Suppose we have a circle inscribed within a square. Now suppose a rectangle extends from one corner of the square to a point on the circle (see diagram).

If this rectangle is 6 inches by 12 inches, what is the radius of the circle in inches?

In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.

UPDATE!

Triangle Formulas : Right Triangle C = A + B = Pi/2 radians = 90 degrees c^2 = a^2 + b^2 P = a + b + c s = (a+b+c)/2 K = ab/2

Given a positive integer k another positive integer x is said to be k-transposable if, when its leftmost digit is moved to the unit's place, the resulting integer is k*x

For example, the integer 142857 is 3-transposable since

Match each of the even numbers between 2 and 30 with a different odd number between 1 and 29 so that the fifteen resulting sums are pairwise relatively prime.

A cycloid is the curve generated by a point on a wheel as the wheel rolls. This diagram shows half of a loop of the standard cycloid, whose parametric form is (t - sin t, 1 - cos t):

Which is larger, the blue area or the orange area?

Consider two semicircles S and T that emanate from the same point with diameters along a common line, with S being the larger.

Draw an isosceles triangle whose base is the part of diameter of S that is outside T and whose apex is on S. Draw a circle inside S that is tangent to S, T, and the triangle.

Prove that the center of this circle is directly above the point common to the triangle and T

Can you get 88 using only the integers 1, 2, and 3, using each of them only once, and using the standard arithmetic operations, including factorials and repeating decimals?

Alice and Bob play a game. A positive integer starts the game and the players take turns changing the current value and passing the new number back to their opponent.

On each move, a player may subtract 1 from the integer, or halve it, rounding up if necessary. The person who first reaches 0 is the winner.

Alice goes first: she makes her choice of move on the starting value.

For example, starting at 15 a legal game (if not particularly well played) could be:

Alice .......................................15 → 8 Bob ......................................... 8 → 7 Alice ....................................... 7 → 4 Bob ......................................... 4 → 2 Alice ....................................... 2 → 1 Bob ......................................... 1 → 0 Bob wins.

For which values of n is there a winning strategy for Alice?

The original problem: The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens attack each other. http://en.wikipedia.org/wiki/Eight_queens

In this version, we need to place as many queens as possible on an NxN board, such that each queen will threaten at most ONE other queen. We ask to prove an upper bound and give a solution matching it for the standard 8x8 board as well as for a 30x30 board.

Prove: if we remove two opposite corners from the chessboard, the board cannot be covered by dominoes. (Each dominoe covers two neighboring cells of the chessboard.)

Look for an “Ah-ha” proof: clear, convincing, no cases to distinguish.

Let S be a set of n + 1 integers from {1, 2, . . . , 2n}. Prove that two of them are relatively prime. First show that this can be avoided if S has only n numbers

Suppose we have 13 real numbers with the following property: if we remove any one of the numbers, the remaining 12 can be split into two sets of 6 numbers each with equal sum.

Rubik's Clock Objective: Get all clock's at both sides at 12 o'clock. By pushing or pussing one of the four buttons different clocks will move at the same time.

Rubik's Dice Object: Re-arrange the metal plates inside the cube in such a way that the dice has only white spots. If red is shown anywhere on the dice even through the small controll holes, the puzzle is not complete. The number of possible combinations is 7! x 4^7=82,575,360. There is only ony one correct solution

Question: What is the relationship between game theory and poker?

Barry Nalebuff: There’s a great example of game theory and poker that is from an economist colleague of mine named Randy Heeb. He teaches strategy and sometimes teaches with me at Yale.

It was I think 2002 when the World Championship, he was playing a fellow named Thomas Preston, who was probably better known as “Amarillo Slim” and they were playing the World Championship Heads Up, which is a form of Texas HoldEm.

It was €40,000 pot, which this day may seem small, but he was assistant professor, so it’s real money. And it turns out the top two people were Slim and my friend Heeb. Heeb have a pair of Knights, and based on the betting and the initial hands, he was convinced that Slim either had a King or an Ace on his hand, but not both.

Now, what happens next is three cards, or the flop, are presented, and then there is another round of betting. And Heeb had decided that if either a King or an Ace came out in the flop, he was going to fold, because basically the chances of the pair of Kings or Aces, which is too high. And clearly if Slim had a pair of King or Aces, Heeb stay in, he was going to leave with that kind of a hand.

The cards came out and there’s a 2 of Clubs and 3 of Diamonds and a King of hearts. And at that point, Heeb sort of realized, damn it, I’ve got big pot, I’m going to lose it, and his eyes just gave it away. He realized that he had not done a good job hiding his emotions, and at the same time, he also realized that Slim had looked into his eyes seeing that dejection and went all in.

