y of x

Last Post on This Site

2009-07-20T02:25:00.000-07:00

As of today, YofX has officially migrated to its new home: www.yofx.org. This site will remain up for archival purposes, but all new YofX posts will take place at the above address. Also, if you are following this via RSS, you can subscribe to the new feed here. Thanks for following and I hope to see you over at the new site.

A Mathematician Goes on Vacation 6

2009-07-17T02:25:00.000-07:00

I don't get too many complaints about the uselessness of negative numbers, but every once in a while in a Prealgebra class somebody will claim that they're never used in the real world. Above is the photographic evidence from within an elevator in Nikola Tesla Airport (Belgrade, Serbia) that negative integers are indeed used. And if one is ever in such an elevator, one surely thinks fondly (for once) of their math teacher. For how else would we know where to go?

**This is the last real post to this site. Next week yofx officially moves to a new home: www.yofx.org. This site will remain up as an archive for us to refer back to.

Which Mathematical Function Are You?

2009-07-16T02:30:00.000-07:00

Lee tipped me off on a great Facebook time-waster: the Which Mathematical Function Are You quiz. You've done plenty of these viral quizzes before, but none so squarely centered on your geek heart. They ask you the standard personality questions. Sometimes the selections are worth a few giggles. Then they determine which mathematical function best fits your personality. I am apparently Gaussian.

Lee is apparently the Riemann zeta-function.

Mystery and wonder are your "prime" delights in life. You seek to apprehend all the hidden, complex patterns in the universe. Some think you are quirky, but they just don't understand you: they have not learned to think outside the box, as you have. You are a misunderstood genius.

I'm curious how many functions the quiz outputs. Let me know what you got in the comments.

**Reminder YofX will be moving to www.yofx.org in 4 days on 7/20**

text2image fun!

2009-07-15T02:28:00.000-07:00

text2image is a fun (and probably totally useless) way to transform any text into an image. Above, you are seeing the visual rendering of 'yofx'. I was thinking that this might be an interesting way for students to conceptualize a function. I hate to say it, but sometimes real-valued functions get boring. Another plus is that we don't actually know how the text gets transformed. Discovering properties of the function by playing around could make for an interesting exercise for students. Is it one-to-one, does an all-caps text string map the same as a lowercase string, what does the domain look like, is there a character limit, are all questions students might generate. I picked this up from Information Aesthetics. Check out that post. There are also some links at the bottom to similar trans-medium mappings.

ABBA = ABBA

2009-07-14T03:00:00.000-07:00

I was dragged to an ABBA tribute concert last Friday night. I'm regrettably familiar with ABBA's music, Mama Mia veteran (even the sing-a-long), but it was the first time I realized the band's name is a palindrome. It is also the first time the name reminded me of a rhyme scheme. (Two revelations, one night.) What is the real origin of the name? According the Wpedia, it's an acronym for the first names of the bandmembers: Agnetha, Björn, Benny, Anni-Frid. I know, you'll never hear Dancing Queen quite the same way again. Anyway, let me know if you can think of any other palindromic bands.

**Reminder YofX will be moving to www.yofx.org in 6 days on 7/20**

Shunt Buster

2009-07-13T02:03:00.000-07:00

Help! This one's killing me.

**Reminder YofX will be moving to www.yofx.org in 7 days on 7/20**

Something to Show Your Algebra Classes

2009-07-10T01:46:00.000-07:00

I actually found a reference to variation that isn't in a text book. The Wired Science blog had a post on high-altitude wind generation (pretty interesting in its own right). They describe the relationship between wind speed and power generated like this...

Wind’s power — energy which can be used to do work like spinning magnets to generate electricity — varies with the cube of its speed.

I wonder how many of the students I tortured with variation would remember what it means if they read this article? Those are the questions that keep a math teacher up at night. BTW, if you're teaching variation. I've always found that students like knowing the world's most famous equation e=mc^2 is a direct variation equation.

Geek of The Week: Nick Hamblet

2009-07-09T01:44:00.000-07:00

Not sure whether you have been following the comments on Prime Time? But, Nick, known to us in the comments as SumIdiot, came up with a nice proof showing that it's impossible to group the digits of a number n in primes so that you wind up with the prime factorization of that number. For example, the digits of 241 can be grouped as (2)(41), but 2 and 41 aren't the prime factors of 241. Another example, 573 could be grouped (5)(7)(3), but again the product (5)(7)(3) is not equal to 573. Enough.

What Nick actually showed was the larger result: any grouping of a number's digits, prime or not, will always have a product strictly less than the number itself. How did he do it? Here's his proof from the comments.

Given d+1 integers between 0 and 9, called a_0, ... a_d, let me write A=(a_d a_{d-1} ... a_0) for the number \sum_{i=0}^{d}a_i 10^i.

Consider the product obtained by splitting this digit string into (a_d ... a_s)(a_{s-1} ... a_0). So this is a product of a d-s+1 digit number and an s digit number (take s >= 1).

Now A=(a_d ... a_0) = 10^s(a_d ... a_s) + (a_{s-1} ... a_0) is at least as big as 10^s(a_d ... a_s). However, since 10^s is strictly bigger than (a_{s-1} ... a_0), A >= (a_d ... a_s)10^s > (a_d ... a_s)(a_{s-1} ... a_0).

So any integer is bigger than the product obtained by splitting it's decimal string into two pieces. It seems the general statement, splitting the decimal string into any number of pieces, follows by induction.

Clever! Nick has also given me the chance to break in a blogroll (Only on the new domain). He runs a real nice blog Sumidiot here.

**Reminder YofX will be moving to www.yofx.org in 11 days on 7/20**

Transform Your Cat with Math

2009-07-08T06:32:00.000-07:00

This XKCD cartoon is a follow up to the Mystery Chord post. Anyone know any other good Fourier jokes?

**Reminder YofX will be moving to www.yofx.org in 12 days on 7/20**

Found Math

2009-07-07T01:52:00.000-07:00

I found this on a notepad. And while it looks like math, I can't figure out how the numbers are related. Help!

**Reminder YofX will be moving to www.yofx.org in 13 days on 7/20**

Sudoku Variation

2009-07-06T02:02:00.001-07:00

My mother sent this sudoku variaton my way. She is a total puzzle junky and found this in the International Herald Tribune. All the normal sudoku rules apply, but you also have to fill the shaded areas with 1 through 9. Thanks, Mom!

YofX Will Move To A New Home

2009-07-03T06:29:00.000-07:00

In the next couple of weeks YofX will begin the move to a new home: www.yofx.org. This will not only be a move to its own domain, but also to a new platform, Wordpress. Here is the plan.

1) For the next two weeks, starting Monday, I will post to both this site www.yofx.blogspot.com and to the new site www.yofx.org. I hope that will give everyone the time to make the transition. (And don't freak out if the new site looks different. I'll be playing around with themes during this period.)

2) After two weeks, starting 7/20, all new posts will be found on www.yofx.org, and this site will be an archive.

If you are reading this blog via RSS, you can subscribe to the new site's posts and comments by following the links. You can even get it via email.

Let me know if there are any problems by commenting. I know this is a pain in the butt, so I want to make sure things go as smooth as possible for you all.

A Mathematician Goes on Vacation 5

2009-07-02T11:56:00.000-07:00

This is the best picture I took all vacation. There is so much going on, but notice in particular how the foot of the table rests on the perimeter of the reflection of the table surface...momement of ecstasy! I recognize not everyone is that moved. (My wife just rolls her eyes.) I'm hoping if you read this blog. You might be geeky enough to appreciate it.

