On August 25, 2012, the world learned the sad news of the death of Neil Armstrong, the first man to walk on the moon.
In this post, we'll take a closer look at the Apollo 11 mission, and just what it took to get Neil Armstrong and Buzz Aldrin to the moon and back.
James Burke, whom you probably know from his documentary series Connections and The Day The Universe Changed, was the main presenter and science reporter for the BBC back in 1969. His style of reporting really brought the Apollo 11 challenge to life. As seen in the videos in this playlist, he shows the cramped, yet functional, command module, the workings of the spacesuit, the escape plans, what zero gravity testing is like, and more!
As James Burke explains in more detail below, what NASA was really trying to do was launch a 36-story building at the Earth in such a way that it doesn't come back down:
One of the most impressive parts of the Apollo 11 mission was the workings of the computer itself. Compared to today's commercially available computers or mobile devices, the specs were amazing primitive. According to Grant Robertson's article, How powerful was the Apollo 11 computer?, the Apollo 11's computer only had 2K of memory and 32K of read-only storage. Simple, but quite capable of doing everything it needed to get to the moon.
For an even more detailed look at the computing power of Apollo 11, check out the Apollo Guidance Computer episode of Moon Machines, available on YouTube (Part 1, Part 2, Part 3). If you enjoy that, check out the rest of the Moon Machines documentary series.
Part of the whole triumph of landing men on the moon was the way the world seemed to stop and hold its breath together. The video below shows not only the landing itself, but also the world reaction to the feat at the time:
Sadly, the Apollo 11 mission, and even the subsequent Space Shuttle program, are now history. Yes, at the time it fired the imagination. Returning to James Burke again, he explains how imagination is not only the source of the program, but also its demise:
The fact that Neil Armstrong survived to return to Earth and live another 43 years is impressive itself. When trying something new and untested like sending a manned mission to the moon, you have to accept that there's a strong possibility of failure. It's not pleasant to think about, but the Nixon administration did have a speech prepared, in the event that Neil Armstrong and Buzz Aldrin found themselves trapped on the moon.
Neil Armstrong, you were a hero to the U.S. and the world. We are truly saddened to hear of your passing, and will continue to honor your memory.
On August 25, 2012, the world learned the sad news of the death of Neil Armstrong, the first man to walk on the moon.
Naturally, everybody makes mistakes. In fact, it's an important part of the learning process.
People are especially quick to learn from mistakes that have large impact. If you ever wondered in school about the importance of doing the math right, today's post will show you many examples of what can happen when the math is done wrong.
When you're in school and, say, working through a word problem, and you make an error, the worst that happens is that you lose a few points on the assignment. Mathematical errors that happen in stories are often interesting and even amusing because they don't have any real world effect.
In Edgar Allan Poe's The Gold Bug, there's an error of trigonometry concerning the discovery of the treasure, but errors like this are easily ignored and don't effect enjoyment of the story. If you're not familiar with The Gold Bug story, there are many places you can read it or watch it online.
Math mistakes get much more serious when they affect our lives, however. The most common example happens in everyday shopping. Many retailers take advantage of consumers' inability to handle math on the stop by employing intentional math mistakes. Just last month, The Atlantic published an article about this called, The 11 Ways That Consumers Are Hopeless at Math. You'll probably recognize many of these approaches from personal experience.
Unintentional mistakes, as you're no doubt aware, can have a great cost in your personal or business life.
In the London Olympics, Tunisian weightlifter Khalil El Maoui was in second place after the first part of weightlifting competition. Unfortunately, because of a math error made by his coach, El Maoui wasn't present for the next part of the competition.
If you think affecting the outcome of a game for 1 person is bad, how about affecting the outcome for 14 people? That's just what happened back on April 30 of this year at the Golden Nugget Hotel and Casino in Atlantic City. Their playing card supplier, Gemaco, had accidentally provided several unshuffled decks, even though they're supposed to provide shuffled decks. Naturally, the players recognized the patterns and took advantage of them to win. The game quickly went from a $10 per hand game to about $5,000 per hand. By the time the mistake was uncovered, the players had won a combined total of $1.5 million!
In this case, the original error wasn't mathematical itself, but the assumptions made on the part of the supplier and the casino did have a mathematical impact on the odds, and thus the outcome of the game. The decision by the casino to not allow many of the players to cash out their winnings may amplify the damage, should the general public see this as the casino being unfair.
