Sunday, 31 January 2010

A Course on Street Science

As they are learning all about Quarks and String theory and the big bang.. maybe we could teach our kid a little about applications of science to solving world problems.. Here is a MIT prof using her students to do just that, and the results are intriguing.. If you watch nothing else, listen to the last five minutes when she talks about her vision... see the world in a whole new way...

Saturday, 30 January 2010

Don't Miss These

Dan MacKinnon over at Math Recreations has a nice blog about Appoloinian Gaskets formed by Ford Circles (what you get when the steering wheel comes off a Model A) and the relationship of course to Farey Sequences..(If you ever added fractions the way they told you was wrong... this is your revenge).

Also Robert Talbet over at Casting Out Nines has a nice video about Sierpinski's gasket as part of a Dorito Ad for the Super Bowl..... Fractals make the big time...

Addendum... Cory Poole, the teacher of the students who created the fractal commented on this post, Nice job Cory, you have ever reason to be justly proud of your students.

After getting Cory's message, I tracked down one of his web sites where he explains the process of construction (and some notes on motivation)... see it here

Fair Division......A Piece of Cake...

Many really complex problems of mathematics are founded in simple social practices, such as the fair division of assets. A company dissolves and the partners need to divide up the assets. The last surviving parent dies without a will and the children have to decide how to distribute the home, boat, car, golf-club membership...etc. Such issues keep lawyers in business and lead to terrible fighting between the parties because of the different ways they value the things to be divided.

One of the most commonly talked about is the idea of dividing a cake fairly between two people. Most people think they know how to do that. Here is a quote from Alan D. Taylor's introduction to The Geometry of Efficient Fair Division, by Julius B. Barbanel
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Discussions of cake cutting almost always begin with the procedure known as divide-and-choose. Historically, this two-person scheme traces its origins back 5000 years to the Bible’s account of land division between Abram (later to be called Abraham) and Lot, and it resurfaces more explicitly two-and-a-half millennia ago as Hesiod, in his Theogony, describes the division of meat into two piles by Prometheus, with Zeus then choosing the pile that he preferred.

Mathematical investigations of fair division date from the early 1940s. The constructive vein was first opened by the Polish mathematician Hugo Steinhaus (see [40]) and his colleagues Stefan Banach and Bronislaw Knaster. Steinhaus appears to have been the first to ask if there is an obvious extension of divide-and-choose to the case wherein there are three participants instead of two, and he derived the scheme referred to in a number of mathematical texts for non-majors (see [18] and [42]) as “the lone-divider method.” But extending this procedure to four or more participants is somewhat complicated, and was not actually achieved until Harold Kuhn [30] did so in 1967. Banach and Knaster, however, took an entirely different tack and devised a fair-division scheme for any number of participants that is known today as the “last-diminisher method.
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The idea that the "cut and choose" solution is "fair" assumes that the object to be divided is uniform and that both people approach the division with the same value system. Consider, for instance, a cake baked in a square pan with frosting on the top and sides. Perhaps the cake has one half that is chocolate and one half is vanilla. Now what if one of the people really likes frosting, or really LOVES chocolate, and other is indifferent to the frosting amount or flavoring(you can play with the implications of levels of these factors). Is there an advantage to being the one who does the cutting or does the choosing? To further complicate the process, consider the different ways you might cut the cake if you were one of these people and you knew the other person's preferences. What if the information access was not equal? How would it effect the choice of who cuts if you knew they had more information about your preferences than you had about theirs? What if you had the information advantage? And what if there were three people.. how would you extend the cut-and-choose method (it can be done)?

And what if the cake is not a cake, but the remains of the family estate, A car, a house, a boat, membership in the golf club...? Now what would be "Fair"?..

Ok, maybe "Piece of Cake" is not such a piece-of-cake after all. Maybe I'll take time to talk soon about the more complicate situation when an estate contains non-divisible items, a family home, the boat in which your dad took you out fishing when you were a kid... which half of your mom's antique Ming Dynasty vase would be more valuable...perhaps not simple when it is your family.

Thursday, 28 January 2010

Baptize Me Brother, I'm a Believer

It started with Dan MacKinnon's blog at mathrecreation one of those blogs I read regularly (he writes, I read, that's regularly).

