While you're celebrating New Year's Eve (meaning you're probably not reading this blog!), or thinking about your favorite math teacher (and probably keeping it to yourself), or considering clicking on the new subscriber chiclets in the sidebar, I thought I would kick off 2008 with something different.
As we were playing family Bingo a few days ago with about three dozen families (my wife was reluctant to go and, of course, she won the first game!), I began thinking about the underlying mathematics of the game and its many variations. I know what my wife would be thinking: "Dave, Why can't you just enjoy the game without analyzing it!"
Here were some thoughts running around in my head, when I should have been concentrating on the two boards I was playing (I won nothing BTW):
(1) Historically: What are the origins of this game? Designed by some brilliant mathematician or was it just some game of chance that evolved?
(2) The number of possible boards must be astronomical. How many different boards are supplied by the companies that manufacture this product?
(3) If I buy a bingo game that comes with, say, 36 boards, are these same 36 boards in every box? If I buy a set from a different manufacturer will the boards overlap or be entirely different?
(4) Are all boards randomly generated by software these days? If a bingo game is to be played in a large hall, with hundreds or even thousands of players, how likely is it that there will be multiple winners in a single game? Do the boards all have different winning lines or are there lots of overlap among boards?
Some probability thoughts:
(5) What is the probability of a winner after the minimum number of balls drawn from the bingo cage, namely four numbers (don't forget the free space!). Since I've never seen this happen, I'm assuming the chances are virtually zero!
(6) More realistically, for about two dozen players, each playing a single board (or one player using 24 boards!), what the expected number of balls drawn before a winner occurs? I was conjecturing less than half of the seventy-five numbers, maybe low thirties.
(7) How did the probabilities change as one increases players and boards? I assumed many mathematicians had already solved all of the intricacies regarding the probability of a winner after 10 numbers, 20 numbers, 30 numbers, etc. Instinctively, I felt that this was a very sophisticated problem, probably beyond my comprehension, but I wanted to know more.
Of course, when we returned home, I did some online research of the game -- fascinating stuff: Origins in Italy, Lotto, Beano, Bingo, Mr. Lowe (the toy manufacturer), Professor Leffler from Columbia University, the fund-raising aspect that started in a church in Wilkes-Barre, PA, and so on...
You can easily find these same sources so I'll leave that for our readers. However, there was a dearth of serious mathematical analysis of the probabilities and the combinatorial aspects. I only found a couple of these and neither went into much explanation of the underlying theory, other than to suggest it is complicated, oh, and data tables generated by some software. Of course, I'm sure I missed some wonderful references that my readers will find.
So I decided to do what I usually do when facing a complicated task (a la Polya): Reduce it to a much simpler problem! Not only to understand it better for myself, but, in the back of my mind, I was thinking of how a middle schooler could begin to understand the complexities of all this.
What could be easier than a 2x2 board - just 4 little numbers on a card and to really oversimplify it, only four numbers will be available: 1,2; 3,4. The semicolon separates the possible values for the first column on the card from the 2nd column. I will use this notation from now on. So here's the first elementary question for the reader and for the student:
STUDENT/READER QUESTION #1:
Assume there is one player with one card using the numbers above. Explain why the probability of winning this simple 2x2 version after two numbers are called is 1, that is, 100%.
You're thinking: Way too obvious a place to start, right! Too boring for the student...
STUDENT/READER QUESTION #2:
Ok, let's dial it up a tad. We'll still keep it a 2x2 card, but, this time, there are three numbers available in each column: 1,2,3; 4,5,6. Remember this notation means that 1,2,3 are the possibilities for the first column and so on.
Again, one player with one card: What is the probability of a win after two numbers are called?
Comment: There are many methods here from listing all of the possibilities to permutations and combinations to multiplication of probabilities (one number at a time without replacement), etc. I believe, pedagogically, it is important for the student to see more than one way!
I could stay with the 2x2 game and add more numbers but the student and our readers are an impatient lot and want to move on to something more interesting, right? So let's move on to a 3x3 board which is much more like the 5x5 board in that it has a free square in the middle. But we have to start slowly here - trust me!
STUDENT/READER QUESTION #3:
Now we have a 3x3 board. The available numbers will be simply 1,2,3; 4,5,6; 7,8,9. Again, one player, one card. Couldn't be easier, right?
(a) What is the probability of a win after TWO numbers are called? That's the minimum number with the free space covered.
(b) A little harder now: What is the probability of a win after THREE numbers are called?
Ok, we'll ask the same two questions with more numbers available:
Suppose the possible numbers are: 1-6; 7-12; 13-18
(c) Now, what is the probability of a win after TWO numbers are called?
(d) What is the probability of a win after THREE numbers are called?
I better stop here! This is enough for Part I. As usual, any results I've stated need to be verified by my readers and don't forget to give proper attribution if using any of this in a classroom setting.
Monday, December 31, 2007
While you're celebrating New Year's Eve (meaning you're probably not reading this blog!), or thinking about your favorite math teacher (and probably keeping it to yourself), or considering clicking on the new subscriber chiclets in the sidebar, I thought I would kick off 2008 with something different.
Sunday, December 30, 2007
HAPPY NEW YEAR!
First of all, I wish to thank Denise over at Let's play math! for recognizing me as one of her favorite math bloggers. The feeling is mutual Denise and, if I were to have my own list of all-stars you'd be one of them.
However, there's something I've been wanting to do for a long time and that's to provide a forum for bloggers and others to acknowledge and pay tribute to one or more math teachers who made a difference in their lives. I know each of us can think of someone and I'd like to provide a place for this recognition. Truthfully, there should be an entire web site devoted exclusively to this -- you never know! In addition to naming the math educator(s) who had this kind of effect, I would ask you to include a short anecdote.
The 3 teachers who most affected me in my life were my dad, Mrs. Hill from Lincoln High School and Dr. Silvio Aurora from Rutgers University. All have passed on but their impact on me is everlasting. As my students will attest, I mentioned one or more of them in virtually every lesson I ever taught. There were many other outstanding educators I could mention here and I certainly don't mean to slight anyone I've omitted. But these individuals changed my life...
