Concurrencies and Coincidences Repost

  Steve Phelps, over at concurrencies, just wrote a "What can you do with three random points in the plane?" blog. Coincidentally, I had just finished an interesting old (1902) article about three random points on an equilateral hyperbola (such as y= 1/x for those not familiar with the term). And by another coincidence, the article happened to involve one of the common concurrent centers of a triangle, the orthocenter where the three altitudes from the vertices intersects. It turns out, that if you pick three random points on a equilateral hyperbola (they can be on either branch), then the orthocenter will also fall on the hyperbola. Stated another way, if you pick the three points all on one branch and make them all free to move, the locus of the orthocenter will be the other branch of the hyperbola.  If two points are on one branch, and one on the other is not, then the orthocenter falls on the branch with two points on it,  

Poncelet had actually written about this as far back as Jan of 1821 in Gergonne's Annales. Oh, by the way, a little "prove this factoid" for my calc kids... the y-intercept of the tangent line to any point on the rectangular hyperbola is always twice the y-coordinate, and the slope is always the square of the reciprocal of the y-coordinate. SWEET!  [Yikes, I've been busted... Keninwa noticed a mistake in the above (thanks guy) actually what I should have said (and this is only true for the basic y=1/x case), the slope is equal to the negative of the square of the y=value .. (and now, head hanging in shame, he wanders off into the sunset, muttering to himself about proofreading)..

Pick Two and Get One of Each?

  Another from my archives: Nice for Alg II level or so, I think.

A colleague from Colorado sent me an interesting probability problem the other day. I like it because it illustrates one of those serendipitous qualities of mathematics. Here is the problem. A Jar has a mixture of Red and White balls so that if you withdraw two, the probability of getting two alike, or two of different color are both equal to one-half. You may want to stop and try it before you read on.
Ok, so we let r = the number of red ones and w be the number of white ones, and the total is r+w. So how could we draw one of each color? Well, red first, and then white, or white first and then red. If we find the probability of each of these conditional events and add them up, that will have to equal 1/2. Ok, the probability of red on the first draw is r/(r+w), and on the second ball the probability is w/(r+w-1) since one of the red balls will be missing. The opposite order is exactly the same with the w and r reversed, so the probability of getting one of each color is 2rw/[(r+w)(r+w-1)]. Setting that equal to 1/2 we get 

If we expand (r+w)² and subtract the 4rw we get 0=r²-2rw +w²-r-w. NOTICE the symmetry, we could exchange r and w and get the same equation. We know right off that any solution (r,w) will have another solution (w,r).

One of the things that is often hard for students is to think of one variable as a constant and the other as a variable. I like to use the word "pronumeral", like a pronoun only instead of him or her we say "that number". It is like a variable that doesn't vary, we just don't know what it is in a particular case. So think of w as if it were fixed. We have that many white balls in the jar and we are wondering how many red can be put in to make the problem work... see it.. w is a "fixed" unknown, but r is going to "vary". That makes the equation a quadratic in r; Ar² +Br+c=0 where A=1, B= -2w-1, and C=w²-w.
We can solve this using the quadratic formula, but if this solution is going to be a rational number, and the number of balls in a jar must be rational, then the discriminant, the expression under the squre root radical in the quadratic formula, B²-4AC, must be a perfect square.
 B²= 4w²+4w+1 and 4AC= 4w²-4w; so B²-4AC= 8W+1. If there is a rational solution, it must be when 8W+1 is a perfect square.

 Wait, I know this one! That's a problem from number theory. The numbers that make 8W+1 a perfect square are called triangular numbers; 1, 3, 6, 10, 15. They are the sum of the first n counting numbers. But a neat thing happens if we plug 1 in for W, the solution for r is 3.... and if we use 3 for w, r=6. Each time we substitute one of the triangular numbers into the quadratic, the next comes out as a solution. So the probability of drawing two balls of the same color, (or of two that are not alike) will equal 1/2 whenever the number of balls of each color are consecutrive triangular numbers. A very geometric solution to a very algebraic question. 

A Parking Problem, and Another Interesting Problem

 

Here are two interesting problems, the second is a little more abstract, but bare with me... there is a purpose...

Suppose there are n parking spaces in a row along a one way street, and n people are driving into town, each with a most preferred spot in the row of spaces. Each of the n people adopts the plan that they will drive along the line until they come to their preferred space. If it is open, they take it, and if not, they continue driving and will take the next available spot. For instance if n=2 parking spaces, then the two drivers would find a parking space if both wanted the first spot, or they both wanted different spots; but one of them is out of luck if they both preferred the last spot, since the one who got there second is out of parking places. Thus for n=2, there are 3 possible solutions out of 2^2 = 4 possible situations. So here are the easy questions, find the number of possible solutions for n=3 and 4, and their associated probabilities of success.
And for the clever people, generalize for n.

If you found that either too hard, too easy, or just want to read on, here is a second problem. For those who have never studied graph theory, or don't remember the terms, a brief introduction. A complete graph with n nodes is called the k_n graph. For n = 4 and 5 the complete graphs look like the ones shown below.




Now a spanning Tree is a graph with N vertices but no complete circuits. You can get from any vertex to any other, but only by one path. Here is an example of a tree with six labeled vertices. Finally, we can give the question... For a complete graph of 3 vertices, 4 vertices, or n vertices, how many different spanning trees are possible?

