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Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever.

 Albert Einstein

I never did very well in math - I could never seem to persuade the teacher that I hadn't meant my answers literally. 

~Calvin Trillin

It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class.  ~Author Unknown

How many times can you subtract 7 from 83, and what is left afterwards?  You can subtract it as many times as you want, and it leaves 76 every time.  ~Author Unknown





a traditional notion that is obstinately held although it is unreasonable; "he still holds to the old mumpsimus that a woman's place is in the kitchen"


SMH Column 8

"I don't think the question is whether five tonnes of budgies (I assume you meant metric tonnes) would make the plane lighter if they all started flying mid flight," insists Carolyn Darrell, of Narrabeen, "It is whether 125,000 budgies would fit in a plane. That is a lot of budgies. Perhaps someone better than me at area calculations could work out how big the plane would have to be to fit in all those budgies [calculated at 40 grams per budgie]"

Maths Tutor ] Articles ] Coincidents ] Scrabble ] Fibonacci ]

The Law of Large Numbers guarantees that one-in-a-million miracles happen 295 times a day in America

Infinity Paradox

Let      x   = 1 + 2 +4 +8 +16 + . . . .

Then 2x   =       2 + 4 + 8 + 16 + . . . .

 x - 2x     =   1

      -x      =   1

       x      =  -1

 ie   1 + 2 + 4 + 8 + 16 + ..... = -1

Thanks Hannah for this one


The demographical status of the world's population

......click here


Benford's law


Benford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is 1 almost one third of the time, and larger numbers occur as the leading digit with less and less frequency as they grow in magnitude, to the point that 9 is the first digit less than one time in twenty. This is based on the observation that real-world measurements are generally distributed logarithmically, thus the logarithm of a set of real-world measurements is generally distributed uniformly.


This counter-intuitive result applies to a wide variety of figures, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). Even more counter-intuitively, the result holds regardless of the base in which the numbers are expressed, although the exact proportions of course change.


It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881 in his paper "Note on the Frequency of Use of the Different Digits in Natural Numbers". The first rigorous formulation and proof appears to be due to Theodore P. Hill in 1988


In base 10, the leading digits have the following distribution by Benford's law, where d is the leading digit and p the probability:
























The Game of Nim


From Wikipedia, the free encyclopedia


Nim is a two-player mathematical game of strategy in which players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap.

Variants of Nim have been played since ancient times. The game is said to have originated in China (it closely resembles the Chinese game of "Jianshizi", or "picking stones"), but the origin is uncertain; the earliest European references to Nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901, but the origins of the name were never fully explained. The name is probably derived from German nimm! meaning "take!", or the obsolete English verb nim of the same meaning. Some people have noted that turning the word NIM upside-down and backwards results in WIN.

Nim is usually played as a misère game, in which the player to take the last object loses. Nim can also be played as a normal play game, which means that the person who makes the last move (i.e., who takes the last object) wins. This is called normal play because most games follow this convention, even though Nim usually does not.

Normal play Nim (or more precisely the system of nimbers) is fundamental to the Sprague-Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games.

Play a variation of Nim called Tax Tix at www.math.com

A more macabre game of Nim



The World of Quantum Physics and String Theory

Part One


Part Two



A Four-Dimensional Tribute to the Late Madeleine L'Engle

Over the weekend, children's author Madeleine L'Engle died at 88. Her most noted work, A Wrinkle in Time, is a story about a girl's journey across the universe in an effort to rescue both her father and the galaxy itself from the evil "Black Thing." The book, which dealt with heady (and un-kid-friendly) concepts like religion, theoretical mathematics and evil, took years to finally find a publisher due to its perceived weirdness. Since finally going into print in 1962, Wrinkle has sold millions of copies and remains a favorite read for young teens today.

One concept L'Engle explored in the book was tessering, a method whereby people could traverse great distances in the universe by "folding" space and time. Although they don't behave in exactly the way L'Engle describes, tesseracts do exist, and serve as important and elegant examples of multidimensional space.

An actual tesseract is best described as a four dimensional cube...and is kind of confusing. So, in memory of L'Engle, we met up with Physicist David Morgan who took a little time out of his day to talk tesseracts with the BPP. Put your measley three-dimensional brains to work on this one.



