## Tuesday, May 24, 2011

### MOBIUS FUNCTION

http://www.gaussianmath.com/misc/mobiusfunction/mobius.html

Introduction to the MÃ¶bius Function

Hat 0

Fundamental Theorem of Arithmetic

Uniqueness of Factorization

Mertens Function

Mertens Conjecture

Identities and Mobidromes

### Mathematical Wormhole 1 & 2

Time travel is possible in mathematics!

Check out ... http://www.gaussianmath.com/
for a more indepth explanation and other interesting topics in mathematics

### The M&M game

It has been said that, "Life is like a box of chocolates—you never know what you're going to get." (Forrest Gump in Forrest Gump, 1994.) In this experiment you can test the "Forrest Gump Chaos Theory" by using M&M's, which are much cheaper than a box of chocolates. What if life is more like a bag of M&M's?

We start with two bags of M&Ms, one with 19 and one with 20.
In her turn, a player has to eat all the M&Ms in one of the bags, and then split the M&M from the other bag between the two, leaving at least one on each bag, and not necessarily evenly. The she gives the two bags to the other player. The player who receives two bags with one M&M each loses as he can no longer move.

### Mathematical Platonism

http://www.lehman.edu/deanhum/philosophy/platofootnote/PlatoFootnote.org/Outreach_files/Mathematical%20Platonism.pdf

James Robert Brown’s Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (Routledge, 2008).

There is a difference between general Platonism and the mathematical flavor. For Plato, each apple, say, is but an imperfect example of the absolute (and perfect) Idea of an apple. But as Aristotle quickly realized, Plato has it exactly backwards: we arrive at the general idea of ‘apple’ by mentally abstracting a set of characteristics we think common to all actual apples. It is we who conjure the ‘perfect’ idea from the world, not the world copying the concept.

But now contrast the idea of an apple with the idea of a circle. Here Aristotle’s approach becomes more problematic, as we don’t find any true circles in nature. No natural object has the precise geometric characteristics of a circle, and in a very strong sense we can also say that the circles we draw are but imperfect representations of the perfect idea of a circle. Ah – but whence does such a perfect idea come from?

Consider another way to put the problem. One major difference between science and technology is that science discovers things, while technology is about human inventions. We discover the law of gravity; but we invent airplanes to allow heavier-than-air flight despite the law of gravity. But where do mathematical objects, like circles and numbers, or mathematical theorems like the Pythagorean one, or Fermat’s Last one, come from? Are they inventions of the human mind, or are they discoveries?

## Monday, May 23, 2011

### Which of the following statements are true?

1. Precisely one of these statements is untrue.

2. Precisely two of these statements are untrue.

3. Precisely three of these statements are untrue.

4. Precisely four of these statements are untrue.

5. Precisely five of these statements are untrue.

6. Precisely six of these statements are untrue.

7. Precisely seven of these statements are untrue.

8. Precisely eight of these statements are untrue.

9. Precisely nine of these statements are untrue.

10. Precisely ten of these statements are untrue.

### Visual.ly

visual.ly is launching soon!

## Sunday, May 22, 2011

### Sum of two dice

2 .................. 1+1 .................................................1/36
3 ...................1+2, 2+1 ..........................................2/36
4 ...................1+3, 2+2, 3+1 ..................................3/36
5 ...................1+4, 2+3, 3+2, 4+1 .........................4/36
6 ...................1+5, 2+4, 3+3, 4+2, 5+1 .................5/36
7 ...................1+6, 2+5, 3+4, 4+3, 5+2, 6+1 ..........6/36
8 ...................2+6, 3+5, 4+4, 5+3, 6+2 .................5/36
9 ...................3+6, 4+5, 5+4, 6+3 ........................4/36
10 ..................4+6, 5+5, 6+4 ................................3/36
11 ..................5+6, 6+5 .........................................2/36
12 ..................6+6 ..................................................1/36

Is it possible to load the dice in such a way that these eleven scores are equally probable?

### A clock puzzle

Imagine a clock that does not have any hands or numbers on it, but it has a chimer.

If the time is 1 o'clock, it chimes once. If the time is 2 o'clock, it chimes twice, and so forth. The time gap between any two chimes is 3 seconds.

How many seconds would it take you to know the time, after the first chime is heard, if it is 3 o'clock?

### Game show: 5 fortune tellers

On a game show there are 5 fortune tellers (A, B, C, D and E)

A has 81% chance of being correct
B ..... 65% .........................................
C ..... 43% .........................................
D...... 35% .........................................
E ...... 8% ............................................

If you were trying to get the best prediction of your future, which fortune teller would you go to?

### n times the sum of its digits

Which number is 5 times the sum of its digits?

In general, to find 2-digit and 3-digit numbers that are (2, 3, 4, 5, ...) times the sum of their digits

And any d-digits (d > 3)?