In this one he mentioned a speech in Ottawa... OK, I'm in England, and not likely to pop over to Ottawa for a quick speech, but he had this quote about what Alfie Kohn was all about..."Our knowledge of how children learn – and how schools can help -- has come a long way in the last few decades. Unfortunately, most schools have not: They’re still more about memorizing facts and practicing isolated skills than understanding ideas from the inside out; they still exclude students from any meaningful decision-making role; and they still rely on grades, tests, homework, lectures, worksheets, competition, punishments, and rewards. Alfie Kohn explores the alternatives to each of these conventional practices, explaining why progressive education isn’t just a realistic alternative but one that’s far more likely to help kids become critical thinkers and lifelong learners."
Wait... that sounds like me talking... no grades (not like we have to give now anyway)... Self directed students making their own educational decisions... HEY... I think I found my Edu-church...

Sooooo, I went to Alfie's web page...wow, this guy is a radical from way back..(you go guy)... and I liked what he was saying... so I read some articles he had published.. and I found one that may save me the price of an international flight from here to there... It may be the same talk he is giving..more or less... If this is not your bag, that's OK too, but here is a clip that I think will help you understand whether you may like this guy or not...

"A friend of mine, who is a teacher-educator, had a daughter in fifth grade at the time of this story. She came home, he wrote me, with a worksheet on simple machines—ball bearings, inclined planes, pulleys, that sort of thing. As he came home from work, she said, “Dad, test me, test me!”

“Well,” my friend said, “Why don’t you just tell me what you’ve been learning about?”

“OK, but first ask me what these things are!”

“OK, if you insist. What is a ball bearing?”

“OK, easy dad! A ball bearing is blah blah blah.” A verbatim repetition of the definition she had learned from the teacher.

My friend said, “But Rachel, what is a ball bearing?”

“I told you, Dad. A ball bearing is a blah blah blah blah.”

To make a long story short, he continues, “I turned over the ottoman, which is on wheels, and showed her the ball bearings, and her eyes got wide.

“‘Cooool! That’s what a ball bearing is? How does it work? Can we take apart the ottoman? Oh, I get it. Why didn’t Mrs. Lambert just tell us this is what it was? Can you buy these at the store? Where do they sell these things anyway? Hey, wanna help me make something that rotates? Hey, cool, watch what happens if I hold one of these things and try to spin this thing. What would happen to this thing if the balls were really big? Would the wheels go faster?’”

A progressive school is not about memorizing the definition of ball bearings, or the date at which an event happened in history, or the difference between a simile and a metaphor. That’s not to say that these topics aren’t covered. It’s to say that questions that kids have drive the education."

That's the kind of education I want to be part of.... Thanks Dan...

Wednesday, 27 January 2010

Geometric Explorations in Three Stages

Here are some geometry explorations I think that you could present to kids in three levels.. For example:

If you had a triangle and marked one of the points of concurrency, perhaps the intersection of the medians or the intersection of the perpendicular bisectors.

First you challenge the student to think about what would happen if??? What if we held two of the vertices fixed, and moved the third along a line parallel to the side joining the first two; what would be the path traced out by the int

It is good for them to ponder the "what might happen" without a definite goal or a right or wrong answer. Maybe doodle with a pencil and paper... Push for a little detail.

Now (this would be stage two) let them actually construct the locus with interactive software and see what happens...this is not trivial, even with the magic of modern software like Geogebra or Cabri or Geometer's Sketchpad... and there are often surprises that lead them to wonder their own "what if" questions.

Now the final stage... prove it analytically. This will be the hardest part. For example, If you picked the centroid, or intersection of the perpendicular bisectors of the sides, then what they will see is that the locus it follows is a straight line parallel to the fixed side; but you can be a little more definite.. Let's call the two fixed points A and B, and put them on the x-axis. Now the third point will have a changing value of x, but the y-coordinate will be constant at some value, call it C. If they have trouble starting, we can ask them to think what is the coordinates of the point M that is the median of AC, and N that is the median of BC..... again the x values will change, but the y-values will be C/2 for both. So now we can challenge them to describe the geometric figure that is AMNB, whose diagonals are the medians, and whose intersection we seek.
So what can we see in the trapezoid??... what do we know??... what about the size of the two parallel bases??... Do you see any triangle relations that would help? Eventually we hope they will figure out that triangles AFB and EFD are similar.. and since the two bases are in a ratio of one to two, the perpendicular distance to the centroid will be in that same ratio... or 2/3 of the way from the lower base to the upper base.... SO??? aha, but the upper base of the trapezoid has a y-coordinate of C/2, so 2/3 of that means the centroid will be at C/3...