My dad: Although not a teacher in name or by profession, he was nonetheless the greatest teacher I ever knew. To this day, I quote his oft-repeated phrase: "If you understand part-whole relationships you can solve virtually any math problem." I'm paraphrasing this somewhat and clearly it's an overstatement of the central ideas of mathematics, but, for K-12 mathematics, it's not far off. My dad always used the Socratic method of questioning and I know that I have always done the same in the classroom. He seemed to know just the right question to ask, just how much to lead me in the right direction, but never giving it away. He wanted me to discover the ideas for myself and feel the satisfaction of doing it on my own. Moreover, he understood that no matter how simply he explained some concept, I was the one who had to internalize it and make it my own. Oh and he believed in 'practice makes perfect!' He cared deeply about me both on a personal level and in my intellectual development. Whatever I was as a teacher was because of you, Dad. Thank you.
Mrs. Hill: My students will immediately recognize this name! I must have mentioned her thousands of times over the years because every time I use her chart/table methods in solving algebra word problems, I'm paying tribute to her! In her algebra 2 class, back in the fifties, a chart had to be used for rate-time-distance problems, age problems, mixture problems (dry vs. liquid!), etc. This was not optional! In the back of my mind I was thinking I could do these without that rigid structure but I complied until it became an ingrained habit. Fifty years later, I'm still teaching those tried-and-true methods that simply...work. Mrs. Hill was the only math teacher who ever gave out composition notebooks at the beginning of a math course and by the end we had filled it up with every known algebraic formula known at that time or so it seemed! In particular, factoring forms for x^3-y^3, x^5-y^5, x^7-y^7, etc. This would be considered a complete waste of time today by curriculum specialists and experts but the grounding I received was invaluable. It's hard to recognize patterns in mathematics when one has no base of knowledge and she provided that base. Thank you, Mrs. Hill.
Dr. Aurora: My general topology teacher... You taught me how little I knew about mathematics! I came into the class believing I was fairly competent with math and within one or two sessions I realized that I had only the most superficial understanding of mathematical proof. You gave us these innocent looking problem sets of proofs, which, if worked through painstakingly, developed the entire foundation and theory of the course. I feel as though I learned more about set theory after one or two these than from all other math courses combined! Some of the questions appeared impossible even with your cryptic clues. You knew we would work together on these late into the night and you knew the elation we would feel if we could actually get a few done, never mind the entire set. We had a week or so to complete each paper and your critique of our work was always incisive and thorough. Later on, we learned that, when you were at Columbia, you had the responsibility to assess the validity of each new proof of the Pythagorean Theorem that was submitted. I'll never forget your florid face or the oversized handkerchief you took out of your back pocket to wipe the sweat off your brow when lecturing. I know that you were directly responsible for my decision to pursue mathematical research. Thank you, Dr. Aurora.
Tuesday, December 25, 2007
The following question(s) are appropriate for grades 5-12. How is this possible? Well, the mathematics needed to solve it is elementary! Of course, as with most math problems of this sort, the meaning/interpretation of the question is the challenge for most. Then there is the issue of going beyond the question to look for deeper mathematical meaning...
To help the younger student get started, we will begin with an example and then proceed with the question. Older students might need the same since the language of the question may be vague or difficult to comprehend:
The number 6 can be written as a sum of one or more odd positive integers in exactly 4 ways:
5+1, 3+3, 3+1+1+1, and 1+1+1+1+1+1. As you can see, we are not considering the order of the summands.
Also, 6 can be written as a sum of one or more different positive integers in exactly 4 ways:
6, 5+1, 4+2, 3+2+1. Again, different orders are not included in our list.
Anything of interest yet? Should students naturally raise their hands and ask questions about these two problems or do we have to cue them? At this point, the educator has many options. Here are a couple:
Pair of Problems:
List all ways to write 9 as a sum of one or more odd positive integers. How many ways?
List all ways to write 9 as a sum of one or more different positive integers. How many ways?
What do you notice?
Investigation (in groups):
Make a table for positive integers from 1 through 10 as follows (I'm abbreviating the column headings). Also, include extra columns for the number of ways for each.
N..........N as a Sum of Odds...............N as a Sum of Different
(1) Can you think of at least 3 benefits from having students doing these?
(2) If class time does not permit further investigation, is it worth assigning these for homework, enrichment, extra credit, etc?
(3) Anyone imagine there might be some highly sophisticated ideas from number theory behind these simple questions? Anyone know who posed these kinds of questions originally and solved the general question?
Thursday, December 20, 2007
Update: And the winner once again is...
Here's her contribution:
Jean le Rond d'Alembert is also famously known for incorrectly arguing in Croix ou Pile that the probability of a coin landing heads increased for every time that it came up tails. In gambling, the strategy of decreasing one's bet the more one wins and increasing one's bet the more one loses is therefore called the D'Alembert system. (WIkipedia)
From my reading of several biographies, it sonds as if he is another Cauchy - cantankerous, with many enemies and few friends.
I've always associated D'Alembert with the Ratio Test for Infinite Series in Calculus but that's one of his minor accomplishments. He also solved the Wave Equation in Physics - not too shabby! On a personal note, his mom left him on the church steps when he was an infant. Later in life, when she wanted to reunite with him, he rejected her...
Did you already notice I changed the image in the sidebar? Before I give a hint, I'll leave it up for another 12 hours as is.
Don't forget to email me at dmarain 'at' 'gee' mail dot com with your answer. As always, the name is not enough. Include a fascinating fact or anecdote about this famous individual and a reference or link that you used.
Tuesday, December 18, 2007
As a result of the numerous views of a calculus problem I published in November, I decided to present the following video mini-lesson. As before, I had to break it up into parts to control the file size for uploading. I hope this has some value for those who were looking for a more detailed discussion of this question. Much of this is highly appropriate for precalculus students.
Note: Before playing the videos below, a correction and comments:
(1) In error, I referred to the cross-section of the cone as an isosceles right triangle. Make that isosceles only!
(2) The video and audio quality is far from perfect. Bear with me on this!
(3) I didn't discuss the case where the height of the cone is less than or equal to the radius. This will not produce maximum volume but should have been noted. I will have more to say about this later.
(4) There is so much more to discuss about this question, in particular, the result that the cone of maximum volume has height equal to (4/3)R or that the center of the sphere divides the altitude into a 3:1 ratio. These may be discussed in upcoming videos. In particular, as suggested in the videos below, there will be a treatment of the 2-dimensional analogue of this problem, namely, the isosceles triangle in the circle problem.
(5) These video 'mini' lessons are designed for the university or secondary calculus student (probably comes too late for the college final exam) or for anyone wanting a refresher. Beyond my personal style of presentation, there are pedagogical issues (instructional tips) that arise in the videos that might be of interest to someone teaching calculus for the first time.