Tintin Calculus

2009-07-01T02:33:00.000-07:00

Trying to figure out a name and some cover art for the calc text you just wrote? Look no further. I found this is the children's section of my library. And before you run to see if your library has it, be warned there isn't any Calculus (as we know it) in the comic. Calculus is a character that is the object of a sinister plot. Tintin then goes on to save Calculus, and probably the rest of the world too. Funny, I was googling Tintin and found an animated version of the comic online. Too lazy to read, watch it. And if you like your calculus affair with a little more calculus, click here.

A Mathematician Goes on Vacation 4

2009-06-30T02:25:00.001-07:00

I saw this symbol on a lot of walls. It is the Serbian Cross. I'm told the meaning is basically equivalent to our motto "United We Stand, Divided We Fall" (although the Wikipedia article says that this interpretation is a common misconception). I was impressed by the interesting symmetry: two lines of reflection and rotational symmetry.

pic by alexmac1

Amoeba Fever Problem

2009-06-29T02:05:00.000-07:00

Here's a very interesting problem sent in by Lee. Have fun playing with this.

Toby has a jar with one amoeba in it. Every minute, every amoeba turns into 0, 1, 2, or 3 amoebae with a probability of 1/4 for each case (dies, does nothing, splits into two, or splits into three). What is the probability that the amoeba population eventually dies out?

If you're wondering, Lee's brother Chip created the image above. I'm guessing it has something to do with the solution.

A Mathematician Goes on Vacation 3

2009-06-26T02:05:00.001-07:00

As I said the first week it rained...a lot. That wasn't all bad. I did get a chance to read a bit. I brought along On Aggression by Konrad Lorenz. I'm not much of a biologist so a lot of the animal behavior descriptions flew over my head. But, I felt right at home with the above diagram. How nice to find a Caley table in the middle of a ten page meditation on the nuances of the duck and drake mating ceremony! Remember your multiplication tables? This is a binary operation. A binary of operation of what? Lorenz call the above diagram a motivational analysis Each picture on the diagram is a combination of the dog's fight or flight instincts. If you go down the left most column, you are seeing the progression of the flight instinct. Going across the top most row, you are getting the progression of the fight instinct. Any other image is some combination of the two instincts. All part of the math of mother nature. Quite interesting!

A Mathematician Goes on Vacation 2

2009-06-25T02:40:00.000-07:00

So, I didn't have such good luck with weather. It rained the whole first week I was there, which meant I watched a lot of Serbian TV on my inlaws' couch. Thank goodness The French Open was on Eurosport or I would have lost my mind. I also caught the above video quite a lot. (Their national stations play music videos between programs.) It seems to be the summer dance jam. There actually is some math in it, and if you can see it betwixt all the half-naked dancers, you know you're a bona fide geek. Good luck!

A Mathematician Goes on Vacation 1

2009-06-24T01:58:00.000-07:00

I just got back from a three week visit of my wife's family in Serbia. I was on the lookout for anything mathematical. Check out this equation on the 100 dinar note (about $1.55). The mug is Nikola Tesla's. He is a Serbian national hero for his contributions to electrical engineering. The equation calculates a unit of measure the tesla which measures magnetic flux density. I tried to figure out what this means. No luck. (It did remind me of the flux capacitor though.)

Mathematical Fortune

2009-06-23T02:43:00.000-07:00

I thought all the statbots out there would get a kick out of this fortune.

Math Donkeys

2009-06-22T02:48:00.000-07:00

I love to give this problem to students.

Two Donkeys are walking down the road loaded down with sacks of potatoes. One looks over at the other and says, "hey, if you give me a sack, I'll have twice as many as you; and if I give you one we'll have the same number. Weird, huh?" The other donkey just says, "Geek" and keeps walking. How many potato sacks does each donkey have?

pic by kaysare

Beatles Mystery Chord Found...by a mathematician

2009-06-19T08:43:00.000-07:00

This is a great story found on Scientific Blogging. The opening chord to the Beatles' song Hard Days Night had long stumped musicians and musical historians. It took a mathematician, Jason Brown, using a Fourier Transform to decompose the chord into its original frequencies. Was it George's 12 string, John's 6 string, or Paul's bass? Find out what the missing ingredient was.

Tweaking Life

2009-06-18T05:33:00.000-07:00

This is a thought on the game of life that Lee introduced me to a while back. John Conway has said that he played around with various rules to see which would produce the most interesting results. I took him to mean that he played around with the survival and birth rules. I was wondering if anyone has played around with tiles having different iteration speeds? Consider if every other column iterated at twice the speed. Basically squares could "play" at different speeds. Has anyone seen anything like this? Lee? Ken, I know you said you've played around with cellular automata?

Geek of the Week: Nerd Nook

2009-06-17T06:41:00.000-07:00

I've seen a lot of math t-shirts. This one on Nerd Nook's photostream caught my attention. (might have to blow it up to read it.) In the 18 million times I've been asked, 'When will I ever use this in the real world?', I have yet to use this technique. It's kind of Dirty Harry meets underpaid math teacher. He also has a bunch of other t-shirts. I love this one.

Mystery Equation

2009-06-16T12:45:00.000-07:00

I was struck by this illustration browsing Drawn's feed. I still haven't figured out what the equation is supposed to relate. The caption on illustrator Mark Smith's site reads

Green idealists fail to make the grade. The people that recycle the most are also the most likely to cause serious environmental damage by taking long haul flights when holidaying abroad.

Is the equation relating atmospheric quantities? Any of you know? Check out Smith's other work. You might enjoy it as much as I did.

Breakout Sessions?

2009-06-15T02:57:00.000-07:00

I remember being at the Barnes Seminar last year about this time and coming up with the following problem. There were probably 150 participants. We convened as a group in the morning and would then split up into breakout sessions of about 6 people. The break sessions were assigned not chosen, and organizers tried to have as little overlap as possible, giving you the opportunity to meet the most number of people. So here is the question, what is the maximum number of breakout sessions until at least two people find themselves in a group again? After you figure that out, what is the maximum number of non-overlapping breakout session of size p chosen from an overall group size of n?

pic by lululemonathletica

Google's Formula

2009-06-12T06:23:00.000-07:00

The cover of Wired's June issue has a three set Venn-Diagram. It looks nice even if it's logical relation to the main three stories are fluff. What I found exciting in the issue was the break out on the innovative way in which Google sells off advertising. Here is my attempt to condense this. For any given set of keywords, let's say like 'math', google holds an auction. All interested parties submit bids to have their add posted when someone searches for math. Who gets the first add on the search page? The highest bidder? No. Google has a quality ranking for each different add based on a number of factors. Google's idea is to keep the add material valuable to users. This is how they do it. They scale each advertiser's bid by the usefulness of the advertiser's add for someone who is searching for 'math'. How they get that quality score for each add involves a lot of data and a few company secrets. What does Google charge the companies for the adds? Not what the company bids. Instead they use the bid of the next highest ranking add, and they multiply it by the ratio of the quality scores of the two adds. The idea being that if your add is a better quality add than the second place bid, you pay proportionally less than their original bid. If it is worse, then you pay more. One interesting effect is that the 1st place add on any search page hasn't necessarily payed the most. If my description isn't working for you, check out pag 112 of the June issue of Wired. Like I said they have a really nice breakout explaining it, complete with an example.