For an advanced challenge, what happens if we follow the intersection of the angle bisectors, or the intersection of the altitudes, the locus will not be a straight line. Can you visualize it? Construct it.. what does it look like... Prove it?

Monday, 25 January 2010

Different Question, Same Idea, Sort of

A couple of days ago I wrote about the probability of a 28 Year Snow and 100 year Rain. It reminded me of another problem that, seemingly completely different, is somewhat related.

Here is a version of the question... Suppose you were looking at a group of n (a large number of numbers) numbers presented one at a time. Your goal is to guess which is the largest number in the list. You get to look at one number, and say "That one." or pass, if you pass, you can never go back and the next number is presented.. How do you decide which number to pick. You could just guess one of the numbers, and when it comes along pick it.. the probability you would win is 1/n. Or you could guess that there will be approximately ln(n)+.58 numbers.. this is the limit for the harmonic series, $\sum_{k=1}^{n}\frac{1}{k}$= [1+1/2+1/3+1/4+...+1/n (for big n...and that doesn't mean huge...at n=20, the actual value is 3.597 and the approximation is 3.572) approaches ln(n) + .0.57721, which will estimate the number of record breaking events in N events, for example, record breaking floods in a 100 year period (look for about 5... ln(100)+.577 = 5.182).. and when that many record results occur, you could pick it... (this would have been my first choice)... unfortunately, the standard deviation of the number of records that occur is pretty wide, so you are not likely to win more than about 20% of the time... Still, in most cases that is better than 1/n.

But what if you decided to wait until a certain number, call it r, of events had passed, then pick the next number higher than any of the first r... But what r would you pick.

If there are three numbers, 1, 2, and 3, they can occur in six orders. If you decide to always pick the first, or second or third you would win 1/3 of the time.. Or you could wait and after the first, pick the next one that is higher... Well if 3 is first, you lose, but if the order is 1-3-2; 2-3-1; or 2-1-3 you would win, hey, that's a fifty percent chance of winning..

If there are four numbers, they can be in 24 different orders. Would it be best to let one number go, or two. If we pick the first number, we know the winning probability is only 1/4, and if we wait until the last number, it is again 1/4. But what if we let one go, then pick the next that exceeds that value... We would pick the right number in 11/24 of the orders.. If we let two go, then picked the next that exceeds those two, we would only win in 5/12 of the trials.

So how do you decide? How many numbers would you let go before you picked the next higher number. If there were ten numbers in all, for example, and you let r of them go by, the probability that the next one is the biggest (and hence you win) is 1/n. But you would also win if the next one was smaller than the max of the first r numbers [prob r/(r+1)] and the one after that WAS the largest in the set (prob 1/n again)... Or ...you could win if the next two after you stopped counting were not greater than the max of the first r numbers [probability r/(r+2)] and the following one (yep, prob 1/n) was the biggest value. Ok, you see a pattern here. So the probability of winning after waiting for any r-values is P(r) = 1/n[1 + r/r+1 + r/r+2 + ...+ r/n-1] but if we factor out the r on top.. this is just r/n [1/r+1/r+1...+ 1/n-1] and that last part is the harmonic series between 1/r and 1/n-1... If we make the assumption that r and n are pretty big then we can approximate the sum of the harmonic series out to 1/(n-1) as Ln(n-1)+ .577 and out to 1/r it would be Ln(r)+.577. Subtracting this last from the first, we see that it will be about Ln(n-1) - Ln(r) or, using the log properties you learned in Alg II, we can write that as the Ln(n-1/r)... so for any number of numbers n, if we let the first r go by and pick the next number greater than any of those, our probability of having that be a winning number is (r/n) Ln(n-1/r). All we have to do is find the maximum value of Ln(n-1/r) and we know how many r to let pass. For calculus students, n is a constant here, r is the variable, so we want find the maximum of (r/n)*Ln(n/r).... and for algebra students, graph it and find the max probability.