If you're getting bored of watching the same chalkboard and my same drab outfit, well, it is a low-budget video! I hope you will let me know if this proves helpful and if you'd like me to continue these. As mentioned previously, I will also be employing other technologies for demonstration purposes.
Happy Holidays Everyone and Peace...
(1) Have you considered subscribing to MathNotations posts? Now you can even subscribe to a feed to those wonderful comments from our readers! Just click on the icon in the sidebar! I'm still not sure how you can comment directly without going to this blog so, if you know how, let us know!
(2) I'm planning another dramatic video, this time on the Mysteries of the Cone Inscribed in the Sphere Problem! I've had so many hits from universities for that post from November that I thought it's time to do a presentation of the problem and how one can relate it to other similar problems.
(3) Some time in January, I will be demonstrating Mimio technology for producing images and/or Quicktime movies of freehand notes written on a whiteboard in vivid color. The company has been gracious enough to send me the equipment for this blog (arriving in January), knowing that I have used it before. It's an alternative to other technologies out there for a fraction of the cost. More info to follow...
(4) I'm also going to attempt to create Demonstrations using Mathematica for this blog. I have a lot to learn here but am anxious to try it. You will be able to view these by downloading the free Mathematica player from Wolfram.com. Those familiar with this outstanding mathematical software know its potential for upper level mathematics, particularly when analyzing 3-dimensional surfaces. One can now produce interactive demonstrations (go to Wolfram to see samples). The company is providing me a temporary license to use their software on this blog! Exciting stuff if I can find the time!
Saturday, December 15, 2007
For the remainder of this post, the statement 0.99999... = 1 will be denoted by S.
Over the course of my math education and my professional teaching career, S has occupied considerable time and provoked much thought on my part and reflection among my students, countless mathematicians and, now, the math blogosphere (see Polymathematic's famous series of posts!). Sane individuals (aka, non-mathematicians) remain skeptical about S, unwilling or unable to grasp the equality in the statement.
They argue: "0.99999... gets closer and closer to 1 but how can you say it EQUALS 1. There's always a gap!" Ah, the mystery of limits!
For many years now, I have been posting my 'proof' of S on various listservs, discussion groups (including MathShare, the one I moderate) and blogs. Here's the reaction I 've generally received: ____________________
That's right - silence. Because I like to put a positive spin on things, I take that to mean no has found a way to refute it! I've even occasionally heard a student say that this convinced her/him.
I don't want to bore the veterans out there who've heard and read all of the well-known arguments, most of which have 'holes' in them (or should I say, discontinuities!). Even using the basic formula for the sum of an infinite geometric series doesn't necessarily satisfy the Odd Thomases (sorry, I'm a Dean Koontz addict) who will continue to question the validity of the statement.
Any attempt to justify S necessarily requires (to paraphrase Liping Ma) a profound understanding of fundamental principles regarding the real number system and my argument is no different.
Enough already -- Here it is:
Non-Rigorous Explanation: If 0.9999... is less than one, then there must be a decimal between it and 1. But this is impossible!
Step 1: Consider the sequence: 0.9, 0.99, 0.999,...
Since this is an increasing sequence of real numbers bounded above by 1, this sequence has a limit, L, namely its least upper bound. As many of you know, I am using the Completeness Axiom for the Reals (known by other names). An excellent reference for the axiomatic structure of the real number system can be found here.
This demonstrates that 0.99999... does exist (i.e., it is a real number). Thus,
0.99999... is the limit L of the above sequence. Verification of the existence of 0.99999... is what is often lacking in other demonstrations of S.
Step 2: L is either greater than 1, equal to 1 or less than 1. We need only consider the last 2 cases.
Step 3: Reasoning indirectly, assume that L<1. By the density property of the real numbers, there must exist at least one real, x, between L and 1. Since L is different from x, it must differ from it in some decimal place. The tenths place? No! Since x is less than 1 and greater than L, it must have 9 in the tenths place. The hundredths place? No, again for the same reason. Need I continue or do you see we've reached a contradiction? Therefore, our assumption that L is less than 1 is false. Thus, L = 1 or, equivalently, 0.99999... = 1. QED!
Ok, your turn! Feel free to critique the proof or present your own favorite argument for or against S. Also, would you consider using this type of argument when teaching this topic?
Friday, December 14, 2007
What is the Largest 3-digit Multiple of 7? - A Middle School Activity on Remainders, Multiples and the Division Algorithm
Please don't forget to give proper attribution when using this activity in the classroom (see sidebar).
STUDENT/READER CHALLENGE #1:
What is the largest 3-digit multiple of 7?
Would you expect most students to reach for the calculator and, without much thinking, test 999,998,997, etc., until they reach 994?
This is an opportunity to review the Division Algorithm, the conceptual meaning of remainder and how important this integer can be when solving problems involving multiples, factors, and various number theory questions (not to mention those repeating pattern problems so popular on standardized tests).
From the calculator, students obtain a result like 1000/7 = 142.8571429. The last digit is rounded which disguises the repeating decimal but some will know that. You ask for the remainder from this division problem and you may get an answer like 6/7 or blank stares or some decimal. Because we live in a calculator environment, students may have already acquired ways to obtain the remainder from the calculator display. Older students with their sophisticated graphing calculators may have a function that returns the remainder (like mod) or some application they downloaded for this purpose.
Here's one way students find remainders on their calculator:
Ignore the decimal, that is, look at the greatest integer value of the quotient, namely 142. Then, the remainder can be obtained from: 6 = 1000 - 142⋅7. This is just another form of how students should check their division:
142⋅7 + 6 = 1000
Students often think about this procedurally, not conceptually. In fact, a thorough understanding of remainders and the division algorithm would make the questions in this post fairly simple.
In general, suppose B is divided by A, producing an integer 'quotient' Q and a remainder R, where R is an integer satisfying A > R ≥ 0. Then the Division Algorithm states:
QA + R = B or R = B - QA
As an aside, some students are taught or discover another calculator approach for finding the remainder:
Subtract (discard) the integer part of the decimal, leaving 0.8571429, then multiply this result by the original divisor 7. Like magic, the display reads 6. Students should realize that the calculator internally stores more places than it displays! It may be worth it to demonstrate why this method works or have students investigate this, so I'll make it part of today's challenge:
STUDENT/READER CHALLENGE #2:
Explain why the above procedure works for finding remainders. You may want to employ an algebraic derivation, using A, B, Q and R as above.