Cover Math Graphic

2009-06-11T18:02:00.000-07:00

I liked this cover illustration from the June 2005 Harvard Business Review. 'It's something in his eyes?'

Geek of the Week How To

2009-06-10T06:02:00.000-07:00

How can I become Geek of the Week? I get the question all the time. I have to admit. There is no formula. There is a lot of luck, and not a little bit of skill involved. For aspirants, MaximumPC has compiled a litmus test for basic geekdom. As they see it, there are 50 fundamental skills. Sue will be glad to know that the recitation of pi to at least 23 digits is on there. Check and see how many of these must have geek-skills you wield. You also might feel that they have overlooked some fundamentals. I'd love to hear what you would add in the comments.

pic by koolbadges

Lobachevsky Chair

2009-06-09T12:06:00.000-07:00

I was at Housatonic Community College some weeks ago for a math conference. They have beautiful new facilities there, but most impressive are the chairs in the cafeteria. They kind of remind me of a hyperbolic plane. The geeky thing was that it was a math conference so I wasn't the only one noticing. Now I just need a set for my dining room.

Gruesome, but Fun

2009-06-08T10:46:00.000-07:00

The following riddle about disabled veterans is from Lewis Carroll's A Tangled Tale.

Say that 70% have lost an eye, 75% an ear, 80% an arm, 85% a leg. What percentage, at least, must have lost all four?

Found this one in Herstein's Abstract Algebra. pic wikimedia

Math and Fragrance

2009-06-05T05:55:00.000-07:00

I went to the mall last weekend with my wife, and was so excited to find an old friend there. She was locked away in glass (like all things of considerable value), but visible among the various other colognes. I couldn't wait to look up the cologne on Givenchy's website. Why did they call it pi? Will it make you smell like pi? Can you really smell like an irrational number? I was so dissappointed when I got home. Here is Givenchy's description of the fragrance.

PI is a fragrance dedicated to adventurers in search of new territories and sensational experiences. The Pi man is a man of action who always surpasses himself.

It's hard to glimpse even a metaphoric relationship between the number and the fragrance.

Topology Question

2009-06-04T03:08:00.000-07:00

Ken's mention of Menger Sponges brought up the association with the Cantor set and the Sierpinski triangle. There must be a name for a topological property of objects from which a congruent copy/s of itself can be removed and the remaining portions of the object are equivalent to the removed piece. The interval, cube, and triangle are all such objects. It seems to me that there is now way that the circle is. Togologists help!

the scarecrow gets his brain (almost)

2009-06-03T05:00:00.000-07:00

I want a doctorate of "think-ology".

781 dots

2009-06-02T11:02:00.000-07:00

This image from the British Library is stunning. Found it on Flickr by sunny-drunk

Alfie's Children

2009-06-01T10:07:00.000-07:00

This is one of my all time favorite problems to give to students.

Determine the ages of Alfie's children from analyzing the following conversation.

Alfie: Can you guess the age's of my children? The product of their ages is 36.

Bernice: That's not enough information.

Alfie: The sum of their ages is the same as our street address.

Bernice: I'll need a little more info.

Alfie: My oldest is a girl.

Bernice: Okay, I've got it.

pic by Alain Gree

7up Math

2009-05-29T06:16:00.000-07:00

Lisa and Marguerite inspired this one. Their chocolate math triggered the memory banks. I ran into this trick a while ago. I'm not sure why Hershey's and 7up are putting out math puzzles. (Geek bait?) The question is, how does it work? This page has been up for a while so googling it is cheating. Don't even think about it. Let the problem burn a little.

Chocolate and Math 2

2009-05-28T02:49:00.000-07:00

I'm glad I can safely combine my two addictions: chocolate and math. British Psychologists at North Umbria University have conducted a study that suggests flavanols present in chocolate (especially dark, my favorite) boost your ability to do certain mathematical tasks. Here's the article in the Telegraph.

pic by mararie

Geek of the Week: Ken Colangelo

2009-05-27T04:41:00.000-07:00

I have received my first ever geek-resumé. It came via email about a week ago. It almost knocked the slide-rule right out of my shirt pocket. Here are the contents of that message.

I’d like to nominate myself.

In my spare time at home (spare time mind you) I’ve been creating code to generate optimized 3D models of Menger sponges.

I used to use huge amounts of CPU time on the VAX at CCSU as a student exploring strange attractors, Julia sets, cellular automata and n-body gravitational simulation.

They even gave me a degree for it.

This rendering I found hangs in my bathroom. 4th level Menger sponges…for the bathroom…ha!
This, combined with my frequent and popular (?!) emails about when to see the ISS passing over and my geektastic panoramic photography I think puts me in the running.

Also, I got my first paying computer job when I was 9. Seriously. I used those monster 8-inch floppy disks and installed CPM on old 8088 computers.

I’ve got a goofy picture of me covered in electrodes and another wearing safety goggles and a breather if you need them for my geek-of-the-week portrait.

I’ve got my fingers crossed.

-Ken

He had me at Menger sponges, but he throws in a follow up email just to make it hurt.

I’m a rabid Scrabble player.
N.S.A.* member number 602852
*National Scrabble Association

The Brain and Math

2009-05-26T02:03:00.001-07:00

Christia sent me this interesting article on new psychological research done on math. Blow it up and read it. You're a mathematician whether you want to be or not. Embrace the inner geek.

The Institute of Figuring

2009-05-25T05:04:00.000-07:00

This is one of the more bizzare websites I've stumbled upon. The kelp-like exhibit you see above is actually a collection of crocheted hyperbolic planes residing at the Institute for Figuring in California. Here's their statement of purpose.

The Institute For Figuring is an organization dedicated to the poetic and aesthetic dimensions of science, mathematics and the technical arts.

The institute is kind of hard to describe. They have a lot of online exhibits. One I thought Jesse might like is called Philosophical Toys. They also will teach you how to crochet one of those cool hyperbolic planes. Time to get out the knit sticks!

Chocolate Math

2009-05-22T04:29:00.000-07:00

Marguerite and Lisa sent this to me yesterday. Hersheys has got a way to figure out your age without you even having to tell how old you are. How do they do it? By chocolate math! Below is the email:

Don't tell me your age; you probably would tell a falsehood anyway-but the Hershey Man will know! YOUR AGE BY CHOCOLATE MATH

This is pretty neat.

DON'T CHEAT BY SCROLLING DOWN FIRST!
It takes less than a minute .
Work this out as you read .
Be sure you don't read the bottom until you've worked it out!
This is not one of those waste of time things, it's fun.

1. First of all, pick the number of times a week that you would like to have chocolate (more than once but less than 10)
2. Multiply this number by 2 (just to be bold)
3. Add 5
4. Multiply it by 50 -- I'll wait while you get the calculator
5. If you have already had your birthday this year add 1759 ..
If you haven't, add 1758.
6. Now subtract the four digit year that you were born.

You should have a three digit number
The first digit of this was your original number
(i.e., how many times you want to have chocolate each week).
The next two numbers are
YOUR AGE! (Oh YES, it is!!!!!)
THIS IS THE ONLY YEAR (2009) IT WILL EVER WORK, SO SPREAD IT AROUND WHILE IT LASTS.