The amazing result is that the two problems seem to be exactly the same, at least they produce the same sequence. The number of solutions for the parking problem is 1, 3, 16, 125, ... (N+1)^(n-1)... For the spanning trees, if we start with a graph of 2 points, there is only one tree. For three points there can be three different trees, and for four... yeah..16 . The image below shows all 16 of them as depicted by Wolfram Mathworld.. the sequence is the same but the index is off by one, so we get 1, 3, 16, 125, ... (n)^(n-2)... Ok, here is the real question for this blog... why are they the same...

Ok, one more curiously beautiful detail about this problem.... the probability of success for the n parking cars at the top has a limit of e/(n+1) as n approaches infinity. I love how often e pops up in these "sorting" type problems. 

Imaginary Numbers and the Imaginary constant

 Imaginary Numbers The word imaginary was first applied to the square root of a negative number by Rene Descartes around 1635.  He wrote that although one can imagine every Nth degree equation had N roots, there were no real numbers for some of those imagined roots. 

Leibniz wrote, "[...neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^3−6x^2+13x−10=0 as having three roots, yet there is just one real root, which is 2, and the other two, however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.] (page 380)

 Around 1685 the English mathematician John Wallis wrote, "We have had occasion to make mention of Negative squares and Imaginary roots".  

Some mathematicians have suggested the name be changed to avoid the stigma that it seems to create in young students, "Why do we have to learn them if they aren't even real."  (To be honest, in thirty years as an educator, I never heard the question.)  Perhaps the weight of history is too much to support the change.  

The first person ever to write about employing the square roots of a negative number was Jerome Cardin (1501-1576).  In his Ars Magna (great arts) he posed the problem of dividing ten into two parts whose product if forty.  After pointing out that there could be no solution, he proceeded to solve the two equations, x+y = 10, and xy=40 to get the two solutions, \$ 5 \pm \sqrt{15} \$  .  He then points out that if you add the two solutions, you get ten, and if you multiply them then the product is indeed forty, and concluded by saying that the process was "as subtle as it was useless."
Cardin also left a seed to inspire future work int he mystery of roots of negative numbers. Cardin had published a method of finding soltuons to certain types of cubic functions of the form \$ x^3 + ax + b \$ .  His solution required finding the roots of a derived equation.  For functions in which the value was negative, his method would not work, even if one of the three roots was a known real solution.  
About thirty years later Rafael Bombelli found a way to use the approach to find a root to \$ x^3 - 15x -4 \$ with the known solution of four.  He went on to develop a set of operations for these roots of negative numbers.  By the eighteenth century the imaginary numbers were being used widely in applied mathematics and then in the nineteenth century mathematicians set about formalizing the imaginary number so that they were mathematically "real."

Some other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors). *MacTutor

Imaginary Unit The imaginary number with a magnitude of one used to represent \$ \sqrt{-1} \$, has been the letter i since it was adopted by Euler in 1777 in a memoir to the St Petersburg Academy, but it was not published until 1794 after his death.  It seemed not to have gained much use until Gauss adopted it in 1801, and began to use it regularly.  The term, imaginary unit, was first created, it seems, by William Rowan Hamilton in writing about quaternions in 1843 to the Royal Irish Academy.  For his three-dimensional algebra of quaternions, Hamilton added two more imaginary constants, j, and k, which were both considered perpendicular to the i, and to each other.  

In most fields of electronics the imaginary constant i is replaced by a j, perhaps to avoid confusion with the use of i for current in Ohm's law.

We Just Don't Talk That Way Anymore

 I love reading old journals and am often struck by the precision and beauty of language in old math and science journals. 

Many brighter than I have commented on the seemingly inevitable reduction in rigor as schools require all students to take more advanced math. I think the same thing has happened across the board to language as everyone is expected to complete a high school education. 

I recently read a note that had two quotes, both saying essentially the same thing. The first is from Leonhard Euler and dates around the American revolution; the second is from George Box and dated around 1987,Empirical Model-Building and Response Surfaces. Both are incredibly literate and knowledgeable people, and yet..

 "Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena" 

"All models are wrong... some models are useful. "

A Brief History of Blackboards and Slates

 

My career in education began with the use of chalk and a blackboard, transitioned into a room with only dry erase marker boards, and finished in a class with only an electronic smart board. Except for the few times I was inconvenience by a projector bulb picking a bad moment to go bad (and if a chalk board had been available, I would have been grateful) I loved the improvements that each transition brought to my ability to more effectively convey my ideas.
But for the period from 1800 to 2000 few things were as ubiquitous in a mathematics classroom as the blackboard. Today modern "white boards" may have taken their place in many institutions, or even an electronic version called a smart board; but board work still seems to be a part of the current classroom procedure. In a recent talk, Keith Devlin began by saying, "I step back from the (now largely metaphorical) blackboard and .. "
Whatever the present state of its demise, the classic chalkboard was so common a classroom presence that it was part of a frequently repeated gag sequence on the popular Simpsons cartoon series.(Not sure if the use shown above could be termed "educational")

It appears that the blackboard first came into American education around 1800. The National Museum of American History website on colonial education says,:

"Mathematics teachers with ties to England and France introduced blackboards into the United States around 1800. By the 1840s, these erasable surfaces were used for teaching a wide range of subjects in elementary schools, colleges, and academies. The Massachusetts educator William A. Alcott visited over 20,000 schoolhouses. “A blackboard, in every school house," he wrote, "is as indispensably necessary as a stove or fireplace."