Students should now use remainder concepts to solve the original problem and the following:
STUDENT/READER CHALLENGE #3
What is the largest 6-digit multiple of 7? Explain your method carefully.
Note: If time permits (like right before a vacation), you may want to introduce modulo arithmetic (congruences, etc.) to solve the remainder problem in a more compact form:
10 ≡ 3 (mod 7) → 106 ≡ 36 ≡ (32)3 ≡ 23 ≡ 1 (mod 7), using the fact that 9 is congruent to 2 modulo 7. Thus, the remainder is 1 and the result follows. Of course, there's a great amount of overhead in developing this much number theory, but if you're looking a holiday challenge...
Q: Why did the mathematician name his dog "Cauchy?" A: Because he left a residue at every pole. Mystery Mathematician #3 Revealed!
What does that say about me that I am amused by that famous pun in the title of this post! Cauchy has always been a favorite of mine and his famous Integral Formula has blighted the youth of many young mathematicians in their Complex Analysis course! I recall one of my high school colleagues annually wearing a T-shirt with this formula imprinted on it on the day of the math finals. I don't believe it gave away any answers to her students!
Anyhow, we did have two winners in this week's contest. I intentionally did not announce this edition of the contest in a post as I wanted to see who would notice the new image in the sidebar! BTW, I certainly didn't expect my Technorati rating or Social Ranking to spike because of this and I was not disappointed!
Here are TC's 'punny' comments:
It is not as if you had exhausted most others that you had to pick a
'residual' mathematician :-)
I first thought it might be L'Hospital.
Lagrange advised Cauchy's father that his son should obtain a good
grounding in languages before starting a serious study of mathematics.
Here's Lynx's comment/anecdote about Monsieur Cauchy:
The mathematician is Augustin-Louis Cauchy.
A quote by him (courtesy of www-history.mcs.st-andrews.ac
Said in 1847
Also, according to Agnesi to Zeno Over 100 Vignettes from the History of Math by Sanderson Smith, he was a teacher at the Ecole Polytechnique. While Evariste Galois was trying to gain admission, Cauchy lost a paper written by the young man.
He was a staunch Catholic and, according to the Catholic Encyclopedia (http://www.newadvent.org
According to this website (http://web01.shu.edu/projects
Okay, that's enough. I'm done researching for now. Thanks for an evening of entertainment.
Thursday, December 13, 2007
From various Google searches reaching this site I often discover some wonderful problems. These often inspire me to extend the question further. Part (a) below was from the Google search. I modified it slightly and then developed it. Thank you to whomever typed in the original search!
Note: For all of these questions, p/q represents an "improper fraction" in which p>q>0 and p,q are positive integers.
(a) How many values of p/q are there in which the sum of the numerator and denominator is 29 and 2 1/3 > p/q > 2 1/4.
Notes: The numerical quantities in the inequality are mixed numerals.
How might one approach this without algebra?
(b) Show there are exactly 16 values of p/q such that p+q = 500 and 5 > p/q > 4.
Is algebra the preferred method here?
(c) Show there are exactly 33 values of p/q such that p+q = 1000 and 5 > p/q > 4.
(d) For Advanced Algebra students: Derive a formula for the number of values of p/q such that p+q = N and B > p/q > A.
N is a positive integer greater than 2 and A,B are positive reals with B >A >2 . Express your formula in terms of A, B and N. The greatest integer function may be needed.
Tuesday, December 11, 2007
It's been awhile but something this good is always worth waiting for!
TC has sent me some fascinating challenge problems for our readers. If you are now sick of watching amateur videos on the Arithmetic and Geometric Mean Inequality, it's time to raise the bar. The following involves a well-known generalization of these means but the results are worth your efforts, particularly parts (c) and (d) below.
If a and b are positive, we can define their generalized mean to be:
GNM = ((ak + bk)/2)(1/k)
This would look far prettier in LaTeX but I'm hoping it's readable. In words, we're looking at:
The kth root of the arithmetic mean of the kth powers of a and b.
(a) What is another name for the result when k = 1? (we're starting off easy here!)
(b) What is another name for the result when k = -1? (slightly harder algebraically)
(c) Ok, now for the real challenge for you Calculus lovers:
What is the limit of GNM as k-->0? The result is totally cool!
(d) TC's Super Bonus: Show that the limit of GNM as k-->∞ is the maximum of a and b.
Note: These have been slightly edited from tc's original problems, but they are essentially the same. Solutions may be posted in a couple of days although the notations will be hard to render. I might just have to do another video or wait for that special technology I mentioned earlier! We're hoping some of you will tackle the harder ones and comment!
Saturday, December 8, 2007
A post from 7-17-07 generated some wonderful comments and solutions. I decided to bring it back using a slightly different diagram. This time I'm asking for a solution path using exterior angles, a tool students often overlook. The most efficient approach may still be the one tc used, involving the sum of the interior angles of a quadrilateral, but the challenge here is to find another way...
Heres' the problem:
Assume Q, S and T are collinear. Determine the value of a+b+c.
Note: Again, there are many wonderful approaches here. Try to use the suggested one...
Friday, December 7, 2007
Mary Everest Boole was a remarkable woman. Her uncle, Colonel Sir George Everest, to whom she was very close, was the Surveyor General of India. He was largely responsible for completion of the trigonometric survey of India along the meridan arc from the south of India extending north to Nepal. The completion of the Indian survey allowed the subsequent survey of Mt. Everest (at the time un-named) and calculation of its summit height. It was later renamed in honor of George Everest.
Through her uncle, Mary met George Boole, an already famous mathematician. Mary enjoyed her time with Boole both socially and intellectually and they soon married. Even though Mary was 17 years younger than George, they were still very close companions and had a very successful marriage. During the next nine years, Mary and George had five daughters. Yet, this happiness would not last for long. Tragically, George caught pneumonia and died leaving Mary alone with her youngest child only six months old.
Mary Everest Boole was a miraculous woman who, widowed for fifty years, raised her five daughters and made countless contributions towards the mathematical education of many girls and boys. Mary considered herself a mathematical psychologist. Her goal was to try "...to understand how people, and especially children, learned mathematics and science, using the reasoning parts of their minds, their physical bodies, and their unconscious processes."