My answers as I worked through were
1) 3
2) 6 (multiplied by 2)
3)11 (added 5)
4) 550 (multiplied by 50)
5) 2309 (added 1759)
6) 342 (subtracted 1967)
Hersey Man, how do you do it!
Let's let the number of times I eat chocolate a week be c. Here's what Hershey Man does with my number.
1) c
2) 2c
3) 2c+5
4) 50(2c+5)
5) 50(2c+5)+1759
6) 50(2c+5)+1759-birth year

Now let's rewrite this expression using the distributive property and then combine some like terms:
50(2c+5)+1759-birth year = 100c+250+1759-birth year = 100c+2009-birth year
Look at the first part of 100c+2009-birth year. Multiplying any single digit number by 100 moves the digit from the ones to hundreds place. That is why the number of times you eat chocolate a week ends up in the 100's place.
The second part of the expression, 2009-birth year, simply calculates your age.

A couple of things. Chocolate math only claims to work in the year 2009, but can be easily updated next year by having people add 1760 or 1759, instead of 1759 or 1758. Also, if you are over one hundred years old, you'll break chocolate math. A woman that is 105 and eats chocolate 4 times a week will be told by chocolate math that she eats chocolate 5 times a week and is 5 years old.

Thanks Marguerite and Lisa!

pic by cursedthing

Math and Dishware

2009-05-21T01:50:00.000-07:00

This page of the CB2 catalog was calling my name. I'll let you guess why. "Excuse me, miss, do you have those in a hexagon?"

Geek of the Week: Vadim Korf (aka KGB)

2009-05-20T02:27:00.000-07:00

Vadim wins Geek of the Week for solving one of the blog problems I wasn't able to solve. Not only solving it, but providing one of those elegant solutions that leaves you saying, "Of course, how come I couldn't think of that?" I hope you enjoy it as much as I do. Here's the problem:

A town creates a committee of six people to solve a new math problem. In the town, everyone is either a friend or an enemy. Prove that there are at least three friends or three enemies on the committee.

And here's Vadim's proof.

Six people: A, B, C, D, E, F

Two kinds of relationships: X, Y
---- r(A,B)=X means A and B have relationship of type X.
---- r(A,B,C)=X means that r(A,B)=r(A,C)=r(B,C)=X

We have to prove that we have at least one r(P1,P2,P3)=X (or Y), where P1,P2,P3 could be anyone from {A, B, C, D, E, F}

Consider A (anyone). It has a relationship of type X (could be either friend or enemy, whichever has a bigger count for A) with at least three other guys, for example B, C, D (and possibly E and/or F as well), so r(A,B)=r(A,C)=r(A,D)=X

If r(B,C)=X then r(A,B,C)=X
If r(B,D)=X then r(A,B,D)=X
If r(C,D)=X then r(A,C,D)=X
If neither 1, 2, or 3 is true, then
r(B,C)=r(B,D)=r(C,D)=Y, then r(B,C,D)=Y

If you are not picking up on his notation, basically he's saying: consider any person in the group of six. That person has a relationship with 5 other people. He/She is either friends with 3 or more people or enemies with 3 or more people. That gives you two cases. The following proof will work in either case, so we just have to look at one. Assume that A is friends with B, C, and D. Vadim has given us a diagram to help visualize the argument. The red line between any two people means they're friends.
Consider the relationship between B and C, if they are friends, we have the picture. (Note: the green lines represent that the relation between them is unknown.) So A,B, and C are all friends. Therefore, there are three friends on the committee. In fact, the same is true if either BD or CD are friends. So what about if BC, CD, and BD are all enemies? Here's Vadim's picture. Remember the red lines represent friendship and the blue lines represent enmity. (Note: ignore the 4. Too lazy to edit it out.) But then B, C, and D are all enemies, so we have at least three enemies on the committee. All done.

Lastly, I always include a headshot of the Geek of the Week. Vadim's a little camera shy and wanted me to include the following in place of his own mug. It clearly goes along with his alias, KGB. I think it's some kind of MIT in-joke that I don't understand? It just makes him all the more geeky.

Mathematical Poetry

2009-05-19T04:46:00.001-07:00

I found a mathematical poetry blog currated by Kaz Maslanka. The equation above is one of many interesting examples in which the language of mathematics is used to articulate a relationship between poetic constants: life, love, affection, sacrifice, and relationship.

Missing Algebra?

2009-05-18T07:42:00.000-07:00

Coming down after your intermediate algebra final? Don't know how you are going to fill up the summer hours without all the homework problems? YofX is here to help. Here's a problem to both reinforce and extend your algebraic superpowers.

Given distinct positive real numbers, x,y,z such that is true. Write x,y,z in increasing order from left to right, and justify your answer algebraically.

Problem stolen from Awesome Math camp's application. Their camps look really fun. I wish I was 15 again!

The Japanese Census

2009-05-15T06:03:00.000-07:00

This is an incredible visualization of Japanese census data rendered by Soma. The first thing to understand is the Japanese census is a little different than our own. Here is Soma's description.

Let's jump right into this: Japan has the absolute best census in the history of my known world. Not only does it include normal things like age, sex, and the height of each of your pets, but it also legitimizes the gossipy question of What Are You Doing Right Now? Japan slapped a bunch of people with notebooks and a sacred Numbers Mission: keep a log of what you do during the day, in fifteen minute intervals. And those people did!

I've lost a lot of hours playing with these graphs!

great pic

2009-05-14T04:03:00.000-07:00

Photo taken by Swisscan that I picked up on the Flickr photo group, Everything's Geometry. (My sentiments exactly.) I take the feed. Join up, geek out.

Geek(s) of the Week: Academic Support Center

2009-05-13T02:43:00.001-07:00

Every semester the tutors in the Academic Support Center miraculously save a good number of my students. I'm not sure how they do it but I know it involves a lot of work. Check out a couple of videos I made for an old course shell project. Who wouldn't want to talk math with either one of these guys? There are plenty of others in the ASC, too. I don't have videos of them, but that is just a matter of time. Keep up the great work!

Mohamed

Jeff

Game of Life

2009-05-12T01:58:00.000-07:00

One of Lee's comments on the mandelbrot set led me to something called The Game of Life. I found lots of interesting sites on the game and tons of videos on Youtube. For the uninitiated, these two videos featuring John Conway, the game's creator, are a good intro to the rules of the game. They also help to explain why the game is so interesting.

Part 1

Part 2

This video is my favorite Life simulation. I was thinking someone has to have mapped the patterns of Life into the audio field. And sure enough, you get both the audio and visual representation of the patterns in this video to I think a much more engaging effect.

A mathematical who-done-it?

2009-05-11T02:12:00.000-07:00

Sue tells me her oldest has discovered the joy of logic problems. Here's a fun one from the exercises in Enderton's A Mathematical Introduction to Logic:

There are three suspects for a murder: Adams, Brown, and Clark. Adams says "I didn't do it. The victim was an old acquaintance of Brown's. But Clark hated him." Brown states "I didn't do it. I didn't even know the guy. Besides I was out of town all that week." Clark says "I didn't do it. I saw both Adams and Brown downtown with the victim that day; one of them must have done it." Assume that the two innocent men are telling the truth, but that the guilty man might not be. Who did it?

pic by martinmaters

The Improbable Power of Exponentiation

2009-05-08T02:14:00.000-07:00

If you are a math teacher, you're probably familiar with this activity.

Get a big sheet of paper and a good paper cutter. Cut the sheet in half, and stack the pieces together. Turn the pile 90 degrees, cut it in half again, and stack the cut pieces again. Keep doing this until you've cut and stacked the sheet 100 times.