James Pillan, a Scottish teacher and education reformer is often cited as the "inventor" of the blackboard, but this seems to be a misunderstanding based on a letter from Pillan which appeared in Jeremy Bentham's Chrestomathia (1815). It was entitled Successful application of the new system to language-learning, and dated 1814; it mentions the use of chalk and blackboard in teaching geography. But Pillan only began teaching in 1810, over a decade after the board made its way to America, and as we shall see, literally hundreds of years too late to "invent" the chalkboard.  He may, however, be the inventor of colored chalk. He is reported to have had a recipe with ground chalk, dyes and porridge.

One of the earliest mentions of blackboards in Colonial America was in 1779 in a letter from John Taylor, a tutor at Queens College (to become Rutgers University) to a graduate student, John Bogart, that he was asking to take over his classes while Taylor was away on military duties. "I have spoken to Mr. Briton to make a blackboard.." *Kidwell, Hastings, & Roberts; Tools of American Mathematics Teaching. (One wonders why, if Taylor thought a blackboard was such an asset in math instruction, he had never had one made for himself.)

Blackboards and slates were seemingly used well before any of the previous examples in musical study. In Composers at Work, author Jessie Ann Owens devotes several pages to the existence of several types of slate and wood "cartella" which were used to write out musical ideas. She describes the discoveries of these with five or ten line staves dating to the 16th century. Much larger wall size examples seem to have been used but have only been confirmed by iconography. The book includes an image from a woodcut by Hieronymus Holtzel of Nuremberg in 1501.

In America they seem to have very quickly become and essential part of daily school life. [From a web page of Prof. Rickey]

Perhaps no one method has so influenced the quality of the instruction of the cadets as the blackboard recitations. Major Thayer (Superintendent from 1817) insisted on this form, although old records show that it was introduced at West Point by Mr. George Baron, a civilian teacher, who in the autumn of 1801 gave to Cadet Swift "a specimen of his mode of teaching at the blackboard."  ( More than a civilian teacher, he was the first superintendent and mathematics professor at what would become the United States Military Academy in 1802, and the founder and editor-in-chief of the Mathematical Correspondent, which was the first American "specialized scientific journal" and the first American mathematics journal, first published May 1, 1804 )

Today it is the prominent feature in Academic instruction. [Quoted from Richardson 1917, p. 25] There is indication that the blackboard was used in a few schools in the US before it was used at USMA. See Charnel Anderson, Technology in American Education, 1650-1900, published by the
US Dept of Health, Education, and Welfare 1961(I have read this document and he credits Frenchman Claude Crozet with introducing the blackboard to the USMA and that he built and painted one to teach his classes.  It may well be that after Baron left under a cloud in 1802, the method was not used by other teachers there until Crozet arrived in 1817. 

Thayer had visited the Ecole Polytechnique in France to study their methods and was heavily influence by the "French" method when he became superintendent, even to the point of extending instruction in French so that the students could better master the French texts in advanced math.. I can't find an early example of the use of chalk or slate in France, but they seem to have been very much a part of the educational process by the time Galois threw an eraser at his examiner in July of 1829.

Galois was not the only one who reacted negatively to some of the innovations in education connected to the blackboard. At Yale, there were two "rebellions" in which students refused to accept some changes in testing practice. Here is a paragraph from Stories in Stone: Travels Through Urban Geology By David B. Williams.

This "rebellion" occurred in 1830, with 43 rebels expelled, including Andrew Calhoun, the son of John C Calhoun, and Alfred Stillé, who eventually did get a degree from Yale and another from  U of Pennsylvania before he became a somewhat famous doctor, and one of the first to distinguish Typhus from Typhoid fever. His rebellious side wasn't limited to college however, as he refused to accept germ theory and laboratory medicine. There had been a similar event in 1825 at Yale, but those students recanted and were readmitted.

One of the earliest mentions of blackboards I have found has nothing to do with education, however. It seems that a custom developed in London's financial district in the later part of the 19th century to list the names of debtors on a blackboard to shame them into paying, and it seems to have persisted for a long time. Here is a description of the practice from Chronicles and Characters of the Stock Exchange
By John Francis, Daniel Defoe; printed in 1850.

From Wikipedia I learned that the Oxford English Dictionary provides a citation from 1739, to write "with Chalk on a black-Board". I know it is common in England for Pubs to advertise with a blackboard outside their doors on the sidewalk, but have no idea how far back this idea originated.
 
Prior to the use of blackboards students learned their early lessons from an object called a hornbook. Here is a description of one from the Blackwell Museum webpage at Northern Illinois University

Paper was pretty expensive once and hornbooks were made so children could learn to read without using a lot of paper. A hornbook was usually a small, wooden paddle with just one sheet of paper glued to it. But because that paper was so expensive, parents and teachers wanted to protect it. So they covered the paper with a very thin piece of cow's horn. The piece of cow's horn was so thin, you could see right through it. That's why these odd books were called "hornbooks."

Hornbooks seem to have been totally imported from England into the American Colonies, and almost all had a cross on the upper left, with the Lord's Prayer at bottom.  The American Revolution seemed to have almost completely eliminated the import of Hornbooks in rejection of all things English at the time.  The education conversion to the blackboard seems to have finished the hornbooks very quickly afterward judging from this quote from the OED about Hornbooks, (a1842 HONE in A. W. Tuer Hist. Horn-Bk. I. i. 7) " A large wholesale dealer in..school requisites recollects that the last order he received for Horn-books came from the country, about the year 1799. From that time the demand wholly ceased..In the course of sixty years, he and his predecessors in business had executed orders for several millions of Horn-books".
.
Early blackboards were usually made of wood, (but some may have been made of paper mache') and painted with many coats as true slate boards were very expensive. Schools purchased large pots of "slate paint" for regular repainting of the boards. The Earliest quotes from the OED date to 1823.