Mary was recognized in England as being an outstanding teacher. One of Mary's pupils was to write later, "I thought we were being amused not taught. But after I left I found you [Mary] had given us a power. We can think for ourselves, and find out what we want to know." Many of Mary Boole's contributions can be seen in the modern classroom today.
tc found this fascinating anecdote:
Niece of a famous George AND wife of a famous George! (Note the
Boolean operator)! A maven of mathematical pedagogy! - A particularly
apt choice for this blog!
I found the following quote particularly interesting:
"I have had homicidal impulse at the touch of other stimuli. When I
was quite young, I used to speculate on the problem why I did not try
to kill someone who worried me. It was not love of my parents that
hindered me; in those moods I was incapable of fear. It was not regard
for God; I considered that God made me as I was and could not
reasonably be angry with anything I did. It was -- I always came back
to the same conclusion -- it was that I thought that if I killed
anyone the police or the hangman or someone would stop my working for
algebra. Besides I felt that all stormy passions in themselves
interfered between me and algebra. Hate and revengefulness,as well as
love and fear, vanished, like burned paper, when they threatened to
interfere between me and algebra (34)."
- from The Forging of Passion into Power (London: C.W. Daniel Company,
1910), found online at http://www.troubling.info
mathmom found the perfect web site to search (sorry, I'm not sharing that at this time!) and commented:
The only woman I could think of before searching was Ada Lovelace and I didn't think the photo was her, though I'm pretty bad with faces. If I'd found a different photo of Boole, I might not have been sure it was the same woman.
I'll leave Mary's picture up there for a couple of weeks. I will probably do this contest biweekly from now on due to 'overwhelming' response!
Thursday, December 6, 2007
NOTE: PLS READ THE COMMENTS FOR A DETAILED DISCUSSION OF THIS PROBLEM, WHICH SHOULD PROVE QUITE CHALLENGING FOR MOST MIDDLER SCHOOLERS.
The previous activity I posted regarding integers that have exactly four factors might lead to some interesting discussion regarding a general description of such numbers. All of these kinds of problems could be handled by simply giving students the general rule for determining the number of factors of any positive integer. I have given this well-known number theoretic formulation in earlier posts, so I won't review that at this time. However, there is a greater benefit to be derived from having students investigate these relationships. The following activity should be adapted to meet the needs of your students.
Some educators react to these kinds of deeper investigations with reactions like:
(a) I have a curriculum to cover. I don't have time for this.
(b) Unless this kind of question appears on state testing, it's simply not practical for me to do this.
(c) My students are just not ready for this kind of thinking.
(d) Dave, you're out of the classroom now, so you're forgetting the realities of most classrooms. Some students don't know their basic facts and you want me to do higher-order thinking! Gee, Dave, are you forgetting we have classified children mainstreamed in our classes? Get real!
(e) Dave, stop suggesting HOW we should teach and just give us the problem. You're trying to impose your style on others - it doesn't work - we each bring our own style to a lesson.
[Comment: I have strong reactions to some of the above, but then I'd be arguing with myself! I'll respond to some of these in the comments section or devote an entire post to these critical issues if my readers decide to respond to this.]
I'm certainly not suggesting that these kinds of explorations should BE the curriculum. There must be a balance between these problem-centered approaches and skills development. I am suggesting there needs to be some time devoted to deeper cognitive processes to foster mathematical development. The following investigation is far from one inch deep! I may continue it later but I'm hoping some will suggest extensions, make comments or report back how it played out in real classrooms (also how it was adapted/revised).
1. The number 6 has 4 factors: 1,2,3,6 (or in paired form: 1,6;2,3).
If we double the number 6, what do you think will happen to the number of factors? Will it increase or stay the same? Will the number of factors also double?
Mathematicians, like scientists, make conjectures or educated guesses, but not wild guesses! We need some evidence or data on which to base our conjectures. With your partner, fill in (and possibly extend) the following table, then formulate your conjecture using correct mathematical language:
Positive Integer.................Number of Factors
Do you think we have enough data to make a conjecture or should we continue the table? Record your observations and then state your conjecture or 'rule'.
Teacher Tip: Depending on the maturity of the group and their experience with these kinds of formulations, you may want to start them off with a prompt:
If we double a positive integer, the number of factors _______________.
Many students will be convinced they have found a mathematical rule that will always work. That's one of the objectives of this investigation: To help them understand that
(a) Pattern recognition does not a rule prove!
(b) The conjecture is based on starting from the number 6. There is no basis for assuming that their 'rule' will be valid if we start from a different positive integer!
This time have students start from a different integer: 18
Make a table similar to the one above, again doubling the integer in the left column.
Again, record your observations and then state your conjecture or 'rule'.
Do you think there is a more general rule that covers all cases or are there simply different rules for different integers? If you were going to investigate this further, what other kinds of starting positive integers would you try?
Teacher Tip: Asking many questions stimulates student thinking and leads to more questions and deeper thought processes on their part. A free interchange for a couple of minutes is invaluable here to have students come to see that mathematical research requires persistence and an attitude of inquiry. As Ms. Fribble from the Magic School Bus would say: ASK QUESTIONS! (or something like this!).
Part III - Start from an odd integer this time: 15
Part IV: To be continued...
Wednesday, December 5, 2007
There are countless problems involving the factors of a positive integer we're seeing in middle school classrooms and on standardized tests these days. They are often used as challenges or warm-ups and questions similar to the one(s) below have appeared frequently on this blog. Students become more proficient with this type of question by doing many variations repeatedly over time. As they mature, they will come to appreciate a more general approach to finding the number of factors of any positive integer. Number theory is now included in most states' standards so there needs to be some time devoted to this topic on a regular basis.
STUDENT PROBLEM/READER CHALLENGE
This problem/activity is often best implemented in small groups. Each member of the group should make their own list and then compare, however, they might want to divide the labor by having some students do the numbers up to 50 and others do the rest.
Suggested Time for Activity: 15-20 minutes (the problem can be explored further for homework or a challenge, then revisited the following day for 5 minutes).
The number 12 has 6 positive integer factors: 1,12;2,6;3,4.
(a) List all positive integers up to and including 100 that have exactly four factors.
(b) Higher-order: These numbers fall into 2 categories. Describe these categories.
Alternate Problem (shorter time needed): What is the largest 2-digit positive integer that has exactly 4 factors?