If your cutter is really good, so that it cuts a thick stack of paper as quickly and easily as a single sheet, and it takes ten seconds to make a cut, stack the pieces, and position them for the next cut, then you'll be finished in 16 minutes, 40 seconds.

If the paper is 0.1 millimeter thick (about 0.004 inch), like ordinary medium-weight letter paper, then a stack of 500 pieces would be 50 millimeters (about 2 inches) thick.

The question is: How thick will the pile be after you cut and stack the paper 100 times?

Jay sent in a nice website to guide students through the activity. Make sure to look at the answer. Jeff Root does an excellent job at getting one to understand in a physical way 2^100.

Feather Knot

2009-05-07T02:09:00.001-07:00

I ran into this Kate MccGwire sculpture via Drawn!. Kind of reminds me of a knot problem. I'm also thinking, how long did it take to place all of those feathers? I thought Algebra was for the patient.

I play Spaghetti and Math Balls

2009-05-06T01:51:00.000-07:00

Sue brought this in for her Number Systems class. Her youngest, Nick, is a budding math geek and this is his game. Here's how you play. One moves along the board by answering basic addition and subtraction questions. Each time you pass start you get a meatball. The goal is to be the first one to fill your bowl with six meatballs. Warning: this game can make you hungry. Bon Appetit!

pics from BoardGameGeek

The Worst Reason to Study Math

2009-05-05T04:42:00.000-07:00

All over China, parents tell their children to stop complaining and to finish their quadratic equations and trigonometric functions because there are sixty-five million American kids going to bed with no math at all.
- Michael Cunningham

A friend forwarded this to me recently. I get sent a lot of quotes and stats of this flavor. I appreciate the aid in trying motivate my students to do math. I'm always looking for another tool in the, "When will I ever use this" battle. However, something about these quotes always seems to turn my stomach. It is either the tatic of fear-mongering or the unanalyzed assumption that the world is ultimately a better place if America continues it's preeminence among nations. I'm wondering what your reactions are? I'm looking forward to some good comments.

May Issue of Wired

2009-05-04T02:31:00.000-07:00

I really enjoyed the new issue of Wired. It's put together by guest editor, J.J. Abrams, who I'm ashamed to say I've never heard of, but whose issue as an homage to puzzling. He comes at it from various angles from a feature story of an artist who has been able to stump the best cryptographic minds in the world to an "enigmatrix" of the network connections between math, board games, plot, mysteries, magic code, and game theory. Best of all, it is full of puzzles. Enjoy the one above!

Geek Goodness

2009-05-01T04:48:00.000-07:00

Mike Z sent me a really a great link that will allow you to use your geek powers to feed the world. The site is called FreeRice. When you enter FreeRice, you get a math question like 13 * 5. If you answer the question correct, you have just donated 5 grains of rice to the UN World Food Program. You can play as long as you want, so fill up your bowl. If you get tired of math questions, you can always switch to something else like vocabulary words or chemical symbols. Who pays for all that rice? I just filled up a bowl and my sponsors were World Hunger Relief 2008, Unilever, and a private donor. Pretty cool, huh. Thanks, Mike!

The Math Curse

2009-04-30T05:38:00.000-07:00

Sue brought in this awesome kids book for math, The Math Curse. I was flipping through it. The illustrations are great and so is the story. The premise is a little girl starts to see everything as word problem. She's getting dressed, word problem. Oh, no! Riding the bus to school, word problem. She can't turn off the vision. That's the curse. Everything is math. If you have ever had to go to a party right after studying for your topology comps? You know exactly what she is going through.

Geek of the Week: Laura

2009-04-29T02:29:00.000-07:00

Laura is our resident Foxtrot expert, which is a surprisingly geeky comic. You might remember the Fibonachos. That was Laura. Below is her latest find. I like this one even better. I can't wait to give it to my students.

I am also happy to report that the caché of the Geek of the Week award has risen, evidenced by the fact that people are now starting to prepare acceptance speeches. Here is an excerpt from the full 20 minute version of Laura's speech.

...Wow! Geek of the Week! I never thought I'd win! It was an honor just to be nominated...I'd like to thank the math department for selecting me..my mom and dad for all their love and support and for giving me my first abacus and then my first slide rule...and let's not forget my Tunxis friends who love to remind me on a daily basis that I am a geek!

Correlation, Not Causation

2009-04-28T04:25:00.000-07:00

This is another one for Sue's stat-heads. I grabbed it off Harvard's Social Science Statistics Blog. Somebody ought to stop the import of those lemons.

Mathematical Frenemies

2009-04-27T02:03:00.000-07:00

This one came up on the problem widget (lower right) a while ago. I've been struggling with it.

A town creates a committee of six people to solve a new math problem. In the town, everyone is either a friend or an enemy. Prove that there are at least three friends or three enemies on the committee.

pic by Alana

Geek of the Week: Robert Devaney

2009-04-24T06:15:00.000-07:00

Last Friday I attended the Spring MATYCONN meeting with my infamous colleague Sue. I was treated to a really excellent tour of the Mandelbrot set by BU prof Robert Devaney. Where most presentations of fractals seem to die in the pictoral realm, Devaney gave the necessary background to appreciate the pictures and inner patterns of the set. He also had some cool visualizations. He's got some of them posted on his homepage.

pic by Fermion

Math Battle

2009-04-23T02:58:00.000-07:00

If you've been reading along, you know I'm interested in finding a spectator friendly math competition. Kate left a provocative comment about the historical precedence of math battles on that post. I didn't run into any historical info, but when I googled 'math battle', I wound up with a Canadian group that has been running an interesting type of math competition it calls a math battle. What makes this more interesting than the usual paper and pencil test taken individually or in a group is that it is not graded. The answers are debated in front of a panel of judges and an audience. It starts in the normal manner with a problem set that each team works. Then the teams challenge each other's answers in a debate-style back and forth between the speakers. So not only can you score points by correctly solving the original problems, but you can also earn points by finding gaps or errors in your opponent's solution. I'm not sure this would work for us. I think John and Steve are thinking of something more immediate, physical, and faster moving, but I would love to see one of these competitions. It seems like an excellent idea. BTW, they also post their old problem sets so it is a great site to go to for a little mathematical recreation. I liked this one.

Two persons play a game on a board divided into 3 × 100 squares. They move in turn: the ﬁrst places tiles of size 1 × 2 lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?

pic by Bre!

Math and Makeup

2009-04-22T04:22:00.000-07:00

This picture was sent in by a student for an enrichment assignment I give in Math for Liberal Arts. Here's the assignment.

Post to the discussion "A picture is worth..." your favorite graph, chart, diagram...etc. You can either attach an image file (.jpg or .gif), embed it, or give us a link. I also want the reason you like it. Could be what it says, could be visual organization/design.

This diagaram is from an independent makeup artist. Brittany Lewis. I am simultaneously drawn and repulsed by it. Who knew makeup could be so analytic? Very impressed.

Speed Pi

2009-04-21T05:45:00.000-07:00

Rob sent this. Duly impressed.

New Tunxis Math Blog

2009-04-20T05:43:00.000-07:00

Former Geek of the Week, Lee Bradley, has started a new math blog geared to helping students pass Tunxis's Intermediate Algebra exam. It's called CCPMath. I'm definitely sending my panic stricken students over to the blog after we get through test 3. Nice work, Lee!