1823 PILLANS Contrib. Cause Educ.    A large black board served my purpose. On it I wrote in chalk. 1835 Musical Libr. Supp., Aug. 77 The assistant wrote down the words..on a blackboard. 1846 Rep. Inspect. Schools I. 147 The uses of the black board are not yet fully developed.

However under "slates" I found other  earlier uses. In "1698 FRYER Acc. E. India & P. 112 A Board plastered over, which with Cotton they wipe out, when full, as we do from Slates or Table-Books" which indicates that boards covered with Plaster or other materials were used to write upon much earlier than the earliest use of "blackboards" in classrooms.

Another early use of slates is given in David E. Smith's Rara arithmetica of a book printed in 1483 in Padua of the arithmetic of Prosdocimo containing a mention of the use of a slate. This led Smith to conclude that at this time the merchants would actually erase and replace numbers (as was originally done by the Hindu mathematicians working in their sand trays) in division rather than showing the cross-outs that distinguish the galley method of division after it was adopted to use on paper.

The very earliest claim for slates I have found is of use in the 11th century. A work called Alberuni's Indica (Tarikh Al-Hind), "They use black tablets for the children in the schools, and write upon them along the long side, not the broadside, writing with a white material from the left to the right."

Chalkboards became so important for teaching that teachers in the 19th century sometimes went to extremes to create one. In Glen Allen, Virginia; a school is named for Elizabeth Holladay, a pioneer teacher who started the first pAublic school in the Glen Allen area of Henrico County at her home in 1886. On a note about the history of the school it says she had, "Black oilcloth tacked to another part of the shipping crate served as a blackboard." 

The slate was used even after paper became a relatively commonplace item. Many school histories report the use of slates into the 20th Century. This use may have been significant. The Binney & Smith company, better known to many for their creation of the Crayola Crayon, began the production of slate pencils, for writing on slate, in the year 1900. As an aside, they also won a Gold Medal at the St. Louis Fair.

Slate pencils prior to 1800 were known as Dutch Pencils in England, but increased slate mining in Wales around 1800 led to more domestic production, and use of slates, and slate pencils in England.   In the journal Australian Historical Archaeology, (2005) Peter Davies reports that in the excavation of a site called Henry Mill that was only operational from 1904 until around 1930 they found 30 slate pencils, remnants of four slates, and a single graphite pencil core. 

In "Slates Away!": Penmanship in Queensland, Australia, John Elkins, who started primary school in 1945, writes that he used slates commonly until around the third year of school.

I think in Prep 1 that we had some paper to write on with pencils, but my memory of the routine use of slates is much more vivid. Each slate was framed in wood and one side was inscribed with lines to guide the limits for the upper and lower extremities of letters. The slate "pencils" were made of some pale gray mineral softer than slate which had been milled into cylinders some one-eighth of an inch in diameter and inserted into metal holders so that about an inch protruded.
Each student was equipped with a small tobacco tin in which was kept a damp sponge or cloth to erase the marks. Sharpening slate pencils was a regular task. We rubbed them on any suitable brick or concrete surface in the school yard. Teachers also kept a good supply of spares, all writing materials and books being provided by the school. It is possible that the retention of slates stemmed from the political imperative that public education should be free.

Slates were advertised in newspapers in the US as early as 1737. Slates, as indicated above, show up as commonplace in quotes from the OED as early as 1698. It seems they may have been used for some artistic or educational purposes as early as the end of the 15th Century. In the famous painting of Luca Pacioli,
Ritratto di Frà Luca Pacioli, Pacioli is shown drawing on a slate to copy an example from Euclid in the open book before him. The closed book, which has the dodecahedron upon it, is supposedly Pacioli's Somma di aritmetica which was written in 1494.

In the Dec 2003 issue of Paradigm, the Journal of the Textbook Colloquium, is an article by Nigel Hall titled, "The role of the slate in Lancasterian schools as evidenced by their manuals and handbooks". A couple of snips from the article appear below:

The Oxford English Dictionary gives as its first citation for slate being used as a writing tool a quotation from Chaucer’s Treatise on the Astrolabe written about 1391. Whether usage began around this time or had begun much earlier is unknown, although as a technology it shared many characteristics with the wax tablet, used extensively from before the time of the Greeks until the 1600s in Europe, and even surviving in some usages until the early twentieth century (Lalou, 1989). Knowledge of the use of slate for writing after Chaucer is limited until one reaches the second half of the eighteenth century. The mathematician Digges (1591) refers to writing on slates and in the new colony of America an inventory (Plymouth Colony Archive, n.d.) made on 24 October 1633 of the possessions of the recently deceased Godbert and Zarah, noted among many items, ‘A writing table of slate’ (table here being a tablet of slate).

Hall goes on to suggest that, in fact, the use of slates may not have been very common in England until the end of the 18th Century because reading (beginning with hornbooks) was much more commonly taught than writing. He credits Lancaster for the promotion of slates for writing and math, but suggests that the slate was a principal element in the "monotorial system" in which more advanced students taught the lower group. 

The blackboard was extended to some specialty uses as well. A "Slated Globe" was advertised in The New York Teacher, and the American Educational Monthly, Volume 6 in 1869 for use in spherical geometry and geography classes. A four inch diameter globe sold for $1.50

Slated Table Stand Globe, manufactured by Rand McNally and Co., ca. 1910s. Courtesy The Newberry Library, Chicago.