As much as I revere Archimedes, life is a series of hellos and goodbyes (Billy Joel has a way with words). Again, if you think you know who the famous woman in the sidebar is, email me at "dmarain at g-mail dot com".
Please do not put your answer in the comments section of this post - we do not want to give it away until the contest is over! I will give up to 48 hours depending on how many emails I receive.
Don't forget to include some curious, fascinating or esoteric fact relating to her mathematical career or some compelling personal anecdote. From what I've read of her, the possibilities are many! Please cite your sources (link) for verification.
If I don't get a response within 12 hours, I will post a hint either in the sidebar or in this post.
Hopefully, this time, one cannot right-click, ctrl-click or view source to find her name!
I know those of you with advanced search skills will find her fairly quickly, but, remember, the name is not enough!
Sunday, December 2, 2007
Ok, we've been discussing the square dissection problem for a few days now. The original problem was here and Solution Method I was here.
Now don't start laughing at these crude, low-tech videos! The 1st video is 2-3 minutes long. It provides a quick overview of the squares problem and a demonstration of why the product of the areas is the same for either pair of non-adjacent rectangles. It ends abruptly with part of the derivation I gave in the previous post.
The remaining presentation is split into 3 segments in order to control the file size (Blogger has a problem with larger files). In these segments, I derive a formula for the maximum product using the Arithmetic-Geometric Mean Inequality (AM-GM) and apply it to the square dissection problem. The first of these clips reviews this important and useful inequality, presenting a standard algebraic derivation. The 2nd of these 3 clips presents a geometric derivation and the final clip shows how we can derive a formula, m4/16, for the maximum possible product of the areas in the general case of a square of side length m.
As I say on the video, there's no SmartBoard technology, no Mimio, no whiteboard with vivid color dry erase markers! Just an old chalkboard my kids use in the basement, some inexpensive chalk and sheets of paper towel to erase. I wore my favorite 'Pop-Pop' sweatshirt - sorry, no formal wear.
Let me know if you find this helpful (once you stop chuckling!). At least you'll be able to match a name to a face and it isn't Archimedes! If I get few if any comments, I'll know that my 15 minutes of fame have expired.
Saturday, December 1, 2007
Since I only received one comment from the squares problem, I guess readers were either bored by this question or are awaiting my solution(s)!
Solution I: Divide side AB into segments of lengths 2+x, 2-x. Here, x can be any real between -2 and 2. Similarly, divide BC into segments of lengths 2-y and 2+y. One can demonstrate that the order here is irrelevant. Then the product of the areas of either pair of non-adjacent rectangles can be expressed (after rearrangement of factors) as
(2+x)(2-x)(2+y)(2-y) = (4-x2)(4-y2).
From the restrictions on x and y, it follows that x2 is greater than or equal to zero and less than 4. Similarly for y.
Therefore, (4-x2)(4-y2) ≤ 4⋅4 or 16.
This also demonstrates that the maximum product occurs when x=0 and y=0! QED
Solution II (using the Arithmetic-Geometric Mean Inequality): We will prove a general result for squares of side m. This will be forthcoming and there may be a visual surprise! Stay tuned!
Friday, November 30, 2007
Ok, the secret's out. Start ordering your
Archimedes and Eureka -- Nature Abhors a Vacuum T-Shirts.
It was a difficult decision, but the results are in (although we still haven''t heard from Palm Beach County). The top 3 submissions based on the logb2(b6) entries in the first ever MathNotations: Name that Mathematician Challenge are -- in no particular order --
I came in a distant 4th (I am going to appeal this). Of course, it was only fitting that I chose the 'Arc-Man' to lead off, since one of my personal favorite posts of all time came earlier this year:
The Genius of Archimedes: Parabolas, Tangents, ....
Anyone recall I attempted my first complicated diagram, suggesting how Archimedes proved that light emanating from the focus of a parabolic surface are reflected in parallel rays (and conversely)?
By the way, one of the better explanations for kids of his discovery of the displacement principle can be found here.
Ok, here are the details:
Mathmom found the following fascinating fact here:
Archimedes invented a puzzle called the Loculus (or the Stomachion,
the Ostomachion, the Syntemachion, or Archimedes' Box). It's like a
huge, complex, tangram. In November of 2003, Bill Cutler used a computer program to enumerate all solutions. Barring rotations and reflections, there are 536
Eric actually used something called a book (for youngsters out there, here is a link to explain the meaning of this obsolete term) to find the following information about Archimedes' perspective on pure math vs. science:
ARCHIMEDES OF SYRACUSE (c. 287-212 B.C.), son of an astronomer, was
Greece's star mathematician. By avocation he desired the pursuit of
mathematics proper, and he was wholly and passionately committed to
mathematics at its "purist." But by world reputation he was an
engineer, especially in the field of military engines, even if he
protested that he derived no satisfaction from this kind of work. And
when confronted with the problem of determining whether a golden crown
was made of pure gold or was alloyed with silver, he initiated the
method of Hydrostatics for the purpose.
Scientist-professors will always be the same. Archimedes, when
heading the Weapons Research Group for the Syracuse Department of
Defense, would write letters to friends that he was yearning to return
to the Campus and do nothing but pure research for its own sake. But
he was apparently doing classified work to his last breath, literally
so. And when he found his theorem on Hydrostatics he was so excited
that he insisted on talking about it to the man in the street.
I won't actually publish tc's joke regarding one of Archimedes' engineering feats (young children might find their way to this blog), so I will leave it up to my readers to invent their own...
Finally, I promise, from now on, not to the use the name of the mathematician in labeling the image file! Duh...
Thursday, November 29, 2007
Before announcing the thousands (or less!) winners of the Name That Mathematician Challenge, I came across a problem about dissecting a square ABCD with lines PQ and RS which are parallel to the sides of the square. (see diagram).
Naturally, I decided to make it into a deeper investigation. Students and/or readers will be asked to find the maximum value of the product of the areas of either pair of non-adjacent rectangles formed. There are many approaches here, one of which uses the famous Arithmetic Mean-Geometric Inequality. As usual you will work from the particular to the general, beginning with a specific value for the sides of ABCD.
STUDENT/READER INVESTIGATION - PART I
The given conditions about the diagram are given above.
For Part I, we will assume each side of the square has length 4.
(1) (Particular) If AP = 3 and RC = 2, determine the product of the areas of APTS and RTQC. Do the same for the other pair of non-adjacent rectangles formed. Do you believe this product is the maximum possible as we vary the positions of segments PQ and RS?