It turns out that the max happens when r=n/e... so for n= 100, we would wait for the first 1/e = 36.7 (ok, 37) numbers and then pick the next number that is bigger than any of those...and the probability you win.... 1/e or about 36.8%

And how is that related to the 100 year snow? Well, 1-.368 = .632... So the probability of picking the max value from a string of 100 numbers by this strategy is the same as the probability that you do NOT hava 100-year snow in the next hundred years... . Bundle up... the snow is more likely.... rule of the day for multiple choice.... Pick e.... ;-}

Sunday, 24 January 2010

Dear Fox News

Joshua Zucker sent this to the AP Stats EDG, and I thought it worth sharing....

It is from the PhD comic site.

It reminded me of this post I made in December.

They did have another nice comic about the recent financial debacle... http://www.phdcomics.com/comics/archive.php?comicid=1077

Saturday, 23 January 2010

The 28 Year Snow and The 100 Year Flood

Europe just went through one of its worst winters in years, and according to the press, the largest snowfall in England for 28 years (and apparently similar effects in the US.. that darn El Nino). The last this big in Europe was 1982. That doesn't actually make it a 28 year Snow, in the sense of a 100 year flood. Most people don't realize that the probability of having a "100 year flood" in any 100 years is only about 63% (go on, say "Huh?").. Officially, a 100 year flood is a flood that has a 1/100 chance of happening in any given year (I think, technically, they say the probability of a flood that great or greater). You can use a little easy probability to see why it would only have a 63% probability of occurring in the next 100 years (even if it hasn't occurred in 200 years, or if it happened yesterday.. the assumption is that the great floods are essentially independent events). So the probability of having (at least one) "100 year flood" in the next 100 years is just 1- (the probability that we don't have one)..which is easier to compute. The probability of NOT having a 100 year flood this year is .99 or 1- the probability that we do.. The probability we don't have one in the next 100 years is just (.99)100 which my trusty calculator tells me is about .366. So it there is a .366 chance of NO great flood, then there is a .634 chance of having at least one.
"You mean there might be MORE than one?" I hear you ask. Well yes, now we just apply the binomial distribution to find, for example, the probability of great floods in two of the next 100 years (and not in 98 others) and accounting for all the possible orders, we get $\dbinom{100}{2}(.01)^2(.99)^9^8$ or about 18.5%; and if you extend that to exactly three great floods, the probability comes out to about 6%... so there is something like a 36% chance of NOT having a great flood, and about a 40% chance of just one. (It takes about 230 years to be 90% sure of having a 100 year flood; and almost 300 years to be 95% sure).

Now if we said, Wait, a "100 year flood" should have a probability of almost one of happening (would you settle for the proverbial 95% standard from statistics?) then we could work backwards and compute the probability of such a flood this year (or any random year). Since the probability of not having one in 100 years is 1-.95, all we have to do is solve the problem (1-p)100 = .05 and we get the annual probability of such a flood. Surprisingly, that puts the probability up to about 3% (.0295..)

So if we assume that this winter was the traditional "28 year snow" by the normal definition (ie, one that has a 1/28 chance of occurring) then the probability that it happens again next winter is 3.57% ( a little greater than our "certain 100 year flood". So how likely is THAT to happen in the next 28 years...... (you are soooo gonna love this) .... about 63%, to be slightly more precise, about 63.87%...

Hmmm (wheels turning in your mind)... that's almost the same answer..... "COINCIDENCE? " I ask my students; and they know the proper reply..." I THINK NOT!". So what is happening here. If the probability of an n-year flood is 1/n, then the probability of one occurring in the next n years is given by Prob(flood)=1-$[\frac{n-1}{n}]^n$.. and if we plot this we see that it very quickly becomes asymptotic to about .63... or 1-(1/e); which is perfectly understandable since the limit as n goes to infinity of $[\frac{n-1}{n}]^n$ is e-1; or about .36.

Which means, for anything more than a ten year event or so, the probability that it happens in the next N years is always about the same...about 63%... Many questions, one answer.... ahhhh ... gotta love it.