Harmonic Mean

2009-04-17T06:04:00.000-07:00

My wife is studying for an actuarial exam. She was doing an online seminar last night and I heard the speaker mention a harmonic mean. I hadn't ever heard of it, so I got curious. For three numbers a, b, and c. We know the average is (a+b+c)/3, but the harmonic mean is the reciprocal of the average of the reciprocals i.e. 3/(1/a+1/b+1/c). So for three particular numbers, 4, 7, and 8, the mean is 6 and 1/3. The harmonic mean is 5 and 23/29. I though cool, but when would I ever use this. Wikipedia gives an nice example:

What is the average speed of a car that goes a certain distance d at a speed of 60 kilometres per hour and then the same distance again at a speed of 40 kilometres per hour?

The (arithmetic) mean is of course 50. If we calculate the average speed directly, we wind up with a different number. The average speed is the distance they traveled divided by the time. The car travels 2d, and the time traveling at 60 kph is d/60. The time at 40 kph is d/40. So we have the average speed as 2d/(d/60+d/40). We simplify 2d/(5d/120) = 240d/5d = 48. This is what we would have go if we'd calculated the harmonic mean of the rates: 2/(1/60+1/40). Cool, huh? I found out there are other means, geometric and quadractic. I'd like to get examples for each of the others.

pic by mince

Using xtimeline in class

2009-04-16T06:00:00.000-07:00

Here is an interesting project my math for liberal arts class started. They used xtimeline to start a history of math timeline. We're a small class so there are only around 10 posts right now. I hope to grow this over the semesters. It was my first project like this. I waffled between using xtimeline and TimeToast. I liked TimeToast better visually, but it doesn't accept BC dates. That's fatal for a history of math timeline. xtimeline also allows multiple editors. The down sides are the appearance and the embed code is slow to load and akward. I'd like to hear in the comments of experience with similar net apps. I'm planning on doing my course schedule on one of these next semester.

Below is Jim R's Tunxis timeline. He tipped me off on this nice set of web tools. Thanks, Jim!

Semester Drag

2009-04-15T02:57:00.000-07:00

It is that time of the semester when everyone is feeling a little bedraggled. My students are looking at me like if I ask them to simplify one more expression they will pass out. I try to put a good face on, but I'm feeling it too. In short, we're all a little fried. Soon the finals adrenaline rush will save us, but in the meantime I figured we all might need a reminder why we take/teach a math class. I found this graphic over at Filckr by Network Osaka. The graphic is really cool. Make sure to blow it up.

Statistically Significant Other

2009-04-14T06:00:00.000-07:00

I thought Sue and her stat-heads would enjoy this one. It's an xkcd comic I ran into on Math Stories.

Bonus problem for College Algebra

2009-04-13T02:31:00.000-07:00

I thought this might be a nice bonus problem for a college algebra class.

Let A be some number. If c = log(A) and c = a+b where a is a non-negative integer and b is a decimal between zero and one, represent A in scientific notation.

Maybe a concrete form of the question would be better for students: If log(A) = 4.182, approximate A in scientific notation.

Tunxis Math Contest Winners

2009-04-10T04:36:00.000-07:00

I want to send out a congratulation to CK and Jeff who tied for first place in the Tunxis math contest. They are both $200 richer at the moment. Who said math will never bring you fame and fortune?

pic by Terren

Mathematical Comedy

2009-04-09T02:13:00.000-07:00

The Second City is comedy venue started in 1959 in Chicago. They now have branches in Toronto and LA. They have fostered the early careers of a lot of comedians (Bill Murray, the Belushi's, John Candy, Dan Aykroyd, Mike Myers...). Sourcebooks MediaFusion released a retrospective of all the great talent that came through SC. I haven't actually read the book, but it is accompanied by two Cd's that I absolutely love. They're filled with some of the best SC sketches. Here are a couple that tickled my mathematical funny bone.

(Warning: the comedy is a little rough. If you are sensitive or a younger reader, it is best to leave this post alone. There aren't any of the holy, holy swear words, but a few minor peccadilloes, and a lot of unpolitically correct humor. Remember, if you can't take a joke, don't.)

Ever feel like you belong to the null set? (Don't think too hard about that.)

And this is a doctors appointment on stats.

Geek of the Week: Bertrand Russell

2009-04-08T03:31:00.000-07:00

My youngest reader Lia sent me this quote from Bertrand Russell for the blog.

There is much pleasure to be gained from useless knowledge.

It made me think of another great quote Jean-Marc introduced me to from Russell's autobiography. (BTW, Russell had a very unhappy youth.)

There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more mathematics.

It suggests that all anyone seriously depressed needs is a little more math. I'm not sure geometry is ready to replace Prozac, but if it worked for Russell, it's worth a shot.

I googled Russell quotes because I forgot the exact wording on Jean-Marc's quote, and I found out that Russell was a master of the form. QuotationsPage had 63 entries for Russell. Here are some good ones.

Everything is vague to a degree you do not realize till you have tried to make it precise.

This beautifully echoes the wonderful failure of his project Principia Mathematica.

If there were in the world today any large number of people who desired their own happiness more than they desired the unhappiness of others, we could have paradise in a few years.

I have to believe he's exaggerating for effect, but let's hear it for a little enlightened self-interest! I could go on forever, but I'll end on this one about teaching.

Passive acceptance of the teacher's wisdom is easy to most boys and girls. It involves no effort of independent thought, and seems rational because the teacher knows more than his pupils; it is moreover the way to win the favour of the teacher unless he is a very exceptional man. Yet the habit of passive acceptance is a disastrous one in later life. It causes man to seek and to accept a leader, and to accept as a leader whoever is established in that position.

pic by Gisela Giardino

Prime Time

2009-04-07T02:35:00.000-07:00

I was thinking about the possibility of defining a subset of the naturals containing numbers whose digits (appropriately grouped) are its own prime factorization. For example, take 32. It would be a number of this kind if 32 = (3)(2). (Of course it's not.) Are there any members of this set? Well, every prime number is trivially a member, 7 = 7, 11 = 11. The interesting question is, does this set have any composite members? I haven't been able to think of one yet.

I quickly checked the numbers less than 100 and there aren't any examples there. The only single digit numbers in this set are the primes. Consider the double digit numbers, 10 through 19. Let A be a digit, 0 through 9, if 1A = (1)(A), then 1A = A. Since no two digit number is equivalent to a one digit number, this isn't possible. (Note, 1A does not mean 1*A.) What about the numbers 20-29? If 2A = (2)(A), then the most (2)(A) could be is 14, because the most A could be is 7. But we know 2A is in the 20's so this is not possible. Any number in the 30's, 50's, and 70's will have the same problem. Any number 4A can only be grouped (4A) because 4 is not prime. So only the boring numbers (primes) in the 40's make it into the set. Same is true for the 60's, 80's, and 90's. What can we conclude? There is no composite example less than 100.

What about 3 digit numbers? We can start to make arguments by thinking of the grouping possibilities. If ABC is a three digit number, then we could group (A)(B)(C), (AB)(C), (A)(BC). Your mission: find a composite number in this set, or prove that there aren't any.

pic by kingfal

Math Contest

2009-04-06T12:03:00.000-07:00

Friday was a big day for math at Tunxis. Both the Pi contest and CCC system-wide math contest were going down. It was a great day. The victor of the pi contest, Alex Bobman, memorized a whopping 302 digits! We are still waiting for the results of the math contest, but I'll post those as soon as I find out.
I was also able to get a copy of the math contest problems. I thought I'd pass along something fun.

