Two of our old friends are back on Grey Matters: Scam School and the game of Nim!
This version uses a dart theme, but you don't even have to be good at the game!
Scam School's 294th episode features an impromptu version of Nim using your fingers as darts, and a napkin as a dartboard:
I'm not going to analyze the play too deeply, as this is simply stanard single-pile Nim, as defined in the first Secrets of Nim post. In addition, it's also essentially the same as Dice Nim, as taught in 250th episode of Scam School, but played to 31 instead of 50.
The Single-Pile Nim tab of the Nim Strategy Calculator can run through the strategy with the following settings:
• Player who makes the last move is the: Winner
• Maximum number of objects (limit): 31
• Nim Game is played: up to limit
• Number of objects used per turn ranges from 1 to: 6
After clicking the Calculate Nim Strategy button, the calculator will return the same strategy in the above video. You can even play around with higher or lower totals, and the numbers you're allowed to use on each turn. What would the strategy be if you allowed the numbers 1 to 7, or 1 to 8? How would it affect the strategy if you play to 32, 35, or 40?
The more interesting aspect of this version of the game, at least to me, is the use of a small dartboard instead of dice. One on hand, this change is merely cosmetic, as it doesn't affect the strategy in any way itself. The psychology on the audience, however, is completely different. Dice aren't as common as dartboards in bars and pubs, so the dart theme makes better sense. In addition, the impromptu nature of drawing on a napkin suggests fairness than someone who brings their own dice, which can suggest that the person suggesting the game has practiced it.
31 also seems to be a favored number for Nim players, most likely because it's high enough to allow you to use your winning strategy, but low enough to keep the game too short for anyone to catch on. In another 31 version using playing cards, the unique nature of playing cards allows you to catch even those who think they know the secret.
As you can see, just examining all the seemingly minor changes in Finger Dart Nim can give you a better understanding of the overall idea behind multiple variations of Nim. If you enjoyed this version, please explore the amazing variety of Nim games taught in other posts here at Grey Matters!
Two of our old friends are back on Grey Matters: Scam School and the game of Nim!
If you've ever done decimal division, you've no doubt run acros numbers that endlessly repeat, such as 0.333 0.666.
If you've ever dealt with decimal division by 7, as taught in my Mental Division With Decimal Accuracy post, you've noticed an unusual phenomenon that goes just beyond the repeating of the digits. It's that odd quality we're going to discuss in today's post.
When you divide 1 by 7, you get the number 0.142857142857... and you notice that the digits repeat endlessly. Even weirder, thought, is the fact that dividing 2 by 7 gives you the same digits in a different order: 0.2857142857142857.... and so on. In fact, as discussed in the mental division tutorial, dividing any number from 1 to 6 by 7 will always return the same digits with only the starting point changing.
Numberphile recently turned its attention to the number 142857, and explains many of the unusual qualities behind this and similar numbers:
Let's take things a step further than in the video above. If we arrange the multiples of 142857 from 1 to 6 in order, we get the following square:
1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2Now, since each row contains the same numbers, it shouldn't be surprising that each row gives the same total (27, as it happens). When you realize that 7 times 142857 is 999999, then it's not difficult to understand that 1 times 142857 plus 6 times 142857 would total 999999, as well. The same is true of 2 times 142957 plus 5 times 142857, and 3 times 142857 plus 4 times 142857. This means that all the columns will give the same total of 27 as well.
This is almost like a magic square! In a true magic square, however, the diagonals would also give the same total. Sadly, in the above square, the left-to-right diagonal totals 31, and the right-to-left diagonal totals 23. This still makes it what it known as a semi-perfect magic square.
Because of the unusual nature of these cyclic numbers, all numbers will have a similar quality. Is it possible, though, that a true magic square can be formed this way?
The answer is yes, it can be done! On page 176 of W. S. Andrews' Magic Squares and Cubes, the author shares Harry A. Sayles' perfect magic square using 19ths:
01/19 = .0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 02/19 = .1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 03/19 = .1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 04/19 = .2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 05/19 = .2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 06/19 = .3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 07/19 = .3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 08/19 = .4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 09/19 = .4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 10/19 = .5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 11/19 = .5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 12/19 = .6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 13/19 = .6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 14/19 = .7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 15/19 = .7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 16/19 = .8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 17/19 = .8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 18/19 = .9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8In this square, all of the rows, columns, and diagonals each total 81!
As an aside, numbers such as 19 have many more interesting qualities for division. To get a better idea of what I mean, check out my Leapfrog Division post.
I'll leave your mind boggled at this point, but I will suggest you do an internet search for the number 142857, as there are still more amazing qualities to discover!
Last month's snippets focused on the simple and straightforward side of math.
I thought it might be fun to dabble in the other end of the spectrum, so this month's snippets focus on taming the seemingly complex side of math.
• One interesting mental challenge is what are known as Fermi estimates. These are estimates which are seemingly done with little or no information, yet give an accurate estimate to the nearest power of 10. The folks at TED-Ed produced the following video which explains the concept in more detail:
Learning how to do this can be a little tricky, as you really have to learn as you go. One of the best online tutorials I've run across as of this writing is this FermiQuestions.com tutorial. The more you learn, the better equipped you'll be to answer the sample questions they give at their website.