The math mistake that are most likely to stick out in people's minds, however, are the ones that cost lives. The TV show Modern Marvels regularly features episodes about engineering disasters, and it's amazing how often the tiniest error can cause the most major disaster. Cracked.com also has an amazing collection of 6 of the smallest math errors, and their horrific consequences.
Many people don't care for studying math because of the relentless focus on precision. When you realize that math interprets and affects the real world, the need for that precision quickly becomes apparent.
In August's snippets, I'm going to take you back in time, and perhaps see the origins of my geeky side.
I'm actually surprised to find out how much of my past is available online, so putting this post together has been fun. The links to videos, books, and magazines in this post all include complete or near-complete archives of each publication, so you can explore them in detail for yourself.
• Back in the late '70s, my father's work had a Commodore PET 2001 computer that could be checked out by employees on the weekends. I didn't see it often, but when I did, playing and learning to program it would consume my weekends. For Christmas '82, my parents saw I had enough interest that they gave me a Commodore 64. Even before that first computer, I got interested in reading early computer magazines, especially Creative Computing. When the Best of Creative Computing books (Vol. 1, Vol. 2, Vol. 3) came out later, I grabbed them quickly.
• After getting my Commodore 64, I stuck largely to magazines that focused on it, mostly COMPUTE! and COMPUTE!'s Gazette. Depending on your particular interests, though, there were an amazing selection of computer magazines available at the time.
• Creative Computing, which started publishing in 1974, had many article that sound quaint today. They included topics imagining what effect computers would have when they were adopted widespread in homes, schools, and offices. They dared to dream that there would be conventions focused solely on computers, tablet computers, and even TV shows about computers! If you were a computer geek in the 1980s, there was one weekly TV shows about computers that you never missed: Computer Chronicles. Amazingly, this show ran from 1983 to 2002 (from before the first Macintosh to after 9/11), and featured reviews, investigative reports, and even minor weekly news updates. It wasn't uncommon to hold off on some new technology until you could find out more about it on Computer Chronicles, because their reviews were so reliable.
• As I moved into the 1990s, I noticed that my interests in computers, magic, math, and memory were starting to come together. One of the first places this became apparent for me was while perusing OMNI magazine, especially their Games column. For example, the November 1981 issue featured Arthur Benjamin's approach to calendar calculation. No, 1981 isn't close to the 1990s, but I didn't see my first issue of OMNI until late in 1989 at my local public library.
One interesting and recurring topic concerned young Michael Weber's discovery of how to easily get items into and out of the locked bank deposit bags in common use at the time. The October 1989 column was the first time the technique was publicly exposed, including the fact that the IRS had already begun to change their bags after seeing Michael's technique. The following month, there was an update, including Michael's technique for getting into and out of plastic bags sealed with glue. Even more than half a year later, there were still companies who weren't convinced that these techniques were legitimate, yet repeatedly refused to witness a demonstration.
Interestingly, from 1988 to 1994, COMPUTE! was bought and published by OMNI magazine. OMNI's own publication run ended in the winter of 1995.
• For the longest time, I remembered seeing a multi-part documentary about the history of computers, but I couldn't remember the name. It was only about 2 months ago when I happened to run across it on YouTube, and discovered it was called The Machine That Changed The World. There were five episodes, all of which are online:
Inventing the Future
The Paperback Computer
The Thinking Machine
The World at Your Fingertips
As it happens, this documentary is another good example of seeing my interests come together. In the Thinking Machine episode, starting at about the 27:30 mark, there's a savant demonstrating an ability to recall calendar dates. This performance long impressed me, and I was thrilled to see it again, since I hadn't seen it for 20 years!
This documentary came out around the same time my trained memory got me accused of cheating in college.
Yes, there were a host of other resources and documentaries that inspired me that weren't directly related to computers. My Iteration, Feedback, and Change series of posts cover many of my more recent influences.
Because computers have been such an important part of my life for so long, I thought it would be fun to share these resources with you. I hope you enjoyed hearing about them, and even better, have fun exploring them!
In the previous post, you learned about a disguised form of tic-tac-toe known simply as 15, and how to avoid losing if you go first.
In this post, you'll learn about the best strategy to use when you go second.
