Two proofs uploaded to my website

I have just uploaded the proofs to the statements made in two earlier posts:

• Curious occurrence of the powers of 4
• Property of a 120 degree triangle. 

Here are the links:

• Powers of 4 (screen)
• Powers of 4 (pdf)
• 120 degree triangle (screen)
• 120 degree triangle (pdf)

Comments are welcome! I am particularly interested in generalizations or variations.

Book lists

I often get asked about lists of books for studying mathematics at various levels. I have compiled such a list, and display it below. I will keep adding to it, as and when I think of suitable titles. 


Readers are very welcome to suggest their favorite titles - I will add these too to the list. 


Remark. 
It is quite difficult to keep such a list to a modest, manageable size - there are so many books worth looking at, and worth studying. One has to decide for oneself how much one wants to spend on books. 


Some of these books may well be available on the Web, free, as pdf or djvu files. But for serious study, it is difficult to study off a computer screen, at least for me :-). 


So either one must print the file in its entirety, or take the trouble to purchase a regular copy.


Here is the list, subdivided into various categories. Some are given in the form of web links.



I. Books by Dan Pedoe

1. Geometry: A Comprehensive Course (Dover)
2. Circles: A Mathematical View (Cambridge)
3. Geometry and the Visual Arts (Dover)
4. Geometry and the Liberal Arts
5. The Gentle Art of Mathematics (Pelican)

See this web page too:

1. [Dan Pedoe]

II. Books by W W Sawyer

1. Mathematicians Delight (Penguin)
2. Prelude to Mathematics
3. Introducing Mathematics: Vision in Elementary Mathematics (Penguin)
4. Introducing Mathematics: A Path to Modern Mathematics (Penguin)
5. Introducing Mathematics: The Search for Pattern (Penguin)
6. A Concrete Approach to Abstract Algebra
7. What is Calculus About? (MAA)

III. Books by Ross Honsberger
See this web page:

1. [Ross Honsberger] 

IV. Books by Martin Gardner

1. Mathematical Puzzles and Diversions
2. Mathematics Magic and Mystery

See these web pages too:

1. [Martin Gardner1]
2. [Martin Gardner2]
3. [Martin Gardner3]

V. Books by Shailesh Shirali

1. A Primer on Logarithms (Univ Press)
2. A Primer on Number Sequences (Univ Press)
3. First Steps in Number Theory: A Primer on Divisibility (Univ Press)
4. Adventures in Problem Solving (Univ Press)
5. Adventures in Iteration, Volumes I and II (Univ Press)

VI. Books for “Plus Two” and IIT-JEE Mathematics

1. Higher Algebra, Hall and Knight
2. Higher Algebra, Barnard and Child
3. Elements of Coordinate Geometry, S L Loney
4. Plane Trigonometry (two volumes), S L Loney
5. Challenge and Thrill of Pre-College Mathematics, Pranesachar, Krishnamurthy et al
6. Calculus (two volumes), Tom Apostol
7. An Elementary Course Of Infinitesimal Calculus, Horace Lamb
8. Combinatorics: Including Concepts Of Graph Theory (Schaum), V Balakrishnan
9. Introduction to Enumerative Combinatorics, Miklos Bona
10. Outline Of Vector Analysis (Schaum), Spiegel, Lipschutz, Spellman 

VI. Other Books

1. One Two Three Infinity, George Gamow
2. Geometry Revisited, HSM Coxeter and SL Greitzer (MAA)
3. Mathematical Circles, Dmitri Fomin & Sergey Genkin & Ilia Itenberg (Univ Press)
4. The Enjoyment of Mathematics, Hans Rademacher & Otto Toeplitz (Dover)
5. One Hundred Problems in Elementary Mathematics, Hugo Steinhaus (Dover)
6. From Zero To Infinity, Constance Reid

VII. Books on History and Culture of Mathematics

1. Mathematics in Western Culture, Morris Kline
2. Great Moments in Mathematics, Volumes I and II, Howard Eves (MAA)
3. Journey Into Genius, William Dunham (MAA)
4. A Mathematician's Apology, G H Hardy
5. The Mathematical Experience, Philip Davis and Reuben Hersh

VIII. Problem Books and Puzzle Books

1. Fun With Mathematics, Ya Perelman
2. The Moscow Puzzles
3. Mathematical Circles, Dmitri Fomin + Sergey Genkin + Ilia Itenberg; Universities Press
4. Problem-Solving Strategies, Arthur Engel, Springer Verlag 
5. The Enjoyment of Mathematics, Hans Rademacher & Otto Toeplitz, Dover
6. One Hundred Problems in Elementary Mathematics, Hugo Steinhaus, Dover
7. Challenging Mathematical Problems with Elementary Solutions, A M Yaglom & I M Yaglom; Dover (in two volumes)
8. USA Mathematical Olympiads 1972-1986, M S Klamkin, MAA