There's also a great book titled How Many Licks? that focuses on teaching you how to develop your Fermi estimation skills clearly and simply.
• If you enjoyed Vi Hart's video on dragons from August, you'll be glad to know she's produced two more videos in the same series. The second one in the series is called Dragon Dungeons:
...and the third in the series is called Dragon Scales:
• Speaking of dragons and other mathematical monsters, the Data Genetics blog has just posted an interesting tutorial on how to escape a monster (using calculus). Ideally, you'll have a basic understanding of calculus to understand what is going on in this fun article. Even if you don't, the article is still readable, and you get a good understanding of how the real world relates to the needed calculus.
Last week, I discussed memorizing the elements on the periodic table.
The big question, of course, is why those elements are important. Since I think understanding is just as important as memorization, I figured it was time to take a closer look at what goes on at the atomic level.
TED-Ed, a learning site from the people who bring you TED Talks, has recently posted several good videos that use animation, metaphors, and other tools that really help make atomic physics much clearer.
The first question is, Just How Small is an Atom?, which is answered below with the help of grapefruits, blueberries, and the Earth:
Now that we have an idea of the scale and density of an atom, what's the deal with those electrons? Surprisingly, The Uncertain Location of Electrons, is a challenging, yet fascinating feature of the atom:
While we can't pin down the electrons, we can define an area where they're 95% likely to be located. Once we know this, we can start getting an idea of How Atoms Bond:
Finally, once we get an understanding of the bonding process, we can further use that knowledge to answer the question, What is the Shape of an Atom?:
For a look at the elements that's just as fun, and may even help explain things deeper, you can also check out my Hunting The Elements post.
For now, though, it's time to journey back to the human scale. Happy atom-smashing!
Let's take a break from the usual math and memory fare to consider a different type of puzzle.
Over at the Futility Closet site, there's a great puzzle concerning relationships, titled All Relative:
A problem from Dick Hess’ All-Star Mathlete Puzzles (2009):
A man points to a woman and says, “That woman’s mother-in-law and my mother-in-law are mother and daughter (in some order).” Name three ways in which the two can be related.
This may seem confusing when you first read it, but it's not hard if you work through it bit by bit.
The first thing which should be made clear is that the woman's mother-in-law can be the mother or the daughter of the man's mother-in-law, so we'll need to work through both possibilities. Two possibilities also suggests only 2 answers, yet the puzzle suggests there are 3 answers, so we'll have to keep an eye out for other possibilities.
Since this puzzle involves in-laws, and therefore marriages, we're going to be assuming all marriages involved in this puzzle are heterosexual, as we'll be using Wolfram|Alpha to work through the family trees, and this assumption will make it easier for that site to process. As you can see by the examples here, Wolfram|Alpha can handle genealogy problems quite well.
Being a man myself, I'm going to work from the man's side of the family tree. The same logic and approach will work from the woman's side of the family tree just as well, however.
Let's start by redefining the man's mother in law to be the man's wife's mother. This approach helps us see each important step on the family tree. We'll also start by assuming the man's wife's mother to be the daughter in the mother-and-daughter relationship between the two mothers-in-law.
Given that, then we've worked up to the man's wife's mother's mother (or, the wife's maternal grandmother). For this top mother to be the woman's mother-in-law, the woman would have to be her son's wife. Putting this all together, we're looking for the man's wife's mother's mother's son's wife (*PHEW!*). If you don't understand that so far, go back and read it again until you do.
So, in Wolfram|Alpha, we enter wife's mother's mother's son's wife, to see how the family tree appears. In the tree that's drawn, self represents the man and wifewife represents the woman.
Unfortunately, Wolfram|Alpha doesn't display the name of this relationship, but it is easy to see from the diagram that the woman could be the man's wife's aunt. Wolfram|Alpha also says that this is the only possibile interpretation of this relationship, so we're done with this assumption.
To get the remaining solutions, we need to work through the man's wife's mother as the mother in the vaguely-defined mother-daughter relationship.
So now, we work up to the man's wife's mother's daughter. For this daughter to be the woman's mother-in-law, this daughter's son's wife has to be the woman. Putting all that together, we're looking for the man's wife's mother's daughter's son's wife.
At this point, we enter wife's mother's daughter's son's wife into Wolfram|Alpha to get a better idea of the relationship. We see from the diagram that the woman could be the man's wife's nephew's wife, as well.
Note that Wolfram|Alpha also mentions there are 3 possible interpretations to this relationship. Perhaps this is where we need to be looking for the tricky part that was mentioned earlier!
Consider that the man's wife's mother's daughter could not only describe the man's wife's sister, but the man's wife, as well! Let's run through the whole thing with this in mind.
This time, the man's wife's mother's daughter will simply refer to the man's wife. For this wife to be a mother-in-law of the woman, the woman must be her son's wife. That makes the woman the man's wife's son's wife.