The game of 15 was written about repeatedly by Martin Gardner. His original Scientific American column on it is available in his book Mathematical Carnival. Gardner also discusses it briefly in Aha! Insight. In both sections, he also discusses other interesting ways to disguise tic-tac-toe.
You can also find out more variations of tuc-tac-toe, including 15, in the Games column of the August 1979 issue of OMNI magazine.
To start, you should understand that going second effectively puts you on defense. In this version of tic-tac-toe, as with regular tic-tac-toe, the first player has roughly twice as many opportunities to win than the second player does.
When going second in 15, the first move is simple. If the other player takes the 5 for their first move, your response is to take any even number. In this post, I'll assume you always take the 4, but the strategies can be adapted to any even number. If the other player takes anything EXCEPT the 5 for their first move, then you must take the 5.
Where do we go from here? Obviously, that depends on the other player's second move. We'll start by assuming they took something other than the 5.
Other person goes first, first move is anything EXCEPT a 5: The first thing you need to watch out for is whether that second card, combined with their first, can make a total of 15. If so, you need to block that potential win by taking that card. For example, if they took the 4 first, you took a 5, then they took a 3, you have to realize that their 4 and 3 can be a win with an 8, so you need to take the 8.
After the other player makes their third move, you need to check for a threatened win and block that, as well. If this move doesn't threaten a win, take any odd numbered card (except 5, of course). At this point, you'll have the 5 and an odd numbered card (such as the 9). If they're smart, they'll see this and block your win (Seeing your 5 and 9, they take the Ace, for example). If they do this, all you can do is block and draw. If they miss it, you've got a win!
Most cases are going to wind up as the game above. If they take two even cards that require a 5 to complete a 15, which would be strange as you've taken the 5 on your first move, take any of the remaining odd cards. They'll either block, or give you the win unknowingly. From here, it's the same as above.
The best possible situation is when their second move gives them two odd cards, neither of which is a 5 (effectively, 2 edge squares, as you have the 5). If their 2 cards are such that your 5 would be required to make 3 in a row, take any remaining available odd card. They'll have to block you, and there will only be 4 even cards remaining.
Take a look at the 2nd card you drew, and think about what two even cards would make 15 with it. This is where it helps to be able to recall the whole board as taught in the first post. If your 2nd card is a 9, the even cards would be the 4 and 2. If it's the Ace, the even cards will be 8 and 6. 7 is in line with 6 and 2, and 3 is in line with 8 and 4.
Whatever two cards you come up with, take either one of those. The other person will have to block 1 of your possible wins, but there will still be 1 way available for you to win, so you take that:
Other person goes first, first move is a 5: As mentioned above, if they take a 5, you simply take any even number. From here, the most likely scenario is that you'll be blocking repeatedly and winding up in a draw, similar to what has already been described. Once again, you can take advantage of any mistakes to win, but otherwise, you'll draw.
To brush up on your tic-tac-toe strategy, check out:
• wikiHow: How to Win at Tic Tac Toe
• chessandpoker.com: Tic Tac Toe Strategy Guide
• Buzzle: Tic-Tac-Toe Strategy Guide
• learnplaywin.com: Tic Tac Toe: Strategy
If you're concerned about not being able to win every time, you can set up the challenge by saying, If you win, I'll...(explain your losing wager here)..., but if I don't lose, I'll...(explain your winning wager here).... That way, if you draw, you can remind people that you bet you wouldn't lose, and since there was a draw, you didn't lose!
I hope you enjoyed this mini-series of posts of 15. Try it out, and let me know what you think of it.
In this post, you'll be introduced to a simple new game to play. Even better, it's a game you'll never lose.
It's not another version of Nim. This time, it's even sneakier!
Here's the rules of the game of 15:
1) Nine playing cards, with face values from Ace to nine, are face up on the table. The Ace always has a value of 1 in this game.
2) Players alternate taking turns, and on a given player's turn, they must take 1 card from the available group on the table. Neither player make take a card that has already been removed from the main pile.
3) The winner is the first person to obtain exactly 3 cards that add up to 15.
What kind of strategy would you use to win this game, or at least prevent losing?