## Saturday, May 21, 2011

### n circles with radius r ...

... are inscribed in a circle with radius 1

Determine the radius r and the area A of the shaded region between the circles for n = 4 and n = 3

Hint: In an equilateral triangle the medians divide each other in the ratio of 2:1

## Sunday, May 15, 2011

### Smallest positive integer that includes among its factors perfect squares

1^2 = 1 ..... 2^2 = 4 ..... 3^2 = 9

4^2 = 16 ... 5^2 = 25 ... 6^2 = 36 ... 7^2 = 49 ... 8^2 = 64

10^2 = 100 ..... 11^2 = 121 ..... 12^2 = 144 ..... 13^2 = 169 ..... 14^2 = 196
15^2 = 225 ..... 16^2 = 256 ..... 17^2 = 289 ..... 18^2 = 324 ..... 19^2 = 361
20^2 = 400 ..... 21^2 = 441 ..... 22^2 = 484 ..... 23^2 = 529 ..... 24^2 = 576
25^2 = 625 ..... 26^2 = 676 ..... 27^2 = 729 ..... 28^2 = 784 ..... 29^2 = 841
30^2 = 900 ..... 31^2 = 961

The factors of 2304 are:

1 2 3 4 6 8 9 12 16 18 24 32 36 48 64 72 96 128 144 192 256 288 384 576 768 1152 2304

The prime factors are: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 or 2^8 * 3^2

2304 is the smallest positive integer that includes among its factors at least 10 perfect squares.

### (sqrt(n)^sqrt(n))^sqrt(n) an integer?

http://www.wolframalpha.com/input/?i=%28sqrt%28n%29^sqrt%28n%29%29^Sqrt%28n%29

For how many integral values of n in the range 1 to 1000 (inclusive) is the following statement true?

### Prime factoization

Find the smallest positive integer with exactly n di fferent factors

### Triangle - Altitude

The figure shows an isosceles triangle ABC with AB = BC

### The 80-80-20 Triangle

Is there anything else that could be said about the 80-80-20 triangle?

Problem 358. Isosceles triangle 80-80-20, Circle, Angles, Congruence.
Problem 274: Isosceles Triangle, 80-80-20, Angles

## Monday, May 9, 2011

### A collection of Math Olympiad problems

(1) Find the smallest natural number which is a multiple of 2009 and whose sum of
(decimal) digits equals 2009.

(2) Find all integer solutions of the equation 3^x - 5^y = z^2

(3) Solve in integers the equation 12^x + y^4 = 2008^z

(4) Determine all pairs of natural numbers (x, n) that satisfy the equation
x^3 + 2x + 1 = 2^n

(5) Determine all pairs (x, y) of integers such that
1 + 2^x + 2^(2x+1) = y^2

(6) Find all primes p such that p^2 – p + 1 is a perfect cube.

(7) Let a, b, c be positive real numbers. Prove the inequality
(a^2 / b) + (b^2 / c) + (c^2 / a) >= a + b + c + 4(a – b)^2 / (a + b + c)
When does equality occur ?

## Wednesday, May 4, 2011

### A Proposal to End the Practice of Gerrymandering

Using a precise mathematical formula, it is possible to redraw district lines fairly and bring competitiveness back to congressional races

Drawn Together: Can Math Nerds Beat Gerrymandering?

Gerrymandering: Why Your Vote Doesn't Count

### Squares of 2 consecutive even numbers using the same digits

206^2 = 42436 and 208^2 = 43264

Find others

### Squares and cubes of 2 numbers using the same digits:

102^2 = 10404 and 201^2 = 40401
102^3 = 1,061,208 and 201^3 = 8,120,601.

### Digits

202^3 = 8,242,408 = (–8 + 242 – 40 + 8)^3

204 = 3^2 + 7^2 + 5^2 + 11^2 = 2^2 + 8^2 + 6^2 + 10^2,
where 3+7 = 2+8 and after adding 3 to each digit: 6+10 = 5+11.
204^2 = 23^3 + 24^3 + 25^3 = 41,616.

206^3 = 8,741,816 = (8 + 7 + 4 + 181 + 6)^3

208^3 = 8,998,912 = (08 + 99 + 89 + 12)^3

209 = 1^6 + 2^5 + 3^4 + 4^3 + 5^2 + 6^1

305^3 = 28,372,625 = (283 + 7 + 2 + 6 + 2 + 5)^3

494 + 209 = 703 and 494,209 = 703^2

88+209 = 297 and 88,209 = 297^2 and 494+209 = 703 and 494,209 = 703^2 and 297+703 = 1000

2,401 = (2 + 4 + 0 + 1)^4 = 7^4

10,000 = (1+2+3+4)^4 = [(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)]/2

### Reversible numbers and their squares:

102^2 = 10404 and 201^2 = 40401.

103^2 = 10609 and 301^2 = 90601.

### Square numbers using the same digits:

108^2 = 11664, 129^2 = 16641 and 204^2 = 41616

148^2 = 21904, 203^2 = 41209 and 302^2 = 91204.

102^2 = 10404 ... 120^2 = 14400 ... 201^2 = 40401 ... 210^2 = 44100.

148^2 = 21904 ... 203^2 = 41209 ... 302^2 = 91204.

130^2 = 16900 ... 140^2 = 19600 ... 310^2 = 96100,
103^2 = 10609 ... 247^2 = 61009 ... 301^2 = 90601.

207^2 = 42849, 222^2 = 49284 and 288^2 = 82944

128^2 = 16,384, 178^2 = 31,684, 191^2 = 36,481, 196^2 = 38,416 and 209^2 = 43,681.

## Monday, May 2, 2011

### Math Education

TWO TYPES OF MENTAL ARITHMETIC
AND THE EMPTY NUMBERLINE

### tan(4*pi/3) and arctan(4*pi/3)

tan(4*pi/3)

tan(4*pi/3) is clearly an irrational number. How to that arctan(4*pi/3), in radians, is irrational?