The Wolfram|Alpha map of this relationship show that the woman is this case is simply the man's daughter-in-law. It also says that there are 2 interpretations of this relationship. What's the other one? If the man's wife had her son by another man, then the daughter is the man's step daughter-in-law.
So, at this point, we've actually found 4 answers to this puzzle:
• The woman is the man's wife's aunt.
• The woman is the man's wife's nephew.
• The woman is the man's daughter-in-law.
• The woman is the man's step daughter-in-law.
Under the assumptions I put forth at the beginning of this article, these are the only 4 possibilities. The answers written up in the original post agree with this, but answer #3 is worded oddly. It says, the woman can be the wife of the man's nephew, whereas it would be clearer to say, the woman can be the man's wife's nephew.
If you think you understand the process and wish to work through the problem with different assumptions, and/or want to work through the problem starting from the woman, rather than the man, I'd love to hear about your logic and results in the comments below!
One of the most feared things students come across is the periodic table of the elements.
They see the sheer number of elements, and all the details such as atomic numbers, weights, and symbols, and it's easy to see why it's considered to be overwhelming. In this post, you can learn to not only tame this beast, but have a little fun with it, as well!
Among the best ways to remember things are by memorizing them in groups, and with a story. YouTube user freegyan gyanlee, with the help of an illustrator friend, is putting out a series of free videos using these techniques to teach the element name, based on the book Memorize the Periodic Table.
Start by learning elements 1 through 10, and when you're comfortable with that story, then move on to the next 10 or so elements. Here is the first video in the series:
The nice thing about the story approach is that, should you desire to remember more about each element, you can attach sillier and more unusual details to the appropriate image. For example, if you wish to remember that boron is the 5th element, you might have the bee being bribed with a $5 bill to bore on the balloon, or perhaps the bee bores on the balloon by accident because he's watching the movie The Fifth Element.
In fact, if you added images to represent just the multiples of 5 to each element (5 to the 5th element, 10 to the 10th, 15 to 15th, and so on), you could make it easy to get the elemental number of almost any element by starting from the named element, and working backwards or forwards to the nearest element with an image for the multiple of 5. For example, what number is nitrogen? Well, we know the night-row general lost his car to a car bomb that resulted when the bee bore on the balloon while watching The Fifth Element, so nitrogen must be 2 elements after number 5, therefore, nitrogen must be element number 7!
At this writing, there aren't video to cover all 118 elements yet. However, there are more than enough to keep you busy:
• Elements 11-20
• Elements 21-30
• Elements 31-40
• Elements 41-50
• Elements 51-63
If you'd like to find out more mnemonics for the periodic table of the elements, check out the Mnemonics Devices for the elements here. As many a memorizer has learned, taking things slow, and making them fun, silly, and unusual is one of the best way to keep things in your mind.
My previous post focused on politics, and this one will focus on the Middle East.
I swear I'm not turning this into a blog about political topics.
Instead of issues in the Middle East, this post will focus on memorizing the names and locations of all the Middle Eastern countries.
Back in January, I posted a series of videos about memorizing regions of the world, produced by YouTube user eveRide. Those videos covered Central America, South America, and Europe.
This past week, he uploaded a brand new video, focusing on the Middle East. Even though India isn't traditionally considered to be in the Middle East, it does serve as a good starting reference point from which to work.
The countries are unlabeled, and their borders can be hard to see in the above video, so below is a reference map that may be of help. Click the map to see a larger version.
From the videos, I gather the producer is a teacher, and tends to make these videos in the fall, presumably for the students. Personally, I enjoy these videos and the creativity used in developing these mnemonics. I'll definitely be keeping an eye on eveRide's channel, and bring you more as they become available.
The hot topic of the week has been, of course, politics.
I generally avoid politics on this blog, since it's far removed from my main focus of mental feats and techniques. However, there are crossroads where politics and math do cross paths.
Generally, the closest I get to politics is the basic stuff, such as memorizing facts about the USA. The advocacy side of things is best left to experienced political bloggers.
I do think it's important to realize that even politics can't escape the effects of math, and vice-versa. This brief introduction to Arrow's Theorem points out that an ideal voting system, as defined by the stated conditions, not only doesn't exist, but is impossible to develop.
For example, let's take a look at First Past The Post voting and its problems, with some help from CGP Grey:
Again, it's not that just this particular system is flawed, but that any voting system with the conditions set forth in Arrow's Theorem will be flawed. For an exploration of other voting systems, check out CGP Grey's other voting system videos.
It's not surprising that politics could be swayed in one way or another, as many accept it as part of the nature of the beast. Math itself, on the other hand, is heavily dependent on things such as proofs and peer-review, so there's no way that math could be corrupted by politics, right?
If you agree with that, James Grime and Numberphile have some bad news for you, as a recent study has findings to the contrary.
Hopefully, this brief post has given you food for thought, and maybe even discussion, in the likely and numerous discussions of politics in the days ahead. If you have any thoughts on mixing politics and mathematics, I'd love to hear them in the comments!