lines as strings, as in Diagram B. Then by disentangling these strings we can clearly present the diagram in the form shown in Diagram C, where the relationship between the buttons is precisely the same as in B. Any solution on C will be applicable to B, and to A. Place your white knights on 1 and 3 and your black knights on 6 and 8 in the C diagram, and the simplicity of the solution will be very evident. You have simply to move the knights round the circle in one direction or the other. Play over the moves given above, and you will find that every little difficulty has disappeared.
This simple puzzle has spawned a number of variations. Martin Gardner offered a simple one that deepens the understanding of the method behind this puzzle. In this version, you start the same way, except that one of the white knights is replaced with a red one (so you have two black knights, a white knight, and a red knight). This time, the challenge is two switch the places of the red knight and the white knight in as few moves as possible. Can you do it?
Nope, you can't do it. Why not? All you have to do is go back to the circle (illustration C above). Lay out your starting point in the circle. for example, with the white knight at 1 and the red knight at 3. Whether you go clockwise or counterclockwise, the red knight and white knight will never switch places, as there is no way for knights to jump each other in this puzzle.
Another variation, which Dudeney mentions in the same book as the original version is known as the Penny Puzzle. In this version, there's a board as shown here. You have to set down a penny on any empty circle, and slide it along either straight connecting line to another empty circle. The challenge is to place 7 pennies on the puzzle in this manner, without getting stuck. You can play an online version at this link.
As noted in the Scam School episode just below the puzzle, Chico Marx has even used a version of this with dollar bills to scam people out of money! Both the Penny Star Puzzle version and dollar bill versions work on the same idea as the Knight Shift puzzle above, but require a little different thinking due to the rules about movement.
Once you've got the hang of the previous versions, here's one last challenging version to try. The version at that link uses a 3 by 4 chessboard, and challenges you to switch the positions of 6 knights! Yes, this one can be done, but can you properly analyze it?
The site gives a solution if you get frustrated, but it doesn't give you the why behind the solution. Even if you break down and peek at the solution, try and analyze the solution in a similar manner to the way we did above.
What makes this one so tricky is that it doesn't break down into a nice neat circle, but it does break down into a simpler shape. Martin Gardner shows the breakdown of the 6-knight version here.
If you challenge your friends with this puzzle, I'd love to hear about any interesting experiences you have, or insights you developed. Write about them in the comments!
Since I redesigned the site, I've been working on redesigning the store.
Finally, the new Grey Matters store is now open!
I've designed to bring products of interest to Grey Matters reader from multiple stores together in one place, to make it more convenient. The store is designed in the same style as the rest of the Grey Matters site, with product descriptions in individual posts.
The heart of the store is the catalog slider (shown below). It works much like the slider on the main page of the other Grey Matters sections, but with a few differences. The first noticable difference is that it doesn't slide automatically, so it's completely under your control. The left and right arrow slide the catalog left and right, and the numbered tabs at the bottom let you jump to any individual panel instantly.
The biggest difference, however, is the filtering and sorting buttons below the slider, which allow you to view just the type of products you want in the way you want. For example, if you want to view just items related to math from Amazon.com, simply click the Amazon.com and math buttons, and the slider will rearrange to contain just those items.
By default, you're shown the newest items available, but you can also view items starting from the lowest priced, starting from the highest price, and also sorted alphabetically.
Did you find some combination of goods you'd like to share? It's easy! Simply copy the address shown in the Share this filter field, and send it your friend.
When you click on any product in the slider, the panel will fill with more information about it, including where you can buy it, a short description, and a link to a post where you can learn even more about it. For example, if you clicked on the book Moonwalking With Einstein, you'd see this:
Clicking on the Buy Moonwalking With Einstein here link, of course, would take you to the Amazon.com page where you can purchase it. The Learn More... links take you to an individual post focusing on that item, so you can learn about it in more detail. If an item isn't what you expected, a quick click on the Return to slider button will take you right back to the items you were viewing.
Many of the items have special preview links, such as book previews and videos. These are included to help you understand the idea behind the individual items, as well as giving you a closer look at the item before you decide whether to purchase.
If you've enjoyed reading Grey Matters, purchasing these items through our store page can help keep it going while getting something for yourself, as well. Check back often, as new items will be added regularly.
Now that I've helped make radian and degree conversion easier in my previous post, it's time to tackle the even more fearful part of the unit circle, the sine and cosine aspect!
Instead of working out sine and cosine in your head, though, you're going to have to work them out on your fingers.