I also recently found this image on a Wikipedia article about Benjamin Pierce. He seems to be standing beside a stand with a spherical blackboard resting on it, but can not be sure that is what it was.

In an 1899 article for the proceedings of the Society for the Promotion of Engineering Education, Professor Arthur E Haynes of the University of Minnesota had an article for, "The Mounting and Use of a Spherical Blackboard, which included this image.

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Recently, J F Ptak posted an article on his Science Books blog from Scientific American, (Sep 13, 1890) about a pen-tip eraser for slate pens meant to be wetted to erase the marks on a slate by the pen.  The article described the invention with credit to the inventor, Mrs Emma C. Hudson

Chain, Chain, Chain, ...Chain of squares..

 

Or "Where Does This Road End?"... and I think the actual next line in that song was "chain of fools", (rock me, Aretha) and yet I continue....

 If you take a number, square each of its digits and find the sum you will get a new number(usually, is there more than one number that produces itself as its own iterate under this process). Do the same to that number, and the sequence continues. But eventually you have to come back where you started. Numbers with more than three digits will always produce a smaller number and three digit numbers will always be less than 9²+9²+9²= 243, so eventually, wherever you start, you end up with a number less than 243 (and 243 --> 4+16+9 = 29 so it gets even smaller).. what would be the last number that produces a number larger than itself? 

Ok, simple stuff, if you start with one, you get one and that's dull so let's go on. If you begin with two you produce the sequence 2--> 4--> 16--> 37--> 58--> 89--> 145-->42-->20-->4 and then repeats the cycle of eight numbers forever. Now the big conjecture... start with ANY integer (oooohhhh, he said you could pick ANY integer, how bold) and eventually, either it gets to one, or it jumps into this chain. Now I wonder; is there a way to prove (short of hacking them all out, which I have done) that there is no "other" chain that some numbers might drop into? I also have a feeling that as the numbers go off to infinity, the proportion of numbers that go off to one has some non-zero limit; in fact, I suspect it might be around 1/7 or just a tiny bit more, but don't have a clue how to prove that (Joshua Zucker said..."I have no rationale as yet, but my computer search seems to tell me that the proportion is more like 21% than 1/7."... Well Josh, I did say "or just a tiny bit more"...).

In 2015 a revision of a paper by Justin Gilmer arrived at a value between d <.18577 and d>.1138.   (arXiv:1110.3836)

Numbers which eventually end on one are called Happy Numbers, and their origin is unclear.  Wikipedia has, " are Happy numbers were brought to the attention of Reg Allenby (a British author and senior lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" .  Both these are cited from Richard Guy in 2004.  

 "Guy asks several questions about happy numbers which can be paraphrased as follows: 

(a) It seems that about 1/7 of all numbers are happy, but what bounds on the density can be proved? 

(b) How many consecutive happy numbers can you have? Can there be arbitrarily many?   (The first consecutive pair of Happy numbers is 31,32.The first triplet  starts at 1880.)

(c) We define the height of a happy number to be the least number of iterations needed to reach 1. How big is the least happy number of height h?   (Happy numbers seem to get there pretty quickly.  the first 100 happy numbers, (up to 694, I think) never takes more than seven iterations)

(d) What if we replace squares by cubes, fourth powers, fifth powers etc., or we work in different bases? 

Some of these questions have been answered at the Online Encyclopedia of Integer Sequences 

I have found the term "happy numbers"used in the same manner back to 1970, in Bulletin (28 - 30),pg 2, of the California Mathematics Council.  (Would love to receive digital copy if one can be had.  Seems to be early plug for computer mathematics in high school.)

 I looked into forming the sum of the cube of the digits... would all roads lead back to one or two then? A quick answer trying only a few numbers... NO, Follow the orbits of 2 or 3 or 7 and they all go to separate self-replicating numbers or one-cycles.. 4 goes into a triplet of 55 250,133,55; so I guess my new question about cubics is...

It seems that any (many?) number 2+3n goes to the same absorbing point as 2 (371).  

All multiples of 3 go to a fixed point of 153, 9, 12, 18, 21, 24, 27, 30, 33.  The others seem to go to the cycle from 4, (133,55,250,133) (not sure if there are more of these). 

19, 34, 37,    (of course 43, 70, 73, 91, 109 .. would go there as well.)  go to the fixed point of 7 (370).

 This seems to explain everything, then you get to 47, and it has a fixed point of 470. And then 49 goes to 1459 and oscillates between 1459 and 919.  

That 370, 371 and 470 all are fixed points makes them Armstrong numbers, numbers that produce themselves when each digit is raised to the number of digits it has (in this case 3).

There is a little more, including some bits abou fourth power sums...enjoy, 

The Cubic Attractiveness of 153

Given Two Points????

 Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can't do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?). 

 They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time." Ok, We'll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3). They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question.... Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not.. I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.

 I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form). They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space. Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension. But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea. (Actually in the last few years I've been working a unit in parametric equations and vectors into my Alg II classes with lots of Bowditch curves (Lissajous?) and three D lines and planes and three D geometry ideas.)

I got this comment from one friend... 

Dave L. Renfro said...

Here's something marginally related to your topic. In 4 dimensional space it is possible for two planes to have exactly one point in common. A very simple example is the xy-plane, namely all points of the form (x,y,0,0) where x and y vary over the real numbers, and the zw-plane, namely all points of the form (0,0,z,w) where z and w vary over the real numbers. It is easy to see that these two planes have only the origin (0,0,0,0) in common, since x = y = z = w = 0 is the only solution to (x,y,0,0) = (0,0,z,w).