Achilles and the tortoise


Zeno's second paradox of motion, of Achilles and the tortoise, is probably the best known of his four paradoxes of motion. In this problem, the fleet Greek warrior runs a race against a slow-moving tortoise. Assume Achilles runs at ten times the speed of the tortoise (1 meter per second to 0.1 meter per second). The tortoise is given a 100-meter handicap in a race that is 1,000 meters. By the time Achilles reaches the tortoise's starting point T0, the tortoise will have moved on to point T1. Soon, Achilles will reach point T1, but by then the tortoise would have moved on to T2, and so on, ad infinitum. Every time Achilles reaches a point where the tortoise has just been, the tortoise has moved on a bit. Although the distances between the two runners will diminish rapidly, Achilles can never catch up with the tortoise, or so it would seem.





Bertrand Russell commented:

This argument... shows that, if Achilles ever overtakes the tortoise, it must be after an infinite number of instants have elapsed since he started. This is in fact true; but the view that an infinite number of instants makes up an infinitely long time is not true, and therefore the conclusion that Achilles will never overtake the tortoise does not follow.






The Ship of Theseus

The ship wherein Theseus and the youth of Athens returned had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.

Plutarch, Vita Thesei, 22-23


M C Escher


M C Escher and the Möbius Strip



Appearance of the Möbius Strip in science and technology

There have been technical applications. Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they allow the ribbon to be twice as wide as the print head whilst using both half-edges evenly.

A device called a Möbius resistor is an electronic circuit element which has the property of cancelling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: U.S. Patent 512,340 "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.  

Since 1930, the Möbius strip has been a classic poser for experts in mechanics. The teaser is to resolve the strip algebraically - to explain its unusual shape in the form of an equation.

In a study that lyrically praises the strip for its "mathematical beauty," two experts in non-linear dynamics, Gert van der Heijden and Eugene Starostin of University College London, present the solution.

What determines the strip's shape is its differing areas of "energy density," they say.

"Energy density" means the stored, elastic energy that is contained in the strip as a result of the folding. Places where the strip is most bent have the highest energy density; conversely, places that are flat and unstressed by a fold have the least energy density.

If the width of the strip increases in proportion to its length, the zones of energy density also shift, which in term alters the shape, according to their equations.

A wider strip, for instance, leads to nearly flat, "triangular" regions in the strip, a phenomenon that also happens when paper is crumpled.

The research may seem esoteric, but Mr van der Heijden and Mr Starostin believe it also has practical applications.

It could help predict points of tearing in fabrics and also be useful for pharmaceutical engineers who model the structure of new drugs.

The Möbius strip was named after a German mathematician, August Ferdinand Möbius, who discovered it in 1858.


Banach-Tarski Paradox.

The theorem of Banach and Tarski which states that if A and B arc bounded sets in a Euclidean space of dimension at least 3 and if both A and B have interior points, then A can be separated into a finite number of pieces and reassembled by moving the pieces by rigid motions (trans­lations and rotations) to form a set congruent to B. In particular, it is possible to cut a solid sphere into a finite number of pieces and to reassemble these pieces to form two solid spheres the same size as the original sphere. No estimate of the number of pieces needed in this case was given by Banach and Tarski, but R. M. Robinson has proved that the smallest possible number of pieces is 5 and that one of these pieces can be a single point; he also prosed that the surface S of a sphere can be separated into two pieces each of which can be separated into two pieces congruent to itself (thus only four pieces are needed to make from S two identical copies of S). See HAUDORFF—Hausdorff paradox.



Visual Perception



Pascal’s Triangle




Taxi Cab Geometry




In taxicab geometry the shortest distance between two points is not a straight line, but rather the number of blocks a taxi has to travel along the streets. So, in the street plan the distance from the blue star to the red star is 4 blocks.


On a grid similar to the one above mark any point as A. Now put an X on all the points on the grid that are 4 blocks away from Point A. Remember this is taxi cab geometry and you have to stay on the streets, no cutting across corners.


On another grid mark two points A and B for two houses, Marcus’ and Christina’s homes. We will use city blocks as our unit of measurement. How far do Cristina and Marcus live from each other in school-bus geometry? How many different ways are there from Christina's to Marcus'? Remember this is school-bus geometry. Stay on the streets.

Marcus and Cristina decide to meet half-way between each other's homes. But wait, what's the problem with that? Put an X on all the midpoints between their two homes.