Friday, 22 January 2010

Math Humor

One of my ex-students, Ali B, sent me a link to the comic web site Spiked Math. Some of the comics are a little ... um...er... "ripe" for my HS students (Ali, I am telling your mom)... but here are a couple I thought were pretty nice..

I like this first one because I have a buzz-lightyear doll on my file cabinet that I can press the button and hear lines from the movie, like the one in this comic...not sure if it has a flaw or if they all work that way, but after you press the button and it delivers a line, it will wait three or four minutes and then do another... really freaks kids out when it happens...

This second, I thought, had the wrong title, it should have been something like non-transitive..(look it up children)...and thanks, Jeffo

Thursday, 21 January 2010

You Can Count on It... A Brief History of Tally Sticks

The term "tally" comes from the name of a stick on which counts were made to keep a count or a score. The Latin root is talea and is closely related to the origin of tailor, "one who cuts". Many math words have origins that reflect back to the earliest and most primitive uses of number. Compare the origins of compute, digit, and score. The first record existing of tally marks is on a leg bone of a baboon dating prior to 30,000 BC. The bone has 29 clear notches in a row. It was discovered in a cave in Southern Africa. It is sometimes called the Lebombo Bone after the Lebombo mountains in which it was found. The exact age of such artifacts is a subject of debate, and their mathematical usage is somewhat speculative. Some sources have stated that the bone is a lunar phase counter, and by implication that African women were the first mathematicians since keeping track of menstrual cycles requires a lunar calendar.

Another candidate for the oldest tally record in history is a wolf bone found in Czechoslovakia with 57 deep notches cut into it, some of which appear to be grouped into sets of five.

In Mathematics Galore by Budd and Sangwin, there is a story of much more recent tally sticks. It seems that until around 1828 the British kept tax and other records on wooden tally sticks. When the system was discontinued they were left with a huge residue of wooden tally sticks, so in 1834 they decided to have a bonfire to get rid of them. The bonfire was such a success that it burned the parliment buildings to the ground. What Guy Fawkes could not do with dynamite the Exchequer did with tally sticks.... The power of math.

The story, as improbable as it seems, is varified by a speech by Charled Dickens 1855. [Charles Dickens, Speech to the Administrative Reform Association, June 27, 1855, in Speeches of Charles Dickens, ed. K.F. Fielding, Oxford: The Clarendon Press, 1960, p. 206, ] The somewhat clipped version below is taken from Number, The Language of Science by Tobias Dantzig (pgs 23&24)

Ages ago a savage mode of keeping accounts on notched sticks was introduced into the Court of Exchequer and the accounts were kept much as Robinson Crusoe kept his calendar on the desert island. A multitude of accountants, bookkeepers, and actuaries were born and died... Still official routine inclined to those notched sticks as if they were pillars of the Constitution, and still the Exchequer accounts continued to be kept on certain splints of elm-wood called tallies. In the reign of George III an inquiry was made by some revolutionary spirit whether, pens, ink and paper, slates and pencils being in existence, this obstinate adherence to an obsolute custom ought to be continued, ..... All the red tape in the country grew redder at the bare mention of this bold and original conception, and it took until 1826 to get these sticks abolished. In 1834 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur ot any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighborhood. However, they never had been useful, and official routine required that they should never be, and so the order went out that they were to be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over-gorged with these preposterous sticks, set fire to the paneling; the paneling set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost therof.
Several images of the fire was painted by J.M.W. Turner who watched the fire from a boat on the Thames. I have a clip that I can not credit that says, "The fire of 1834 burned down most of the Palace of Westminster. The only part still remaining from 1097 is Westminster Hall. The buildings replacing the destroyed elements include Big Ben's tower (oooh, side bar... Big Ben is not the name of the tower at Westminster, it is the name of the great Bell in the Chimes there.. admit it, you did NOT know that), with it's four 23 feet clock faces, built in a rich late gothic style that now form the Houses of Commons and the House of Lords. These magnificent buildings are still the subject of many paintings, including my own Parliament, with the grand Westminster Abbey on their north." The one below hangs in the Tate Gallery; while another, I believe, is in a gallery in Cleveland, Ohio.