(2) (General) Show that the product of the areas of either pair of non-adjacent rectangles formed is less than or equal to 16. For example the product of the areas of APTS and RTQC is ≤ 16.
(1) Do you think many students would guess what the configuration would be for the maximum product to occur? Is proving the conjecture much more difficult?
(2) The challenge here is to find an effective use of variables to denote the segments. There are many possibilities, some much more efficient than others.
(3) I will add additional parts to this challenge after receiving comments on Part I. How would you generalize this result further? More interestingly, there is a way to prove Part I using the AM-GM Inequality?
Tuesday, November 27, 2007
Update2: Only one submission thus far (and a good one!) so I will give our readers another day to email me some fascinating fact about 'A'! Ironically, Isabel, over at God Plays Dice, has a humorous post on attaching names to faces of current mathematicians. What are the odds! As with all experiments, we'll see how this challenge is responded to before making it a regular feature!
Update: Mathmom, reminded me that one can right-click on a PC and see the name of the image file. Working on a MAC, I forgot that. Well, this first picture might be a freebie, but you still need to wow me with some esoteric fact about him! Next time, I'll rename that file!! I also removed the personal info requirement (again, thanks to mathmom's astuteness!).
Starting today, we are introducing a different kind of challenge. Look at the image near the top of the sidebar. Here are the rules (read these carefully before submitting):
DO NOT NAME THE FAMOUS MATHEMATICIAN IN A COMMENT! (If you did already, I will delete it!).
Instead, send me an email at "dmarain at gmail dot com" with the following information:
(a) The name of the person
(b) One fascinating fact about her/him; it doesn't have to be what made that person famous -- I'm looking for the unusual or curious here, be it mathematical or something personal...
(c) Please cite your source for this fact (include the link) - I need to verify its authenticity
I will choose the top responses received within 24-48 hours of the time the picture first appears. Earlier submissions may receive higher ranking than later submissions. I will announce the winners in a couple of days and, of course, include the fascinating facts as well.
Note: I realize that students today have extraordinary online research skills (although one cannot enter an image in Google!), but, remember, the winners are not only determined by giving a name. Feel free to share this with students. They can enter too!
Sunday, November 25, 2007
If you were looking for a challenge here in higher math using combinations and permutations, sorry to disappoint you! I felt compelled to write this essay after watching my wife patiently attempting to teach one of my children how to open a combination lock. She doesn't think of herself as a teacher, but, she is, and, in many ways, far more skilled than I ever was.
One of the rites of passage for many middle schoolers is mastering the intricacies of the combination lock for their lockers, somewhat akin to elementary schoolers learning how to tie their shoes. Do you remember the frustration you felt the first few times you tried to solve the puzzle of these locks? Do you recall your euphoria when it magically opened? Consider all of the 'skills' involved and think of the parallels to mastering the algorithms of mathematics:
(1) Fine motor skills required to precisely turn the dial and stop at the correct number
(2) Memorizing the 3 numbers in sequence
(3) Understanding the difference between Right and Left when rotating the dial and retaining the R-L-R sequence
(4) The absolute discipline and precision required - close is not good enough
(5) The dreaded second step of the process needed in going 'past zero'
(6) The extreme feelings of frustration from failing repeatedly and the inclination to give up, yet driven to continue
(7) The elation felt in getting it the first time all by yourself, only to be followed by despair when you can't seem to duplicate the feat!
(8) The feeling of accomplishment when you can do it almost every time without anyone helping you
(9) Is there any substitute for independent practice in achieving mastery here?
(10) How important is motivation here in driving the child to continue in the face of adversity?
What about the challenges faced by the 'instructor' here? If you were the one who helped someone succeed, did you find it frustrating or did you have 'unlimited' patience? Did you have to practice it yourself first and think about breaking this 'automatic' process into simple discrete steps? Did you have to try different verbal instructions (for example, using 'down' and 'up' vs. 'left' and 'right') or different techniques of one approach failed? Did repeated demonstrations in front of the child suffice? Did the child say, "Let me do it by myself?" If you've helped several children learn to 'unlock' the combination, did you use the same approach successfully with each child? Are some youngsters simply unable to 'solve the problem' at that time and need to be given a key lock instead as an accommodation? Is making this concession detrimental to their self-esteem and eventual development or is it reasonable at that time? Will some of these youngsters be able to succeed later if given the opportunity to try again (when developmentally ready)?
Is there a metaphor here for teaching children mathematical algorithms? By the way, can you think of others skills or concepts involved in opening the lock that I overlooked? Pls share!
Now, parents, extrapolate this 'teaching' process to dozens of unique math students every day with a myriad of different algorithms over the course of a school year? Anyone can teach, right?
I realize some of you will see the flaws in this metaphor and will point out all the differences between opening the lock and solving a mathematical problem? I know the parallel is far from perfect but this is something that just struck me and I had to put my thoughts down. You know, like a journal, a diary, a blog... Your thoughts?
Thursday, November 22, 2007
Many math educators use warmup problems to review, challenge or set the tone as students walk in the room. Routines like this are effective in having students 'hit the ground running'. These mini-problems can be on the board, while an overhead transparency of selected answers are displayed. The instructor then has time to circulate, check homework, engage students, and get a feel for the difficulties they had with the assignment. By the way, do most of your students take these warmups seriously?
Here is a warmup that requires more active participation on the student's part. I may suggest a different one in a later post, but I'd really like readers to share some of their favorites!
All students stand at their seats. They are told they will be have to give the next number in sequence, according to some rule that will be explained. They will have 3 seconds to respond (can be adjusted but no more than 5). If incorrect or time runs out, they will be instructed to sit down and the next person will have to give the correct answer and so on. You may want to start them off by giving them the first number (judgment call here). I suggest you allow a maximum of 5 minutes for this activity.
Here's the problem I gave to a group of high schoolers but it is highly suitable for middle schoolers as well:
Using positive integers, think of primes ending in 1 or 3 (you may want to use the technically precise phrase 'whose units digit is 1 or 3'). For example, 21 'ends in' 1, but it is not prime. You must go in order and you are not allowed to ask the number the previous person gave! You will have 3 seconds to respond. If incorrect or time runs out, I will ask you to sit down and the next person will need to give the correct number. We will continue until there is only one star shining or time runs out and we have co-champions!