A farmer had a daughter who spoke in riddles. One day the child was asked to count the number of goats and the number of ducks in the barnyard. She returned and said, "Twice the number of heads is 76 less than the number of of legs." How many goats were in the barnyard?

pic by ADoseofShipBoy

Math and Dance

2009-04-03T04:00:00.000-07:00

I have a friend who is getting her doctorate in dance. I asked her if she knew of any intersection between math and dance. She dug up this link to Mathdance. There are a couple of professors that have written a series of instructional dance activities to teach math concepts. Pretty wild. I can't ever imagine leading my students in a ensemble dance of irrational numbers. Maybe I should. I went looking for inspiration on Youtube and found the Dance of Parallel Lines...

and some kind of wild trig dance...

Math, the Chaste Science

2009-04-02T02:16:00.000-07:00

Check out this snarky math comic. Would anything outflank math on the right? Could there be a logician out there saying, mathematics is just applied logic?

Wine and Math

2009-04-01T06:15:00.000-07:00

Who knew math would help you drink better? Enologix is a Napa Valley consultant using data analysis to help vineyards make better wine. I caught their profile in Wired. Not surprising in an industry dominated by connoisseurship, the analytic approach to production decisions has been quite controversial. I guess the only thing to do is open a bottle and find out. (You know the geeks are going to win, right. The geeks always win.)

pic by Buson

Math Boo-yah

2009-03-31T06:42:00.000-07:00

I just got done with a round of midterms. I'm always fascinated by the scribbles students write on their test. My favorite scribble among the recently corrected is a student that got a little over zealous when checking his answer. I'm always harping on students to check. I say it so many times I'm even sick of hearing it. Well, I finally reached someone. Boo-yah!

Probability Problem

2009-03-30T05:40:00.000-07:00

Lee found this nice problem in the Minitab manual that accompanies "Elementary Statistics" by Mario F. Triola. The problem posed is:

Two points along a straight stick are randomly selected. The stick is then broken at these two points. Find the probability that the three pieces can be arranged to form a triangle.

pic by flowers & machinery

Geek of the Week: Colleen

2009-03-27T03:22:00.000-07:00

Listen to this:

I think the geekiness explains itself. If you need more convincing...Colleen took time over vacation to memorize it, just so she could recite it for Sue and me after break. Not only is it geeky, it's sweet and geeky. Thanks for making my pi-day, Colleen!

100 Greatest Theorems

2009-03-26T02:57:00.000-07:00

Jay sent me an unbelievable link. Paul and Jack Abad presented this list of 100 greatest theorems at a math conference in 1999. Their criteria for the greatness of a theorem was
1) the place the theorem holds in the literature
2) the quality of the proof
3) the unexpectedness of the result
I had fun looking for my favorites: denumerability of the rationals, the non-denumerability of the continuum, and the incompleteness theorems. Jeff and Lee will be happy to see that the infinitude of the primes made it (#11). The list has dates for all the theorems, and links to proofs for some of them. Any guess as to what's number 1?

pic from Boston Bill

Using Selectricity

2009-03-25T06:43:00.000-07:00

I'm entering a unit on elections in my Math for Liberal Arts class. There are a lot of slick tools out there to run online polls like PollDaddy. I was having trouble finding any tool that allowed user's to fully order their preferences... until Selectricity. It's a really great tool, particularly if you are studying elections. Check out this poll, it allows you to see current results, and the current winner under a number of common election systems: Borda, Condorcet, etc. Anyone can create an election without registering. For more extensive security measures, register for a full account. All free!

pic by Enrico Fuente

19th Annual Tunxis Math Contest

2009-03-24T06:05:00.000-07:00

Rob Clark is hosting his annual Math melee. You can problem solve for money. Here are the details.

Tunxis Annual Math Contest: Friday, April 3, 2009 from 12:00 – 2:00 p.m. in Room 6-201. Cash Prizes Awarded! $250, $150, $100 for 1st, 2nd, and 3rd.
OPEN TO ANY STUDENT CURRENTLY ENROLLED IN ANY COURSE AT TUNXIS. QUESTIONS ON THE EXAM ARE MOSTLY ALGEBRAIC. THERE IS NO TRIG OR CALCULUS ON THE EXAM. SIGN UP WITH R. CLARK IN OFFICE F-32. SAMPLE EXAMS AVAILABLE IN OFFICE F-32.

My New Favorite Puzzle: Tentaizu

2009-03-23T05:49:00.000-07:00

I found a new puzzle. New to me anyway. It's called Tentaizu. I found it in the in-flight magazine on the way home from vacation. Btw, Southwest's magazine has the best puzzle section that I've ever seen. Not a reason to buy a ticket, but a definite perk. Anyway, I was completely charmed by this game. I thought we would play one this week. Maybe if we get excited we can try to automate a solution technique. I also tried to find a website that generates these like those I've found for Sudoku and KenKen. No luck.

Here are the instructions from Southwest's magazine:
In each of the grids on this page, stars are
hiding in 10 of the 49 squares. Your task is
to determine the positions of the stars. The
numbers in the grid provide clues: A number
in a square indicates how many stars lie next
to the square—in other words, how many
adjacent squares (including diagonally adjacent
squares) contain stars. No square with
a number in it contains a star, but a star may
appear in a square with no adjacent numbers.

YofX on Spring Break

2009-03-13T11:38:00.000-07:00

Tunxis is going on spring break this week. I'll be enjoying the vacation, but will be back posting on 3/23. Until then, stay geeky, and solve some good problems.
Already, bored? Check out this Geeks Gone Wild Vid

Jean-Marc's Angle of Fire

2009-03-13T02:36:00.000-07:00

Jean-Marc passed on this problem to me. Here's the story. You're on one side of a pit of fire. You see two walls protruding from it. You must figure out how to build a wall so that if it were continued past the pit of fire it would bisect the angle. I solved this with trig the other day during a meeting and told Jean-Marc. He said the fun solution is purely geometric. So I'm back at work on this problem with my weak geometry. Help!

Understanding Instant Runoff

2009-03-12T10:00:00.000-07:00

I've been crawling the web, gearing up to teach voting systems in my Math for Liberal Arts course. I want to pass on this interesting site.

The voting that we do in the US is a modified plurality vote. One problem with our system is that often the winning canidate has not garnered a majority. One popular alternative used frequently in other countries is run-off voting, or what we call the Hare method in class. I found a video (below) and a great site, Choiceranker to explain how run-off voting works. Choiceranker even gives you the chance to participate in opinion polls that use the run-off method. Check them out.

Sign Up for Tunxis Pi Contest

2009-03-11T02:55:00.001-07:00

Just wanted to let you know that the 3rd annual Tunxis Pi recitation competition is coming up Friday, April 3rd at 10:00 am. Sue has even designed M&M's for it. The sign up sheet is outside her door (office F-7). There are cash prizes for the top three finishers: $150, $100, $50. Come eat chocolate, make money, be geeky!

Rolling Stones + Math

2009-03-10T06:00:00.000-07:00

Christina found a math course at MIT called Street Fighting Math. That's what we need at Tunxis! Contact math. (Plenty of elbow/knee pads though.) Until we offer it, you can follow the link and get all of the MIT course's course materials: lecture notes, video, etc.

What Would a Spectator Friendly Math Competition Look Like?