Around 1960 an ancient mathematical record on bone was uncovered in the African area of Ishango, near Lake Edward. While it was at first considered an ancient (9000 BC) tally stick, many now think it represents the oldest table of prime numbers. Here is a link with a picture where you can see and read more about the "Ishango bone"

Wednesday, 20 January 2010

Typing Monkeys

More observations stimulated by John Barrows new book (see my recent blog)

Everybody has heard the suggestion that a million (or some other number) of monkeys typing continuously for many millenia would eventually produce a) Shakespeare, b) all of known science, c) the bible, d) all of the above).
It began with Jonathon Swift and Gulliver's Travels, 1872, according to Professor Barrow. In the tale "a mythical professor of the Grand Academy of Lagado who aims to generate a catalogue of all scientific knowledge by having his students continuously generate random strings of letters..." (I think, see emphasis in the excerpt below, that it was random strings of words).. Anyway, according to the good Professor Barrow, the story was embellished in different forms until French Mathematician Emile Borel{there is a street and a square named for him in the 17th District in Paris} suggested that random typing monkeys could duplicate the French national library. A few years later(1929), Arthur Eddington Anglicised that to "books in the British Museum."
By 1972, Arthur Koestler writing in The Case of the Midwife Toad, New York, 1972, page 30, refers to Monkeys
typing Shakespeare as "proverbial":"Neo-Darwinism does indeed carry the nineteenth-century brand of materialism to its extreme limits--to the proverbial monkey at the typewriter, hitting by pure chance on the proper keys to produce a Shakespeare sonnet."

Ok, so eventually someone had to put this to a more scientific test, and they did. "A website entitled The Monkey Shakespeare Simulator, launched on July 1, 2003, contained a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from Henry IV, Part 2, reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:"RUMOUR. Open your ears; 9r5j5&?OWTY Z0d... "

Even more impressive, to me, is the fact that 'in another part of that book, Swift tells of how the astronomers on the flying island of Laputia had: "discovered two lesser stars, or satellites, which revolve around Mars, whereof the innermost is distant from the center of the primary exactly three of his diameters, and the outermost five: the former revolves in the space of ten hours, and the latter in twenty-one and a half".

Swift wrote this in 1726, but it was not until 1877 that Asaph Hall discovered the two moons of Mars.'... I just did a little checking on the orbit and periods he "predicted?" and the actual periods are about 7 hours for Phobos, and about 30 for Deimos... and their distance from the planet were about 9 x 103km and 23.5 x 103km. Mars has a diameter of 6.794 x 103km so they are closer to 1.5 and 3 radii away it seems, but wow, for 100 years before the actual discovery??? Don't you wonder what made him use Mars instead of Venus or ??? Wait, maybe authors typing randomly can describe the true nature of the universe (with some limits of error)...

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If you have your copy, here is what I found in Chapter Five of Gulliver's Travels. The whole thing is available at the Guttenburg Project.

"The first professor I saw, was in a very large room, with forty pupils about him. After salutation, observing me to look earnestly upon a frame, which took up the greatest part of both the length and breadth of the room, he said, "Perhaps I might wonder to see him employed in a project for improving speculative knowledge, by practical and mechanical operations. But the world would soon be sensible of its usefulness; and he flattered himself, that a more noble, exalted thought never sprang in any other man's head. Every one knew how laborious the usual method is of attaining to arts and sciences; whereas, by his contrivance, the most ignorant person, at a reasonable charge, and with a little bodily labour, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study." He then led me to the frame, about the sides, whereof all his pupils stood in ranks. It was twenty feet square, placed in the middle of the room. The superfices was composed of several bits of wood, about the bigness of a die, but some larger than others. They were all linked together by slender wires. These bits of wood were covered, on every square, with paper pasted on them; and on these papers were written all the words of their language, in their several moods, tenses, and declensions; but without any order. The professor then desired me "to observe; for he was going to set his engine at work." The pupils, at his command, took each of them hold of an iron handle, whereof there were forty fixed round the edges of the frame; and giving them a sudden turn, the whole disposition of the words was entirely changed. He then commanded six-and-thirty of the lads, to read the several lines softly, as they appeared upon the frame; and where they found three or four words together that might make part of a sentence, they dictated to the four remaining boys, who were scribes. This work was repeated three or four times, and at every turn, the engine was so contrived, that the words shifted into new places, as the square bits of wood moved upside down.