Newcomb's Problem

A being, who you believe to have superior predictive powers, makes you an offer. Before you are two boxes, identical in appearance. Within one box, let us call it 'Box A', there is either one million dollars or nothing. Within the other box, 'Box B', there is definitely one thousand dollars.

The being, whom in deference to tradition we will call the Newcomb Being, will allow you to either take both boxes, or just Box A.

Twenty-four hours ago, the Newcomb being made a prediction about what you would choose. At this point, it placed the money in the boxes. If the Newcomb Being predicted that you would take both boxes, it left nothing for you in Box A and the $1000 in Box B. If the Newcomb Being predicted that you would only take Box A, it would leave the million dollars in it, as well as the thousand dollars in Box B. If the being predicts you will base your choice on a random event (like flipping a coin), it will leave Box A empty.

Now you have grounds to believe that in such matters, the Newcomb Being has a success rate on predictions of about 90%.

Depending upon you choice and the being's anticipation of your choice, you could walk away with $1,000,000, $1,001,000, $1000, or nothing. If you choose both boxes, the least you could receive is $1000 and the most would be $1,001,000. This seems to be the decision which would maximize your return, while minimizing risk.

On the other hand, if you pick both boxes, chances are that the Being would have predicted this and left you nothing in Box A. Giving up an almost certain one million for a certain one thousand seems awfully foolish.

Working out the expected return, based on the Being's 90% success rate, you come up with following decision matrix:


You Choose

Box A Contains:

Box B Contains:

Expected Value

Only A

.9 x $1,000,000



Both Boxes

.1 x $1,000,000



It seems like choosing only Box A gives you a lot higher expected return on average, even if there is the risk of nothing. But there you stand, before the boxes, and the Being has already placed the money. There either is or is not $1,000,000 in Box A. There certainly is $1000 in Box B. Nothing you could do at this point will change what is in the boxes. Why shouldn't you just take both to be on the safe side?

The being awaits your response. What do you do?

Since it was originally posed by William Newcomb, The Newcomb Problem has generated a great deal of commentary in philosophic circles on both sides. In fact, this problem is an oft-used example in the free will debate.


Non Euclidean Geometry 1


Lunes and Triangles in Spherical Geometry



Euclidean Geometry
Geometric Constructions

Billiard Balls



Smart man + smart woman = romance
Smart man + dumb woman = affair
Dumb man + smart woman = Marriage
Dumb man + dumb woman = pregnancy

Smart boss + smart employee = profit
Smart boss + dumb employee = production
Dumb boss + smart employee = promotion
Dumb boss + dumb employee = overtime

A man will pay $2 for a $1 item he needs.
A woman will pay $1 for a $2 item that she doesn't need.

A woman worries about the future until she gets a husband.
A man never worries about the future until he gets a wife.
A successful man is one who makes more money than his wife can spend.
A successful woman is one who can find such a man.

To be happy with a man, you must understand him a lot and love him a little.
To be happy with a woman, you must love her a lot and not try to understand her at all.

Married men live longer than single men do, but married men are a lot more willing to die.

A woman marries a man expecting he will change, but he doesn't.
A man marries a woman expecting that she won't change, and she does.

A woman has the last word in any argument.
Anything a man says after that is the beginning of a new argument.

Old aunts used to come up to me at weddings, poking me in the ribs and cackling, telling me, "You're next." They stopped after I started doing the same thing to them at funerals.



The Square Wheel

Stan Wagon, a mathematician at Macalester College in St. Paul, Minn., has a bicycle with square wheels. It's a weird contraption, but he can ride it perfectly smoothly. His secret is the shape of the road over which the wheels roll.


Stan Wagon rides his square-wheeled trike over a special roadway.
Courtesy of Stan Wagon

A square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary.


A hanging chain.


A catenary is the curve describing a rope or chain hanging loosely between two supports. At first glance, it looks like a parabola. In fact, it corresponds to the graph of a function called the hyperbolic cosine. Turning the curve upside down gives you an inverted catenary—just like each bump of Wagon's road.

What about the drill that can drill a SQUARE HOLE

You can check out the mathematics of this square hole...

Reuleaux Triangle  Wikipedia

mathworld  a great mathematics site


The Fastest route is not always the most direct path.... a curious demonstration : The fastest path