For what it's worth, I first learned about this possibility in a "side-bar diagram" in the following calculus book, which I was trying to work through during the 1974-75 school year (I managed to cover about the first 1/4 of the book), when I was in the 10th grade. Incidentally, the first college math class I took (an ODE course, about a year later) was from one of the authors (Embry) of this text.

Embry/Schell/Thomas, "Calculus and Linear Algebra. An Integrated Approach" (1972)

At this point you might try exploring other possible intersections between two four spaces, intersecting at a single point other than (0,0,0,0) , or a in a line, or a 3-plane.  Send me your best ideas, especially for sharing this with students.

This was all back in about 2009 and I retired two tears later and never followed up on this idea with a class, but I have some thoughts and will share them by adding to this post as time goes on, so check back...and share your ideas> I will pass them along as I go.  

I did come up with one more idea I wish I had thought of while I was teaching.  Somewhere in their HS math education most students who stay at it will learn to solve systems of equations and understand how to use matrices to find the point where three planes intersect IF they intersect in a single point. 

Planes in Space- part One

Take your basic bright kid in alg II or pre-calc, or often in calculus, and ask, "What is the intersection of two lines?"... They say, "A point."... good answer.

"Can you write the equation of a line?" Again they are on target.
"If I give you the equation of two lines on the plane; can you find their point of intersection?" The good ones can, and know they can.

Now ask the same bright kids, "What is the intersection of two planes in space?". They answer correctly again, "A line."

So far everything is great, but now we ask them to write the equation of a plane.... uhhh... gee..... and at this point, when asked about one of the fundamental structures of plane geometry, their analytic geometry skills are exhausted.

My experience was that after solving systems of three equations in three unknowns, they remembered how to do it (mostly) but didn't know what it was they had done???? 

but thankfully after va brief review a light goes on, and they understand.  

 Still, a very few may actually be able to produce x+y+z=1 or some other for the equation of a plane. Now we ask about the intersection of two planes, and almost none of them can do it. The scary part, is that very (very) few of the teachers of alg II and above that I have questioned about this could provide an answer either .

I would begin by recalling that an equation in three variables, such as 2x+3y+z=6 can represent a plane in space. When students had three such equations that intersected in a unique point, they found the solution by one of several methods. Most students learn to solve such equations by the methods called elimination and substitution at the very least. Others may have also been introduced to Cramer’s rule for solving systems with determinants and perhaps two methods using matrices.

The most commonly taught matrix method is to write a matrix equation and then solve it using the inverse matrix method. A second, and as I would point out, more efficient and general method is the Gauss-Jordan reduced row-echelon form (RREF) of an augmented matrix. 

We begin with three planes determined by the equations {x + y – 2z = 9; 2x – 3y + z = -2; and x + 3y + z = 2} This same system of equations can be expressed as the matrix equation.

Notice that the left matrix is made up of the coefficients of the three variable terms in each equation, and the right matrix contains the constant terms. We can find the intersection by taking the inverse of the left matrix and multiplying on the left of both sides of the equation. The simplified result gives

This seems to be the most commonly taught method, and the one that students and teachers seem to prefer, and yet it has two major disadvantages. The first disadvantage is that it tells you little or nothing about systems which have a solution, but not a single unique solution. In fact, it seems most students (or teachers?) can not distinguish between the cases (and there are several different ones) with no solutions from the ones with an infinite number of solutons. This same defect applies to attempts to use Cramers Rule. The second problem is that the inverse method is more computationally complex, that is, it takes more operations for the solution than the alternative RREF method, and the difference grows as problems reach higher orders of magnitude. For the problems that are generally assigned at the high school level, the difference in computability presents no real problem, but the difference in the range of applicable questions can be very significant in a students understanding of general systems of three equations.

In contrast with the Inverse method that will only work if the three planes intersect in a single point, the RREF form will allow us to work with systems which do not even have the same number of equations as unknowns. This is the type of situation created when we try to find the line of intersection of two planes.

RREF for two planes

So let's talk about the situation where two plains intersect in a point.  

We will use the equations 2x + 3y – 3z = 14 and –3x + y + 10z = -32. When we write an augmented matrix for the system of only two equations we get a 2x4 matrix, shown here:


When we reduce this system, by matrices or otherwise, we get

which is a matrix expression of x - 3z = 10 and y + z = -2.

Well, what can we do with that to help us find the line of intersection..... baby steps.... let's find one point on the intersection. 

 

We notice that both equations contain a z variable, it might occur to us to ask, “What happens if we substitute different values in for z?”. For example, if we try z=0 we note that from the first equation we get x=10 and from the second we get y=-2. What does this tell us about the point (10, -2, 0). If we check it against the two original equations we notice that the point makes both equations true, 2(10) + 3(-2) – 3(0) = 14 and –3(10) + (-2) + 10(0) = -32. So the point (10, -2, 0) is on both planes and therefore must lie on the line that is their intersection.

Can we find more points? What happens if we try z=1 or z=2 or other values. Using z=1 we get x – 3(1) = 10 which simplifies to x=13 ; and y +(1) = -2 which simplifies to y=-3. Checking the point (13, -3, 1) we see that it also makes both equations true, and so it must also be on the line of intersection.

And now we have come full circle, and we are back to the starting point, and now they really write the equation of a line through two points, in pretty much and dimension.  but what surprises may jump out in 4-D????? (cue dramatic music)

----(More to come)

BEWARE

Four Space Ahead

Not Quite Equal

 ***** WARNING!!!!!********** Repeat of some very old jokes

Darryl Brock, A fellow teacher at school sent out some puns today... and I rewrote some of them as .... dare I call them... equations... 