Six hours a day the young students were employed in this labour; and the professor showed me several volumes in large folio, already collected, of broken sentences, which he intended to piece together, and out of those rich materials, to give the world a complete body of all arts and sciences; which, however, might be still improved, and much expedited, if the public would raise a fund for making and employing five hundred such frames in Lagado, and oblige the managers to contribute in common their several collections.

He assured me "that this invention had employed all his thoughts from his youth; that he had emptied the whole vocabulary into his frame, and made the strictest computation of the general proportion there is in books between the numbers of particles, nouns, and verbs, and other parts of speech."

Tuesday, 19 January 2010

Lewis Carroll, Happy Birthday, and RIP

January was the month in which Lewis Carroll was both Born (27th) and Died (the 14th)

It seems a fitting time to remember some of my favorite (not all true) stories about him.
Carroll was of course the pseudonym by which he wrote, but in his day to day life he was an instructor of mathematics at Oxford by the name of Charles Lutwidge Dodgeson (Lewis Carroll is an alteration of the Latinization of Charles into Carroll, and the replacement of Lutwidge with Lewis). He was a good (but not great) mathematician, but in the way of the world, he is remembered most for his children's stories... and had he not written them, he would probably be remembered for his photography... And if he had avoided that also... Maybe people would know he was a mathematician, but probably not; how many people on the street have heard of Euler?.

An interesting, but seemingly false, story circulated about a gift of a book on determinants to the Queen of England by Lewis Carroll. Here is the version as it is told on the Mathworld page.
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Several accounts state that Lewis Carroll (Charles Dodgson ) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants. Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland, expressed a desire to receive the author's next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author's other books, and was sent one of Dodgson's mathematical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worth while to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson).
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Ok, but still a neat story...

I love this letter, written by Dodgeson to a young man named Wilton Cox.
"Honoured Sir,
Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.

If x and y are each equal to 1, it is plain that

2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0.

Hence 2 × (x2 - y2) = 5 × (x - y).

Now divide each side of this equation by (x - y).

Then 2 × (x + y) = 5.

But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5.

Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.

I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll."

Another interest of Dodgson's was the analysis of tennis tournaments:

"At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself." Carroll set out the guidelines for a seeding system well before the good folk at Wimbledon ever thought of it. "Good on ya" as they say over here.

And from a letter he wrote in 1868 with suggestions for essentials of math instruction

Here are links to several other blogs I have written that involve Lewis Carroll in some way..
A Brief History of Logic Diagrams

Searching for Snarks

and Two times Two is Five Enjoy...

Monday, 18 January 2010

The Tale of the Creole Pig

This is not a math blog, except that it has to do with logic, or the lack thereof, but I teach kids, and this is a story my bright kids need to read.....and thanks to JD2718 in New York for passing this along.

Just before the recent storm hit Haiti, Kendra Pierre-Louis wrote this blog about the Creole Pigs that were once literally everywhere in Haiti, and how they came NOT to be there. It is a story of the worst indifference to sustainable development, and needs to be shared. ... This is me sharing. Since Blogs come and go, I am copying the whole thing below, but I do encourage you to read the original:
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Growing up in the United States, I grew up listening to my Haitian father speak longingly of two things that he said we couldn't get quite right in the US. The first were mangoes. Most of the mangoes that are found in the US are vaguely round like a Nerf football, and have a mostly deep reddish hue when ripe. They are beautiful, but to hear my father speak, are to the mango what the Red Delicious is to the apple: overproduced and vaguely generic.

The mango of his childhood, the Madame Francis mango, is flatter and green - like an overgrown lima bean. Even at its ripest it only hints at a dusky yellow color. It is also unique to Haiti. I've had it, and he's right it is delicious a queen among mangoes.

My father's other long lost food craving, pork from the Creole Pig, was also unique to Haiti. Unlike the pink pig encapsulated in the image of Wilbur, the pig from Charlotte's web, the Creole Pig was not pink. It, like the population of Haiti, was black and thus unlike American pigs did not sunburn. Raised by eighty to 85% of rural households, the relatively small but dense Creole pig subsisted not on grain, but on the detritus of the island's human population. It could thrive on the husk of rice, the cob of corn. In a nation without consolidated trash pickup the Creole pig acted as the nation's garbage men playing a key role in maintaining the fertility of the soil. And, because it was not dependent on feed for its survival, it functioned for the peasant population as a sort of mobile, literal piggy bank - the animals were sold or slaughtered to pay for school, for marriages, for unexpected medical expenses.