You might be surprised to learn that you probably already know this game. It's a disguised version of (or in math terminology, it's isomorphic to) tic-tac-toe! How exactly does this relate to tic-tac-toe? Imagine the numbers 1 throught 9 arranged as a classic 3 by 3 magic square, so as to total 15 horizontally, vertically and diagonally:
This might seem like a hard arrangement to keep in memory, but it's easier if you picture the arrangements of even and odd numbers separately:
Now you can clearly see how the game relates to tic-tac-toe, and why it's played to 15. Since the other person doesn't realize what they're playing, this gives you an advantage.
In this version, however, the Ace through nine are laid out in a straight line in order, not the traditional crisscross pattern, so you can't see things as clearly as you would in a regular game. So, exactly what strategy should be used?
The proper strategy depends on whether you're going first or second. Let's start by assume you're going first. Following the classic strategies for X, as taught at chassandpoker.com and Wikihow, you'll want to take a corner square, which in this game equates to any even card (2, 4, 6, or 8).
When first learning this version of the game, always take a 4 when you go first. As you become more proficient in the game, you can start with any even card, but always starting with a particular card at first will help you get familiar with the essential.
There's only 2 different replies the other player can make:
1) They choose a 5: This is akin to taking the center square. You must reply by taking the 6 (the diagonally opposite corner). This might seem strange, as you'll have a 6 and 4 with no possibility of a 5, but you're setting a trap for them. If their 2nd move involves taking either the 8 or the 2 (a corner square, in other words), you've just won!
How? You take the sole remaining even number, which simultaneous blocks their possible win, and opens up 2 ways to win for you! When they block 1 way, you simply play the other to win.
Below is an animation of how the game looks in the standard form of tic-tac-toe. If you arrange the cards in the form of a magic square above, you'll be able to better follow along as I teach the strategies.
Remember, in actual play the cards are laid out in straight line Ace through 9, but laying cards out in the magic square form during practice will help you learn the strategies more quickly.
There's another possibility here. If you've take the 4, they've responded by taking the 5, then you've taken the 6, they could possibly take an odd card (equivalent to an edge square). In that case, you'll have to block by taking the 1 card that would total 15. For example, if they now have the 9 and 5, you'll want to take the Ace (9 + 5 + 1 = 15).
Here's how that kind of game looks in tic-tac-toe:
However the game proceeds from this point, just make sure you either wind up with the 3rd even card and block as needed, unless you hav take advantage of any mistakes they make by completing a row of 3. As you can see, the second player's best move is to take the 5 followed by any odd-numbered card, as it's possible to play you to a draw.
2) They choose anything EXCEPT a 5: In response, you need to take either the 8 or the 2, whichever one they haven't blocked. If they took the 3 or the 8, then the 4-3-8 (leftmost) column is blocked, meaning you have to take the 2. If they took the 9 or the 2, then the 4-9-2 (bottommost) row is blocked, so you'll need to take the 8. It's also possible that neither the 8 nor the 2 is blocked, and you have a free choice.
They should recognize that you need a particular card at this point, and take that card next. If you've take the 4 and the 8, it's not hard for them to figure out that they need to take the 3. If you've take the 4 and the 2, they'll go for the 9. If they don't make either of these proper responses, they've just handed you the win by mistake!
Assuming they don't hand you the win by mistake, first ask yourself if they can win by taking the 5. If they can win with a 5, take it! This will block their win, and set up two possible wins for you. All they can do at this point is to block you in one corner, and you win by taking the other:
If you don't need to block them with a 5, you'll need to take an even card (corner). If there's only 1 even card remaining, take it. Otherwise, you'll have two possibilities and you'll need to make sure that the one you take isn't blocked. To do this, simply ask yourself whether it's possible to make 3 in a row with your cards and the remaining cards. If so, then it's not blocked, and you can safely take that even card.
At this point, you'll have 3 even cards (corners), and two ways to win, so you simply wait for them to block one way, then you play the other to win:
That covers all the possibilities for when you go first. In the second part of this series, I'll delve into what happens when you're the second player. For now, simply practice as the first player. Remember, try the strategies out with cards in the magic square arrangement above, then get used to playing with the numbers in line, as you would in a real game.
In today's post, I've gathered a few free goodies for you!
They're all about mental math, so you may wind up learning something if you go through all of them!