IX. Biographies

1. The Man Who Knew Infinity, Robert Kanigel

X. Journals

1. Crux with Mayhem (Canadian Mathematical Society, CMS)
2. Mathematics Magazine (Mathematical Association of America, MAA)
3. College Mathematics Journal (Mathematical Association of America, MAA)
4. Mathematical Gazette (Mathematical Association, UK)
5. Mathematics Teacher (National Council for Teachers of Mathematics, USA)
6. Arithmetic Teacher (National Council for Teachers of Mathematics, USA)
7. Resonance (Indian Academy of Sciences)

XI. Websites

1. [Mathworld - Wolfram]
2. [Art of Problem Solving]
3. [Mathematical Excalibur]
4. [Math Forum]
5. [Math Pro Press]
6. [Plus]
7. [Cut the Knot!]
8. [Euclid's Elements]
9. [Math History 1]
10. [Math History 2]
11. [Math Celebration]

A curious occurrence of the powers of 4

Here is a curious way in which the powers of 4 turn up in a recursively defined sequence. 


Let the sequence of positive integers a(1), a(2), a(3), ... be defined using the following recursive formula:


a(1) = 1,
a(n) = a(n-1) + Floor(Sqrt(a(n-1)), for n > 1.


The "Floor" function is defined as follows: If x is any real number, then Floor(x) = the largest integer that does not exceed x. For example, Floor(3.1) = 3, Floor (10.7) = 10, Floor(-1.8) = -2, and so on. And, of course, Sqrt simply means "square root". 


I have used these notations because normal mathematical notation does not display properly in a blog.


Using the definition  one can compute the entire sequence recursively. Thus: 


a(2) = 1 + Floor(Sqrt(1)) = 2,
a(3) = 2 + Floor(Sqrt(2)) = 3,
a(4) = 3 + Floor(Sqrt(3)) = 4,
a(5) = 4 + Floor(Sqrt(4)) = 6,


and so on. Continuing this way, we can enumerate the entire sequence (it is particularly easy to do it using a computer). Here are the first 50 terms:


1, 2, 3, 4, 6, 8, 10, 13, 16, 20, 24, 28, 33, 38, 44, 50, 57, 64, 72, 80, 88, 97, 106, 116, 126, 137, 148, 160, 172, 185, 198, 212, 226, 241, 256, 272, 288, 304, 321, 338, 356, 374, 393, 412, 432, 452, 473, 494, 516, 538, ...


From this list, let us sift out just those numbers which are perfect squares; we get the following:


1, 4, 16, 64, 256, ...


Why, these are just the powers of 4. How very curious!


How is one to explain this?


Can we modify the defining rule so that we get some other power sequence (or some other sequence of interest)? Yes we can!  - if we replace Sqrt by CubeRoot we get something equally striking. 


So let us define the sequence of positive integers b(1), b(2), b(3), ...  using the following formula:


b(1) = 1,
b(n) = b(n-1) + Floor(CubeRoot(b(n-1)), for n > 1.


As earlier, we can enumerate the entire sequence, recursively. Here are the first 50 terms:


1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 133, 138, 143, 148, ...


The perfect cubes in this sequence turn out to be these numbers:


1, 8, 64, 512, ...


and these are all the powers of 8.


Many variations are possible, within the same theme. 


Proofs, anyone? Here is a link to one possible line of analysis, written by my friend Ramana: http://ramana.posterous.com/an-intriguing-sequence-contain. Do have a look at it.


I'll write the proofs and post them to my website, later this week.

Triangle with a 120 degree angle

Here is a beautiful result concerning triangles in which one angle measures 120 degree.


Let ABC be a triangle in which angle A measures 120 degree, and let the internal bisectors of all three angles be drawn; let them meet the opposite sides at P, Q, R (so that AP is the bisector of angle A, BQ is the bisector of angle B, and so on).

Now a beautiful fact emerges: angle QPR is a right angle! 


There are many nice proofs of this fact. Here is a figure showing the result.

Just as beautiful is the fact that the statement has a converse: 


If points P, Q, R are constructed as described above, starting with an arbitrary triangle ABC, and angle QPR is a right angle, then angle A measures 120 degree. 


But this is less easy to prove.


I invite the reader to find both the proofs!

Powers of 2 and 3

Exploring power sequences can be a lot of fun! Many famous problems have arisen from playing with such sequences; for example, Fermat's Last Theorem (proved three and a half centuries after it was first stated by Fermat), Catalan's problem ("The only two perfect powers that differ by 1 are 8 and 9"; see http://mathworld.wolfram.com/CatalansDiophantineProblem.html; the theorem was proved very recently, in 2002, by the Swiss mathematician Preda Mihăilescu), and Beal's problem (which is still open).

On my website www.mathcelebration.com, which I launched more than a year back, and which I use to upload articles I have written for teachers and students, I have an article all about the powers of 2 and 3. Here are two links to it:

• http://www.mathcelebration.com/PDF/PowerTwoScreen.pdf
• http://www.mathcelebration.com/PDF/PowerTwoPDF.pdf.

Do have a look at it!