At that point, Heeb said, “Ah, Slim wasn’t betting based on his cards. He was betting based on the fact that I was giving in, and therefore it means he had the Ace not the King, so he doesn’t have the pair and so he went all in and won.”

And it was this great analysis of, okay, what is he thinking, because if you truly have the pair then what he is happy about is the cards, not about my rejection and that wasn’t what Slim was betting on.

But, typically, poker players spend a lot more time watching the other players than they do watching their own cards and also that’s why a lot of people wear sunglasses.

Stephen Miles: When you interview people and you ask them about their careers retrospectively what most people will say is stuff just happened to them. “I was here.” “I was lucky.” They’ll attribute it to all sorts of things

If you think about your career and think about it broadly in terms of what am I learning, what is this position doing for me, how am I going to get to the next one, what is the most important next move for me and you think two or three moves ahead you can start to put yourself in advantageous positions to accelerate your career disproportionately. As opposed to sort of creeping along the floor you can start to sprint.

Before you set out on your own personal game theory you have to assess your own playing field and who is on the playing field:

Your Boss:

Does your boss want to be known for creating talent and being a net contributor of talent inside your company or organization or does your boss want to be known for really delivering outstanding results because although subtle it really matters for you as an employee because if they want to be a net contributor of talent they’re willing to rotate people through positions, coach mentor them and then if you will, set them free in other parts of the organization. If they’re solely focused on delivering results then what they’re going to do is over hire people for the job and keep them in the job for as long as they possibly can because that is how you’re going to deliver results. You’re not going to take any risk on people. So you as a person playing on that field needs to think about what are your boss’s motives and what are they known for.

Your Peers:

There is only one job, so you’re competing for that job with your peers and you have to assess what are their strengths, what are they doing that is disproportionate to what you’re doing and how do you position yourself to be the best candidate for the next job, so you have to assess horizontally if you will, the peer playing field.

Your Subordinates:

If they choose you to go to the next job, do they have somebody to backfill? Have you created a successor for yourself because if you haven’t, if you’ve made yourself indispensible, guess what, you’re probably going to be indispensible and, therefore, not moved. So you have to think about below are you creating the conditions to make it easy for you to move and allow your boss to make that move with you?

Take two identical discs and slot them together along diameters so that the discs are perpendicular. There is one separation of centres for which the compound shape has a centre of mass which remains a constant height above the surface of a table. The resulting shape rolls along in a most intriguing way.

Historically, the first straight line linkage was described by Sarrus in 1853. It differs in that its parts move in three dimensions. It is applied widely in jacks, elevating platforms and similar devices.

Prof. John Conway proved that there are uni-stable polyhedra. That is to say, polyhedra which are stable on only one face. More information is available from http://www.howround.com/

The width of a smooth shape is the distance between parallel tangents. When a shape is not smooth we must talk about parallel support lines instead. If we have a circle then the width is constant. If the width is constant, then is the shape a circle? These solids have constant width.

You have been captured by an evil wizard. However the wizard also likes maths, and sets you a challenge. You have two identical vases, and 100 white beads and 100 black beads. You must arrange the beads however you like between the two vases. The only condition is that no vase can be empty. And all the beads must be in the vases.

The wizard will then get his assistant to choose a single bead from one of the vases. The assistant will pick purely at random, and will not peek! If the assistant picks a black bead, you will go free. If the assistant picks a white bead... well, let's not go there. Obviously you would like the assistant to pick a black bead.

The question is this. How do you arrange the beads so as to give yourself the best chance of freedom?

For example, 5, 14, 17, and 23 give a collection of four positive integers. Now, form the differences of pairs of these numbers by subtracting the smaller number from the larger in each pair (although this won’t really matter). In our case, this process would give us the following:

Now, form the product of all these differences. This gives us 9 x 12 x 18 x 3 x 9 x 6 = 314,928, in our case. Let’s call the result the Prod-Dif of the four integers. So, the Prod-Dif of the collection {5, 14, 17, 23} is 314,928. We can form the Prod-Dif of any collection of four positive integers. Then we have a very interesting problem to ponder:

What is the largest integer that divides evenly into all Prod-Difs? That is, what is the greatest common factor of the collection of all Prod-Difs? And of course, WHY?

Question : Can the board be set such that no legal move is possible?

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Bejeweled Blitz is a game played on an 8x8 space with 6 different color gems. See the picture below.

The object of the game is to move the gems so that combinations are formed, hence gain points.

1) The board is set such that the same color gem does not occupy 3 consecutive spaces in any column or row.