(a) Reviews primes (How many of your students do you think would be eliminated early by starting with 1? You can always start them off with 3 if you feel this will help. Some students will begin with 11, assuming that you meant 2-digit numbers. By the way, 51 and 91, in particular, typically knock out many, if they get that far! Finally, are you thinking this question is not particularly relevant for high schoolers? Count how many questions relate to primes on the SATs!)
(b) Improves listening skills and concentration (How many of your students do you think will forget the last number given either because their minds are wandering or from trying to think ahead to their turn?)
(c) Learning how to think under pressure. (Although we know some students will 'freeze up' or be embarrassed if they are eliminated, they will not be alone! Typically, about half of the students in an above-average class will be sitting down on the first pass through the class! With a high-achieving class of very strong students, you may need to make several passes to reach a winner. If there are a couple left after 3-4 minutes, proclaim co-champions.
Let me know how this goes if you try it after the Thanksgiving break. What variations would you use to make this more effective for your students? What other kinds of problems are suitable for this 'Bee?' Were my predicted statistics way off? Did you predict that students would get past 100?
Again, please share some of your favorite warmups!
Wednesday, November 21, 2007
[Reprinted from Jan 3rd, 2007]
I'd like to share a sentiment that was emailed to me by a close friend. I decided to re-publish this on the eve of Thanksgiving. You may have already seen this, but it does send a message about giving thanks for what we have. (Thank you, Elisa, for sharing this):
One day a father of a very wealthy family took his son on a trip to a farm with the firm purpose of showing his son how poor people live. They spent a couple of days and nights on the farm of what would be considered a very poor family.
On their return from their trip, the father asked his son, "How was the
trip?" "It was great, Dad." "Did you see how poor people live?" the father asked. "Oh yeah," said the son. "So, tell me, what did you learn from the trip?" asked the father.
The son answered: "I saw that we have one dog and they had four.
We have a pool that reaches to the middle of our garden and they have a creek that has no end.
We have imported lanterns in our garden and they have the stars at night.
Our patio reaches to the front yard and they have the whole horizon.
We have a small piece of land to live on and they have fields that go beyond our sight.
We have servants who serve us, but they serve others.
We buy our food, but they grow theirs.
We have walls around our property to protect us, they have friends to
The boy's father was speechless. Then his son added, "Thanks, Dad, for showing me how poor we are."
Isn't perspective a wonderful thing? Makes you wonder what would happen if we all gave thanks for everything we have, instead of worrying about what we don't have.
Appreciate every single thing you have, especially your friends!
Life is too short and friends are too few.
HAPPY THANKSGIVING TO ALL MY READERS!
Sunday, November 18, 2007
[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]
The cone in the sphere problem led me to an interesting relationship in the corresponding 2-dimensional case with a surprise ending. (Only a math person would compare a math problem to a mystery novel!). The following investigation allows the student to explore a myriad of possibilities: from similar triangles to the altitude on hypotenuse theorems to Pythagorean, to chord-chord or secant-tangent power theorems, coordinate methods, draw the radius technique, etc. Sounds like this one problem might review over 50% of a geometry course? You decide for yourself! Just remember -- one person is not likely to think of every method. Open this up to student discovery and watch miracles unfold...
STUDENT ACTIVITY OR READER CHALLENGE
In the diagram above, segment AF is a diameter of the circle whose center is O, BC is a tangent segment (F is the point of tangency), BC = AF and BF = FC. Segments AB and AC intersect the circle at D and E, respectively. Lots of given there! Perhaps some unnecessary information?
(a) If AF = 40, show that DE = 32.
Notes: To encourage depth of reasoning, consider requiring teams of students to find at least two methods.
(b) Let's generalize (of course!). This time no numerical values are given. Everything else is the same. Prove, in general, that DE/BC = 4/5.
(c) So where's the 3-4-5 triangle (one similar to it, that is)? Find it and prove that it is indeed similar to a 3-4-5.
Saturday, November 17, 2007
Update: View the series of videos here explaining the procedure for solving the cone in the sphere problem below as well as related questions.
Many Algebra 2 and Precalculus textbooks have begun to include those challenging 3-dimensional geometry questions involving 2 or more variables and/or constants. However, we know from the difficulty that calculus students continue to have with these, that we need to do more before students do their first optimization problems in calculus. You know the kind: Determine the radius of the __________ of maximum volume that can be inscribed in a _________ of radius R. These problems have fallen out of favor somewhat with the AP Development Committee, perhaps because they lack that real-world flavor or perhaps because they had become predictable or perhaps too hard. I would argue they have been part of the rites of passage for calc students for many generations for a reason - they blended the spatial reasoning of geometry with the need to identify variable relationships and reduce the number of conditions down to one function of one variable if possible. In other words, they help to develop mathematical sophistication. I 'cut my teeth' on these -- did you? Any calculus teachers reaching this topic yet in AP Calc?
Anyway here's an activity for you Algebra 2 or Precalculus students to prepare them for these challenges. As usual we proceed from the concrete (i.e., given numerical dimensions) to the abstract. Rather than attempt to draw the diagram, which is fairly challenging for me given the tools I have, I will describe the problem verbally. Good luck!
(1) A right circular cone of height 32 is inscribed in a sphere of diameter 40.
Note: Students need to learn how to make a diagram of this problem situation.
(a) Determine the radius of the cone.
(b) Determine the volume of the cone. [Imagine asking students to memorize the formula!]
(c) Keep the diameter of the sphere at 40. This time, determine both the radius and volume of the inscribed cone whose height is 80/3. The numbers are messy but try to work in exact form (fractions, radicals) before rushing to the calculator to convert everything to decimals. Oh well, we all know what will happen here!
(d) Try another value for the height of the cone, keeping the diameter of the sphere at 40. See if you can produce a volume greater than in (c). Any conjectures?
(2) We could throw in an intermediate step by using a parameter R to denote the radius of the sphere, and use numerical values for different possible heights of the cone, but I'll leave that to the instructor. Instead, we'll jump to the abstract generalization:
A right circular cone of height h is inscribed in a sphere of radius R.
(a) Express the radius, r, of the cone in terms of R and h.
(b) Express the volume, V, of the cone as function of h alone (R is a constant here).
(c) Use your expression for r and your function for V to verify your results in (1).
(d) Calculus Students: You know what the question will be! Oh, alright:
Determine the dimensions and volume of the right circular cone of maximum volume that can be inscribed in a sphere of radius R. Anything strike you as interesting in this result?