All of this is spoken in the past tense because between the 1970s and the 1980s the Creole pigs were systematically eradicated under pressure of the US government.

Like most of development history some of the facts are in contention, but this much is certain. In the 1970's the African Swine River Virus had spread from Spain to the Dominican Republic and then to Haiti by virtue of the Artibonite River which straddles the two countries.

Now comes the contentious part.

By 1982, says the United States government almost 1/3rd of Haiti's pig population was infected. A lot of Haitians (and many independent organizations) argue otherwise. What is not in contention is that the US in fear of the virus spreading to its own pig population pressured Haiti's government to seize all of the pigs and kill them.

Everyone who had pigs seized were supposed to be compensated in the form of replacement pigs - fat, pink pigs from the American Midwest, deemed 'better' by the USDA. These pigs needed clean drinking water (which 80% of Haitians did not have access to), $90 dollars a year in feed (in a nation where per capita income was$130 dollars a year), vaccination, and special roofed pens to serve as protection from the harsh Caribbean sun.

Does anyone see a problem with this?

Never mind the fact that many Haitians who had their pigs seized were never actually compensated (more on that in a second) - they couldn't have afforded the compensation anyway. In fact, many of those who received pigs found that their new pigs rapidly died.

So much for 'better'.

The eradication of the Creole pig only served to further impoverish Haitians. It forced many children to quit school, forced small farmers to mortgage and eventually lose their land, and forced many Haitians to cut down trees, rapidly increasing the Island's rate of deforestation, to create cash income from charcoal. All simply to save an already rich country from the small risk (and by most independent accounts the number of pigs infected in Haiti was much smaller than the 33% cited by the US) posed to it by a poor, tiny isla
nd nation.

It was, however, a boon to US pig farmers who generated millions in revenue according to grassroots international offloading these ill suited pigs on poor Haitian peasants. How?

Sunday, 3 January 2010

Weird Al, doing the Palindrome song..

Thanks to Ron Dirkse, A great Stats teacher who still resides in his adopted homeland of Japan..(huge envy here) This is great.. Thanks Ron...

Which leads me to my next question.... Who can come up with the best math palindrome... If you are in my math class, you can get extra credit for this one.... (don't be greedy, take the challenge, then we'll discuss what it is worth)..They don't have to be numbers, but they can... 12(63)= (36)21...

Saturday, 2 January 2010

A Palindromic Date, and A Problem

01/02/2010.... or 01022010 Ok, technically that is not a Palindrome, or else numbers like 110 would be palindromes. But still,close enough to be special as pointed out by my lovely sweetheart.

How unusual is that??? Well, there are only 19,998 numbers less than 100,000,000 that are palindromes...I just came across this at the Wolfram Mathworld page.. The sum of the reciprocals of the palindromic numbers ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212...) converges to a constant...... approx 3.37018..
Strange, but as numbers get bigger, prime palindromes get very rare it seems...2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, and it seems that after eleven, there are no palindromic primes that have an even number of digits (Every palindrome with an even number of digits is divisible by 11).

Next year on Nov 20, we get a real palindromic date, 11/02/2011.... Now that IS unusual.. Ok, and the next one after that is?????

Addendum, Professor Charles Wells of Case Western Reserve suggested a follow up, "Count the number of dates in the American system that are palindromes by something cleverer than brute force. " Well, I didn't actually do that, but I did take a moment to figure out that, if my math doodles are right, there was not such an event from 12/31/1321 (oops, think that should be 09/31/1390... "mia culpa") until 10/02/2001... then after the Jan 2nd this year, and Nov of next year... (and the one you have to figure out that comes after that)... well, there just are not very many of them...

also... I just read somewhere that we who are alive today (those of us over 20) are kind of blessed because we have lived through two palindromic years... 1991 and 2002, and that won't happen to anyone again for about a thousand years unless someone does discover a way not to die in anything like what we consider a normal lifetime now-a-days... lucky me.. lucky you???