• 2011 “Lightning Calculation” Calendar: Yes, August 2012 is probably a strange time to be recommending a 2011 calendar, but this calendar still has a valid purpose. Each month teaches new mental math techniques, and the calendar itself then gives you related daily challenges to practice! Naturally, there are calendar calculations, but there's multiplication, squaring, trigonometry and more. If you like this calendar, you might want to explore the rest of the author's site, as well.
• Short-Cut Math: This is a clearly-written guide to the mental math shortcuts you're most likely to need during your lifetime. Various publishers still market this book, but the original at the link is still public domain. The section on the under-utilized technique of casting out 9s always makes me think of this amusing anecdote, in which an adult learns the technique for the first time, and it actually provokes a violent reaction!
Ars Calcula: This is a simple blog about mental math, and each entry teaches one specific mental math technique. It's not much else than that, but it neither claims nor needs to be any more. The multiplication guide is especially handy, as it gives a sort of hierarchy of when each multiplication technique is appropriate.
Vedic Mathematics: My Trip to India to Uncover the Truth (Alex Bellos): Back in 2011, Gresham College hosted a series of lectures about early mathematics in various cultures. In my opinion, Alex Bellos had the most intriguing lecture of the group, in the video below. Vedic Mathematics is said to be an ancient approach to mathematics, but Alex Bellos questions that history, and then even questions whether the answer is important or not. His powerpoint slides, a transcript, and both audio and video of this presentation are available for download at the link (scroll down to Crore blimey! My trip to India to Uncover the Truth about Vedic Mathematics).
Arthur Benjamin at Etech: You've more than likely seen Arthur Benjamin's Mathemagic performance at TED, but that video only gives a taste of his complete show. This paragraph's main link, however, is a more complete, 37-minute-long version of his show at Etech. There are many aspects you wouldn't expect, such as opening with a card trick, and even teaching the audience some mental math they can use!
Everything above is free and legal to download if you wish to save it to your hard drive. This may not seem like much, but if you take the time to really explore each item, you'll have enough food for thought for quite some time!
(NOTE: Check out my other Nim posts by clicking here.)
Looking around the web, I've discovered many new things related to Nim, a favorite game here at Grey Matters.
Instead of discussing a new variation of Nim, this post goes back and takes a closer look at some versions of Nim that have been discussed in previous posts.
Futility Closet just posted a puzzle called Last Cent, which is simply single-pile Nim played with 15 pennies and moves limited to taking 1, 2, or 3 pennies. If your Nim skills are rusty, try and work out the best strategy for this before moving on.
Moving on to multi-pile Nim, there's an attractive new table-top version of Nim now available, called Abacan. Below is a video review of Abacan, so you can get a better idea of how it looks and works.
It's described in the video as a game where the last player to make a move loses, but readers of Grey Matters know that you could change that rule to the last player to make a move being the winner, and only a minor change in strategy would be required.
With information from my posts on multi-pile Nim, you could work out the strategy for yourself, or you could just take the direct step of using the multi-pile Nim Strategy Calculator for a 5-pile game of piles consisting of 1, 3, 5, 7, and 9 objects.
Back in the March 1962 issue of Scientific American, Martin Gardner wrote about a game called Hexapawn, and how anyone could build a simple computer out of matchboxes and a few tokens that could learn how to win the game on its own! The article was reprinted in his book The Unexpected Hanging and Other Mathematical Diversions, and later in chapter 35 of The Colossal Book of Mathematics.
In that chapter, Martin Gardner briefly mentions that such a computer could also be built to play Nim, but doesn't give much in the way to details. Over at Tony's Math Blog, Tony discusses a Nim matchbox learning computer in more detail, if this strikes you as a fun an interesting project. He called his set A Nim's Game Experience Learning Automaton, or A.N.G.E.L.A., for short.
Besides Martin Gardner's classic writings on this game, there are also some excellent papers and lectures about Nim available on the web. Paul Gafni has a multi-part video on YouTube, and here's the first video:
The complete series can be found at the following links:
Intro (This is the above video.)
David Metzler also has a lecture video series on Nim, and this one is a bit more technical, with computer graphics used to help explain the concepts. Here's the first video in David Metzler's Nim lecture series:
David Metzler's complete lecture can be found at these links:
Part 1 (This is the above video.)