 The roundest knight at King Arthur's round table = Sir Cumference.

 

 eye doctor on an Alaskan island =an optical Aleutian 

 A grenade thrown into a kitchen in France = Linoleum Blownapart 

 Atheism = a non-prophet organization

 A chicken crossing the road = poultry in motion 

 short fortune-teller who escaped from prison = a small medium at large

 WWI soldier who survived mustard gas and pepper spray = a seasoned veteran

cannibals eating a missionary = a taste of religion

 joining dangerous cults = Practicing un-safe sects!

*** Please do NOT throw things...  

But do send your examples of more of these...

An 18th Century quick Approximation for Angles in Right Triangle (Repost)

 

The triangle above is a right triangle, and almost no one who reads this blog didn't know that. In fact for most people who have had a pretty good math background, you would be surprised to read a property or theorem about right triangles that you didn't know, or at least hadn't heard, especially if it dates back to the 18th Century.

Ok, so if I give you the three side lengths of a right triangle, how would you find the smallest angle..... Hold on, No calculator, and No tables...
I had no idea how to achieve such a result when I came across such a method by a Hugh Worthington in "An essay on the Resolution of Plain Triangles, by Common Arithmetic." It was in an anthology of math writing from the 1500's to the present in "A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing" by Benjamin Wardhaugh (pg 97), credited to Hugh Worthington in 1780.


So How is it done? Well, in the words of the author, "half the longer of the two legs added to the hypotenuse, is always in proportion to 86 as the shorter leg is to its opposite angle. "

As a math equation we with c as the hypotenuse and a as the smallest leg, he states that \(\frac{b/2+c}{86}=\frac{a}{A}\)

In the common 3,4,5 triangle that produces \(\frac{7}{86}=\frac{3}{A}\) and A would be equal to 36.85714286 (Ok, I divided it out on my calculator). Then I checked using Arcsin(3/5) and got 36.86989765...... OK, That seems to be a really good approximation, and I checked with a much smaller angle (the 7, 24, 25 triangle) and it was also very close.

I've been working on this for a while and can't come up with how he might have arrived at this approximation, nor can I find any other use of this. Would love to know if it was explained somewhere, or other works using it.

Addendum: Some really good math in the comments leading to the idea from Matt McIrvin and Paul Hertzer that "I think the leading error in this rule actually comes from the use of 86 degrees as an approximation to 1.5 radians." So using 85.94 or so as the constant improves the already awesome (my view) approximation accuracy. Thanks guys.. now can anybody find another publication using this idea?

Shortly after I wrote this, John Golden, an amazing teacher of math educators created a geogebra demo to greatly improve this, so I immediately stole it and added it here. Thanks John.

All That Glitters is not Golden

  

Over many years of teaching, I realized that most students, and many teachers had extensive misunderstandings about the "Golden Mean" and it's history.

I want to try to dispel, and expand, on some of these common misunderstandings.  For example, many think that the "Golden Mean" was known to the early Greeks (it was) by that name (it wasn't).   The idea that Euclid labeled the idea, which was found in geometric constructions such as the pentagon (and pentagram), as "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The "extreme and mean ratio" is still frequently used to describe the idea.


Others believe it did not exist until Fibonacci created the Fibonacci numbers from which it was derived as a limit of the ratio of consecutive terms. First, Fibonacci did not create the, so called, Fibonacci sequence. It was known to the Indian Mathematicians as early as Pingalia before 200 BC. Fibonacci's Liber Abaci (1202) included both the means and extreme property, and the famous sequence, but it seems he never realized that the ratio of consecutive terms of the sequence would approach the well known ratio. Luca Pacioli gave the name "Divine Proportion" to his 1509 book about the ratio, illustrated by Leonardo da Vinci. Leonardo first used golden for the ratio by using the latin "secto aurea" (golden section)  The first use in English did not occur until mathematician James Sulley used it in 1875, according to Alfred Posamentior.  And it was "mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;this was rediscovered by Johannes Kepler in 1608." *Wik





Perhaps the most common misconception of students is that the Golden ratio is some how a one-off.  A very special number with nothing else like it.  In truth, it is part of a class of numbers known as Pisot Numbers.  In fact, it is one of an infinite set of  numbers that are solutions to a quadratic equation, sharing many of the "special" qualities of the golden ratio.

The golden ratio is the smallest of these, but others include the "silver ratio" 1+2–√$1+\sqrt{2}$ , and the "bronze" ratio, 3+(√5)2$\frac{3+\sqrt{(}5)}{2}$.  The three numbers are roots  of the quadratic terms x2−x−1$x^2-x-1$, x2−2x−1$x^2-2x-1$ and x2−3x−1$x^2-3x-1$. (There is a pattern here, it WILL be back)


I will try to point out how some of those "special" qualities of the golden mean are shared by these other metalic means.

One of the things that impress students is the continued fraction for the golden mean repeats the same number over and over:
ϕ=1+11+11+11+…$\varphi =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\dots }}}$



but very few realize that there is a very similar expansion for the "silver mean",

ϕ2=2+12+12+12+…${\varphi }_2=2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\dots }}}$

and I bet most of them can figure out how to write the bronze (or third metallic) mean, and all the ones that come after it.


How about looking at a different way to write the positive roots of each of the metallic means I gave above.