2009-03-09T02:39:00.000-07:00

John and Steve have given me the challenge of creating a math competition that is actually fun to watch. There are tons of competitions out there. Rob runs one every year at Tunxis. However, almost all of these involve students sitting at a desk solving problems. What would a spectator friendly math competition look like? I have been thinking in two directions on this.

1) Taking math problems and making them physical. Classic bucket problems could be fun to watch if you took away paper and pencil and gave competitors actual buckets. (Think Die Hard with a Vengeance minus the bomb.) Or, here's a pack of M&M's, add these two numbers in base 6. Compass and protractor constructions would be interesting, particularly because hardly anyone knows how to do them anymore.

2) Structure the competition so that the audience can evaluate or judge the participants. Math contests are usually only judged for correctness and speed by experts. John mentioned the example of the poetry slam. How could this happen for math? Here are some ideas to get started with. Contestants solve a problem at home before the contest, come to the contest, and explain their solutions in 3 minutes or less. Then the audience judges them on the clarity/creativity of their explanations. Or, contestants prepare a 3 minute lecture/demonstration about math and the audience judges. Another wilder idea is to have them perform a creative task that doesn't have a correct answer. I use an enrichment activity in Math for Liberal Arts that might work. I have students create a poem using the power set structure on a set of words. The audience would definitely be able to evaluate this.

I'm not sure exactly what John and Steve had in mind for the bigger picture. I think they plan to have contests like this in all disciplines. They definitely gave me an interesting thought problem though.

Solving The Rectangles Problem

2009-03-06T05:06:00.000-08:00

I keep posting problems and you do all the solving. I figured I better walk the walk, so I worked on Lee's problem from last week's post. It was really a nice one. The idea was to give the dimensions of five rectangles such that
1) the sides have unique integer dimensions between 1 and 10
2) they can be arranged into a square.

The first thing I did was narrow the size of the possible squares. I did it originally with an area argument. The minimum area of 5 rectangles that meet criteria 1 would be 1*10+2*9+3*8+4*7+5*6 = 110. The max would be 10*9+8*7+6*5+4*3+2*1= 190. So, 110<11^2, 12^2, 13^2<190. That limits the squares I was considering to only three different dimensions. (Later I found another way to argue for these as the only three possibilities.)

I have to admit I had a little more information than you guys did on the blog. Lee is a puzzle maker and he brought in a wood version of one of the solutions to show me. I was playing around with it on my desk one day and found it surprisingly hard to arrange into a square. It took me around 5 minutes to arrange 5 rectangles into a square.

Seeing the correct configuration was the insight that broke the problem for me. The rectangles looked something like this when assembled.

Note that this isn't drawn to scale, but essentially you have one rectangle in the middle with the others spiraling around it. The dimensions of the middle depend on the dimensions you choose for your outer rectangles.

That's when I started playing with me three possible areas.

1) If the overall square is 11 by 11, then what are my options for the inner rectangle. 10 can't be one of the sides of the inner triangle because I have to add two numbers to it to get 11. 9 also can't be it because I'd have to add two one's to it to get 11, and I'm only allowed one one. 8 is the highest possible side of the inner rectangle, because the outer rectangles could have 2 and 1 as sides.

But then since the sides of the whole rectangle are 11, we can fill in...

Now only 3,4,5,6,7 are left, and the only pairs that add to 11 are (5,6) and (4,7). That means 3 has to be the other dimension of the center rectangle.

What two remaining numbers could you add to 3 to get 11? There aren't any. Conclusion: 8 can't be the largest dimension of the inner rectangle. Try 7. And so the process repeats through all possible cases. I really didn't do it that systemattically, but this makes for a better demonstration.

When I got through with the process I only found two solutions: An 11x11 rectangle like this

and a 13x13 like this

At this point, I freaked out. I thought my logic was impeccable but Lee promised 4 solutions. At some point I thought to go back and look at the solution he gave me and saw that the sides were identical to mine, but his solution had flipped two of my sides giving different dimensions to the surrounding four rectangles. I'll show you with the 13x13,

So that's it, Lee also has a solution page on his website check it out. What I want now is a proof that 5 rectangles with unique sides cannot be arranged into a square in any other way (than the way I described above). There has to be a simple topological proof of this. Help?

Fun Sent in by Steve

2009-03-05T05:17:00.000-08:00

How many seconds are there in 6 weeks. Write the answer in 3 symbols only.

This is a another problem neat problem sent in by Steve L. Thanks!

(pic by in touch)

Happy Square Root Day 3/3/09

2009-03-03T09:02:00.000-08:00

I bet you didn't know it was Square Root Day. Don't feel bad. I didn't either. Rob clued me in. Here is the Wikipedia entry explaining the holiday. There are only 9 in each century. The next Square Root Day is not until the year 2016, so party really hard. That might begin by learning how to take square roots by hand, extract responsibly.

pic by _ahn

Question: Anyone seen these numbers before?

2009-03-03T06:07:00.000-08:00

There is a long history of identifying subsets of the natural numbers particularly based on their divisibility properties. There are friendly numbers, extravagant numbers, deficient, perfect... The other day in class we were doing prime factorizations and I wound up with something like 2^2*3^3. I started thinking about all numbers n such that if n has a prime factor p, p^p divides n. Even more, I was thinking about adding the stipulation of sequential prime factors. So here are the first 4 numbers...
2^2=4
2^2*3^3=108
2^2*3^3*5^5=337,500
2^2*3^3*5^5*7^7=277,945,762,500
Anyone ever heard of numbers like this? What are they called? Are they used for anything?

Graph of the Day

2009-03-02T08:45:00.000-08:00

We're snowed out today! (except for maybe evening classes). Enjoy the day off, solve a math problem.

Vision Under the Influence of Math

2009-02-27T06:15:00.001-08:00

Here is a pic from the Flickr photo group called "Everything's Geometery". It is a continuous stream of people posting photos that reflect a geometric/mathematical aesthetic. I'm not sure what that means. Maybe clean lines. Traditional figures. Order. I'll ask Carianne later. I do know that the photos tend to be similar to the types of visuals that arrest me as I go through a day. One of the most common photos types in this stream involves stairs, particularly the play of shadow on stairs. I'm not sure why that is so powerful, but if you have ever stopped to admire a staircase climb the side of a building, you might be a math geek.

pic by Juilien'

Shirley Ellis's Name Game

2009-02-26T03:22:00.000-08:00

Probably one of the coolest algorithms of all times. I love the fact that the name game is not a song, but a means to generate a whole series of songs given a cede. Formally, the name game is a lot more interesting (if not a little sillier) than a normal song. The Name Game is the brain child of Shirley Ellis. She recorded it in 1965. (Somehow I thought the song was much older.) The placard below is from some promotional material of the time, and actually explains the rules (or algorithm). Note the exceptions so we don't end up swearing.

Geek of the Week: Lee Bradley

2009-02-24T17:51:00.000-08:00

Welcoming the newest member to our math community Lee Bradely. Lee has just been hired on as a full time math tutor in the Academic Support Center. He gave me a great problem. You have to create 5 rectangles, such that the length of each side of one of the rectangles is an integer 1-10. You have to use all 10 numbers, so no sides of the same length. Furthermore, the resulting rectangles have to tile together to form a square. He has sworn there are 4 solutions and has given me one. Stop over in the ACS to say hello and get a hint from him. Also check out Lee's website. He uses mathematics to make beautiful puzzles.