2) A legal move consists of exchanging 2 neighboring gems (horizontally or vertically NOT diagonally) such that combinations of 3 or more are formed.

In the picture above, the highlighted yellow gem can exchange positions with the purple above it to form a combination of 3 horizontal yellows. There are other possible moves e.g. moving white to the bottom right position to form a combination of 3 vertical whites

What is the smallest number that can be expressed as a sum of six different squares but cannot be expressed as the sum of five different square numbers?

UPDATE!

Take,

1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91

Could you express 91 as a sum of five different square numbers?

n^4 + 4 is composite (i.e., not prime) for all n > 1

Well, n^4 + 4 = (n^2 - 2n + 2) (n^2 + 2n + 2) is a proper factorization, since the smaller of the two factors is greater than 1 when n > 1.

This type of factorization is often called Aurifeuillian (sometimes also spelled Aurifeuillean) in honor of the French mathematician [Léon François] Antoine Aurifeuille.

Rearrange the key caps of the 1 through 9 on a numeric keypad so that no cap is on its correct key, in such a manner that each of the three rows forms a 3-digit perfect square.

Now an ugly issue: what happens if in the prime decomposition of the numbers involved in the Ruth Aaron pairs, one or several of these prime are powered? Will you take all the prime factors involved (with repetitions) or will you take only the distinct primes involved (without repetition)?

As a matter of fact you can do it in one way or another, with the result that you will generate two kind of sequences: Ruth-Aaron pairs, prime factors

Ruth-Aaron triplets do exist. That is to say, three consecutive numbers, n, n+1 and n+2 such that the sum of the prime factors of each number adds up to the same quantity a) without repetition or b) with repetition.

The first example, prime factors with repetition, is the triplet 417162, 417163 and 417164:

Show that Gauss's discovery that every number is the sum of three or fewer triangular numbers implies that every number of the form 8k + 3 can be expressed as the sum of three odd squares

How slinkies get entangled, and how to untangle them?

To get a better feel for it, get hold of a beaded necklace (like the ones they fling around at Mardi Gras). Lay it out flat and straight. Twist the two strands together at one point, then let go. They'll stay together. If you try to pull them apart, likely nothing will happen; you have to untwist them in exactly the same way, at the same point. Now, do the same thing with a slinky. The reason that undoing it again is harder, is that where they intertwine wanders up and down the slinky as the spirals shift against each other. However, if you pull apart what you can, you'll find that there are a few specific points (or, in the case you're describing, one specific point) where they're actually locked together. Treat each end of the slinky, from the last knot onwards, like a giant bead. Twist them around each other in the right direction, and they'll come right apart.

A conversation between a clinical researcher and a statistician about the meaning of a p-value (this is loosley based on the beginning of an article, "What your statistician never told you about p-values", by Blume and Peipert).

Reference:

Blume, J. and J. F. Peipert (2003). "What your statistician never told you about P-values." J Am Assoc Gynecol Laparosc 10(4): 439-444.

Power of the test, p-values, publication bias and statistical evidence A discussion of statistical evidence and why you might not get results significant enough to reject your null hypothesis even if your alternative hypothesis is correct.

This video is related to the article "Statistics notes: Absence of evidence is not evidence of absence" by Douglas G. Altman and J. Martin Bland. Video created by Matt Asher. For more statistical insights please visit statisticsblog.com

How not to collaborate with a biostatistician. This is what happens when two people are speaking different research languages! My current workplace is nothing like this, but I think most biostatisticians have had some kind of similar experiences like this in the past!

The symbols + and –, referring to addition and subtraction, ﬁrst appeared in 1456 in an unpublished manuscript by the mathematician Johann Regiomontanus (a.k.a. Johann Müller). The plus symbol, as an abbreviation for the Latin et (and), was found earlier in a manuscript dated 1417; however, the downward stroke was not quite vertical.

In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred (1574–1660) in his book Keys to Mathematics, published in London. Incidentally, this Anglican minister is also famous for having invented the slide rule, which was used by generations of scientists and mathematicians. The slide rule’s doom in the mid-1970s, due to the pervasive inﬂux of inexpensive pocket calculators, was rapid and unexpected.

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture”

(Bertrand Russell, Mysticism and Logic, 1918).

"In our private life as in our collective life there is no other truth than a statistical one."

-- Simone de Beauvoir

“We now know that there exist true propositions which we can never formally prove. What about propositions whose proofs require arguments beyond our capabilities? What about propositions whose proofs require millions of pages?

Or a million, million pages? Are there proofs that are possible, but beyondus?”