On the last page of my unit circle tutorial, I include the following video, which teaches the basic finger trick we'll be discussing in this post:
We're not going to worry about tangents and cotangents in this post. However, note that this video is referred to as part 1, and the teacher mentions that she'll cover the other quadrants in another video. Unfortunately, as near as I can tell, that second video was never posted.
That's OK, because inspired by this video, I developed my own approach and mnemonics for covering the other quadrants. Make sure you've got the basics down as taught in the video above before dealing with the other quadrants.
The mnemonic I developed to help deal with this approach involves imagining that you live upstairs from a mom-and-pop store, where you work during the day for your parents (Mom and Pop). Before I explain it, here's the mnemonic itself, starting with Pop waking you up for the day:
- First, Pop comes upstairs to get you up.
- Second, He asks you to mop down the floor in the living area.
- Third, Mom comes upstairs to update you with some news.
- Fourth, You're taken downstairs to meet the new employee, whose name is Pam.
The video above teaches the hand trick for quadrant 1, which is in the upper right, so let's start there. The hand trick is simple there, as both the cos and sin are positive, and the fingers start palm towards you, with the left pinky being 0° up to the thumb being 90°. The important factors here are that the cos and sin are both positive numbers, and that you count in an upward direction on your hand.
How does this relate to the mnemonic? Remember who comes to get you up out of bed? It's Pop. PoP = Plus, Plus! In which direction is Pop heading? He's heading UP to get you UP and out of bed. This will help you remember to go UPward on your fingers.
Let's try a quick problem with this knowledge before we move on, such as the cos and sin of 60°. We're working in quadrant 1, so we recall that PoP goes UP first. The fingers go in upward order 0° (pinky), 30° (ring finger), 45° (middle finger), 60° (index finger) - Ah! That's the one we're looking for! We fold that in, and we see 1 finger above (giving up part of the cos), and 3 fingers below (giving us part of the sin). That translates into:
A quick adjustment for the square root of 1, which is 1, is made. Since PoP helps us remember that both signs are positive, we're done here.
Let's move onto quadrant 2, which is the upper left one. Remember, when dealing with circles, you count the quadrants in the order as if you're making a C (for circle!). We're in quadrant 2, so what happens second in our story? You're handed a MoP and told to MoP DOWN the floor.
This means that you're going to be working downard on your hand, from the thumb to the pinky. The thumb will still be 90°, but now the fingers will represent the angles as follows: 120° = index finger, 135° = middle finger, 150° = ring finger, and 180° = pinky. MoP, of course, reminds us that the signs will now be Minus, Plus.
With this knowledge, what are the cos and sin (respectively) of 135°? We head down from the thumb to the middle finger, since the middle finger represents 135135°. We fold that in, noting that there are 2 fingers above (helping us with the cos), and 2 fingers below (helping us with the sin). This would seem to suggest that the answer is:
However, this neglects to take the signs into consideration. Remember the MoP? Since that means Minus, Plus, we adjust our answer appropriately:
Got the idea? We're already half done! Remember, through all these quadrants, you're always working with the left hand, and with the palm facing you.
We're moving to quadrant 3, in the lower left. What happens in the 3rd part of the mnemonic? MoM comes UPstairs with an UPdate. You're already ahead of me, aren't you?
The pinky, of course, is still 180°, and we go UP (naturally) from there: 210° = ring finger, 225° = middle finger, 240° = index finger, 270° = thumb.
Think you can handle the cos and sin of 210°? Try it on your own before reading further. Don't forget to get help from MoM!
What did you come up with? We bend in the ring finger, the 210° finger. That 3 fingers over, and 1 below. This, along with the signs hinted by MoM, gives us:
To round this out with quadrant 4 (correct - the lower right one), we'll figure out 315°. What happend in the 4th and final part of the mnemonic? You go DOWNstairs to meet PaM, the new employee. Since you probably have the idea by know, which fingers go with which angles?
Going DOWN from the 270° thumb to the 360° pinky, we have a 300° index finger, a 315° middle finger, and a 330° ring finger.
How about that 315° cos and sin, not forgetting our new employee PaM? Folding the middle finger in, we have 2 above and 2 below. Which gives us what for an answer?
PaM is Plus, Minus, right? That's why the signs are placed where they are.
Now that you've got the idea of how to use the mnemonic, here it is again with the important parts in bold:
- First, PoP comes UPstairs to get you UP.
- Second, He asks you to MoP DOWN the floor in the living area.
- Third, MoM comes UPstairs to UPdate you with some news.
- Fourth, You're taken DOWNstairs to meet the new employee, whose name is PaM.