If you liked my Chocolate Nim post, there's an online paper focusing on approaches to different versions of Chocolate Nim at virtualsciencefair.org. Even if you don't understand it right away, trying out the included Java applets and practicing with real chocolate when possible will help you pick it up quickly.
If you remember my post on fractals from summer 2011, you might be startled to learn that Nim can be examined using fractals! It turns out that the humble Sierpinski triangle turns out to be an excellent tool for examining effective Nim plays, as explained in this PDF.
For even the most ardent Nim fan, there's plenty of material here to study for quite some time. If you've made your own interesting Nim discoveries, let me know about them in the comments!
Last month, Wolfram|Alpha introduced a cool new interactive front page, featuring icons you can click to learn more about its related topic. For example, clicking the atom picture brings up a query for nuclear power production in France.
Since they're sharing some of their favorite searches, I thought I'd share some of mine, as well.
Pythagorean triples search: A Pythagorean triple is any group of three integers (whole numbers) a, b, and c, that satisfies the Pythagorean theorem equation:
a2 + b2 = c2.
Probably the most famous Pythagorean triple is 3, 4, 5, which qualifies because:
32 + 42 = 52.
Wolfram|Alpha displays several examples here.
What if you want to find all Pythagorean triples involving a given length, however? You can use this query, setting a to any desired leg length. In that example, we're looking for a leg length of 25 by setting a to 25, and we find that a = 25, b = 60 and c = 65 is one possible Pythagorean triple, and a = 25, b = 312 and c = 313 is another. The inspiration for this query came from this Pythagorean theorem slideshow on slideshare.
However, the above search only works for legs of a given length, and won't find a hypotenuse of a given length. So, I re-worked the formulas and came up with a version that searches for any given hypotenuse length. In the example search for 25 (done by setting c=25), we find a = 20, b = 15 and c = 25 for one set, and a = 24, b = 7 and c = 25 for the other.
Quizzes: Sometimes I use Wolfram|Alpha to generate sample data to help test myself. For example, if you're practicing either the Day of the Week For Any Date feat or my Day One version, you can generate a random month calendar in the 20th century using this query. If you prefer a full calendar of a random 20th century year, use this search. Obviously, you'll either need someone to help by generating this data out of your sight, or find a way to cover part of the screen.
For practicing the square root estimation feat I recently posted, it's a simple matter to have Wolfram|Alpha generate random numbers from 1 to 1000. If someone else is helping you, you can also have the answer on the screen for their verification. For example, if the randomly generated number is 862, your estimate should be 29 and 21/59ths, which is reflected on the screen showing a=29, b= 21, c=59.
If you like the cat stuck on a pole presentation for the square root estimation feat, Wolfram|Alpha can generate random numbers in the right range for that, as well.
Faro Shuffle Simulation: If you're not familiar with the faro shuffle, it's a challenging shuffle that seems to be a regular riffle shuffle, but has a predictable, even mathematical, outcome. Below is a demonstration of the fact that 8 faro shuffles will return the deck to its original order:
Note how, in the video, the Ace of Spades begins as the card on the face, and remains as the card on the face throughout all the shuffles. In card slang, this is referred to as an out-faro, since the face card (or top card, if the deck were held face-down) always stays out. If it went in under a new face/top card, it would be referred to as an in-faro.
I developed this Wolfram|Alpha query to research what happens in faro shuffles of decks of various sizes. In this, you set y to the number of cards involved (52 in the link). The variable z is set to either 1 for a in-faro (as in the link), or 0 for an out-faro. This is easy to remember, since 1 looks like I, for In-faro, and 0 looks like O for Out-faro.
I explain the use of this algorithm in more detail in this Magic Cafe post and this one.
Birthday Problem: In the following video, James Grime asks among the 23 people on the field in a soccer game (11 from each team plus the referee), what are the odds that two people have the same birthday? (We're not including the year, and we're assuming that we know that none of the people on the field were born on February 29th.
It's one thing to hear about this, and another to check it out for yourself. If you want to see the data for any amount of people from 2 to 60, check out this query. As you can see, by the time you reach 57 people, the odds are better than 99% of them sharing a birthday! To really understand this data, however, check it out as a graph instead. You'll see it rises surprisingly quickly, and begins to level off as it starts to become a near-certainty around 50 or more people.
That's all for now. If you have any unusual and favorite Wolfram|Alpha queries of your own, please share them in the comments!