The Goilden mean is 1+4+(12)√2$\frac{1+\sqrt{4+(1^2)}}{2}$ ; The Silver mean is 2+4+(22)√2$\frac{2+\sqrt{4+(2^2)}}{2}$; and the third metallic mean is 3+4+(32)√2$\frac{3+\sqrt{4+(3^2)}}{2}$... the ones following are equally evident.

Students confusion with history is somewhat confused by the fact that they are often introduced to the Golden mean in association with it's relationship with the Fibonacci sequence; 1, 1, 2, 3, 5, 8, ... and the ratio of consecutive terms approaches the Golden mean.
For the Silver mean, there is another well known(but not often to high school students) sequence that is mistakenly attributed to English mathematician John Pell. The sequences is 0, 1, 2, 5, 12, 29,... and bright students can quickly guess the Fibonacci-type recursive formula for this, and will probably anticipate that the sequence for the third metallic ratio would be 0, 1, 3, 10, 33,... and probably all the metallic ratios after that.

In Geometry students are familiar with the fact that the Golden mean can be found in the pentagon, between a diagonal and a side, or between the two sections of the intersection of diagonals.

The Silver mean is found in the ratio between a side and the second shortest diagonal

Unfortunately, that's where the sequence ends.  There are no regular polygons with ratios of sides and diagonals that are in the ratio of any other metallic mean.  As I will point out later, there are non-quadratic numbers that are Pisot numbers (or cubes and higher order) that I have not checked.

Some lesser known facts about the Metallic Means is that there powers approach "almost-integers" as higher (and not so much higher for many) powers.  For example ϕ7≈29.03444${\varphi }^7\approx 29.03444$ and ϕ13≈529.0019${\varphi }^{13}\approx 529.0019$  as you might expect from experience, odd powers overshoot the mark a smidge, evens undershoot.  The error diminishes logarithmetically.

IF we go to the other metallic ratios, they demonstrate the same behavior more quickly.  For example the silver mean  ϕ72≈478.00209${\varphi }_2^7\approx 478.00209$ and ϕ132≈94642.000010${\varphi }_2^{13}\approx 94642.000010$  .

And the third metallic mean gives ( ϕ73≈4287.00023)${\varphi }_3^7\approx 4287.00023)$


Another interesting, and not well known fact about the Fibonacci sequence is that the digits Mod (n) have a repeat period.  For the Fibonacci period, they repeat their last digit, (mod (10) ) in a 60 digit cycle.

011235831459437 077415617853819 099875279651673 033695493257291

It turns out that this is true of all the metallic sequences, but it may be easier to spot in the shorter binary cycles.  The Fibonacci digits mod(3) cycle 0,1,1 repeatedly, (Even, Odd, Odd).   For the Pell sequence, the cycle is 0,1; and these two sequences alternate between the odd and even metal ratios.

 But base three is not too hard, so let's look at that cycle of remainders on division by three:  

For the Fibonacci sequence the cycle is 0, 1, 1, 2, 0, 2, 2, 1.

For the Pell Sequene the cycle is this cycle sort of the reverse of this, 0,1,2,2,0,2,1,1.  

And the third metallic sequence, cycles 0,1, similar to the binomial cycle.... (can you figure out why there is never a remainder of 2 when a bronze sequence is divided by 3?)  

After I first wrote this post, I came across a page called Goldennumber.net which had a nice compass rose illustration of the 60 cycle or the Fibonacci sequence in base ten. As noted, they credit a copyright to Lucian Khan.

Of interest is that the zeros occur equally spaced at the NESW compass points.  This is a pattern of many repeat cycles with metallic ratios, the zeros are equally spaced.  It is also easy to pick up from this that all the numbers at 30 degree multiples are fives, showing that F(5n) is divisible by five.  The page also pointed out that any two non -zero remainders that are 180 degrees apart on the wheel sum to ten. Students might want to explore similar patterns in wheels of  Fibonacci or other metallic sequences for remainders by other divisors. 

Other Pisot Numbers, including a Super-Golden Number, 

Wikopedia gives this description of the Pisot Numbers: 

In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938.

There are other sets that are roots of cubic (and higher order) equations, and the smallest possible Pisot Number is called the "Plastic Number" and is the real root of x3−x−1$x^3-x-1$,

or approximately 1.324717957...


Just as the golden mean has it's value as the limit of the ratio of consecutive terms of the Fibonacci sequence, the plastic number can be derived from the Padovan sequence, first developed around 1994(?), P(0)=P(1)=P(2)=1; and P(n)=P(n-2) + P(n-1)  which begins, 1,1,1,2,2,3,4,5,7,9,12,16...

This sequence has a longer mod(2) cycle.  Like all Pisot numbers, they approach almost-integers, but they do so much slower than the powers of the Golden Mean.


There is even a Super-Golden Ratio which is the real root of \( x^3 -x^2 - 1\ )

or approximately 1.4655712318...

It has its related sequence also, Naryana's cows, which dates back to the 14th Century. Unlike Fibonacci's rabbits, the Cows go through three stages, immature, adolescent, and then mature, so only the matures reproduce. The pattern looks like The first few terms of the sequence are as follows: 1, 1, 1, 2, 3, 4, 6, 9, 13, 19,... . (students could have fun creating four or five stage maturation sequences and look for the limit of their ratios as a limit, and compare the qualities to those from these ratios.  )

As always, comments (and corrections) are welcomed.

The sequence on the far right is a variation of the Padovan Sequence which begins with 3, 0, 2, and f(n) = f(n-3) + f(n-2). Named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay