bonanova

Clubs [lowest] Diamonds, Hearts, Spades [highest]. Or just assume they're numbered 1-52. No ties. My bad: rank was the wrong term to use.

I shuffle a deck of cards and deal them into two piles. . The first card I place in front of me If the second card is lower in rank, I start a pile to the left of the first; otherwise to the right. Now I have two piles [one card in each pile, with the lower ranked card on the left.] I continue, until they are all dealt, as follows: If a card is lower in rank than the previous card, I place it on the left pile. If a card is higher in rank than the previous card, I place it on the right pile. . When I'm done, the left pile will tend to have lower ranked cards, and the right pile will tend to have higher ranked cards. But not completely: a J that follows a K will be in the left pile and a 5 that follows a 3 will be in the right pile. What is the expected highest ranking card in the left pile? Of the lowest ranking card in the right pile? To make the rankings clear, I numbered the cards: from 1 [Club deuce] to 52 [spade Ace]. You can give the answer as a number or card, as you like.

Be careful not to assume a value for p - only that it's in [0, 1]. That is, the Thinkers might be bad thinkers. This is a special trial, so the jury can be any size. If you like, let it be the choice of a three-person jury or no jury at all and the judge decides. The judge, by the way is a Thinker and also has a probability p of getting it right.

You're suing for damages and have a slam dunk case on the evidence. So you want a jury that will get it right. The jury pool comprises Thinkers and Gamblers. All Thinkers have a probability p of getting the right answer. All Gamblers flip a fair coin. You have a choice of juries: Two Thinkers and one Gambler. You need two votes out of the three.One Thinker. You need his vote. Which jury do you pick? This is a deceptively easy problem, and you should be able to do it in your head. Have fun.

Actually I was thinking of the interleaved digits approach. It demonstrates the 1-1 correspondence most clearly I think. Even though it is monstrously discontinuous. Aside: (line segment cardinality = ray cardinality) I don't have photoshop on this machine, but this is an easy sketch to make. Draw the point P at (-1, 1) Draw the unit line segment from (0,1) to the origin. Rays drawn from P through the line segment give 1-1 correspondences of its points to the points in the ray which is the positive x-axis.

Simpler than multiplication / division is to just count: 2 4 6 8 10 12 14 16 ... 1 2 3 4 5 6 7 8 ... [/code] Now does anyone care to place the real numbers in a 1-1 correspondence with the points in a unit square, or infinite 3-space? Assuming we accept they both are all Aleph[sub]1[/sub]. [/font]

You can count things like the people in a room, apples in a basket and letters in this sentence. These are examples of finite sets - their cardinality is just the number of objects they contain. If a dinner party has 13 guests, and there are 13 place settings at the table, we can say the cardinality of the guests is the same as the cardinality of the dinner plates or of the soup spoons. When the guests are seated, each guest has a plate and a soup spoon. If one of the guests does not show up, the cardinalities would then differ. You can also count the positive integers. In fact you use the positive integers to do the counting, so they're sometimes called the counting numbers. There are an infinite number of them. So their cardinality can't be a number like 43 or 125. So we give this cardinality a symbol instead: Alepho Any collection [set] whose elements can be placed in a one-to-one correspondence to the counting numbers has the cardinality Alepho We call such collections countably infinite. That might sound like a contradiction. Other collections are infinite in "number" [cardinality] but are not countable. An example is the real numbers. Their cardinality is given the symbol Aleph1. They cannot be placed in a 1-1 correspondence to the counting numbers. Other sets have even higher cardinality. So we've mentioned four classes of sets: . finite sets, the positive integers [countable] Alepho the real numbers [uncountable] Aleph1 higher infinities Aleph2... .Just for fun, classify the following ten sets by cardinality: 1, 2, 3 or 4. . UnicornsNegative integers Odd integersNeurons in your brain Rational numbers [p/q where p and q are integers] Atoms in the universeFunctions of real numbers Points in the line segment [0, 1]Points in the square having diagonally opposite corners [0,0] and [1,1]Points in infinite 3-space. .Remember, if you can construct a 1-1 correspondence, you can say two sets have the same cardinality. And here's a clue - the list has at least one of each type.

If that were true, the space program could be run without resorting to rockets. What a fuel savings! It requires energy to lift an object at rest above the earth's surface.

More a humorous story than a joke. Caveat: the truth of the story has not been confirmed. Yesterday I was at my local Costco buying a large bag of Purina dog chow for my loyal pet, Shelby, the Wonder Dog. I was in the checkout line when a woman behind me asked if I had a dog. What did she think I had, an elephant? So since I'm retired and have little to do, on impulse I told her that no, I didn't have a dog, I was starting the Purina Diet again. I added that I probably shouldn't, because I ended up in the hospital last time, but that I'd lost 50 pounds before I awakened in an intensive care ward with tubes coming out of most of my orifices and IV's in both arms. I told her that it was essentially a perfect diet and that the way that it works is to load your pants pockets with Purina nuggets and simply eat one or two every time you feel hungry. The food is nutritionally complete so it works well and I was going to try it again. (I have to mention here that practically everyone in line was now enthralled with my story.) Horrified, she asked if I ended up in intensive care because the dog food poisoned me. I told her no, I stepped off a curb to sniff out an Irish Setter, and a car hit us both. I thought the guy behind her was going to have a heart attack from laughing. Costco won't let me shop there anymore..

Gorgeous pics, araver. Bravo! They have names, actually, following the guy who first publicized them. Wikipedia has a nice animation here. Along with the math that proves they're circles, altho it could be hazardous to your health. Aside: some other relevant and pretty pics.

The relative size of the triangle and sphere determines the angles. More precisely, the solid angle subtended by the triangle is what matters. On a sphere the size of the earth or the size of a marble, either one, a triangle with vertices all on a great circle has a 540o sum, for both the internal and external angles.

Johnathan, Thank you.

In geometry, "regular" has a particular meaning with regard to polygons: equal length sides and angles. With that meaning, for example, regular rectangle refers to a square. . There is a geometry of the plane, denoted as plane geometry or Euclidean geometry. The interior angles of all Plane triangles, regular or not, sum to 180o. . There is a geometry of the sphere denoted as spherical geometry. The interior angles of Spherical triangles sum to a value between 180o and 540o. . There is also a hyperbolic geometry. The interior angles of Hyperbolic triangle sum to a positive value less than 180o. . The OP did not specify that the triangle was be a plane triangle, nor could it, sensibly. As peace*out notes, on the surface of a sphere no plane figure can be drawn.

If 3 is false, was it wrongly derived from 2? Or is 2 false also? Of the five statements, which is the last true one?

Not quite. sqrt[1] = 1 or -1. It's always real. Check your answer by squaring it. [-i]2 is not 1.

Actually, they're the same. Let x = 1/i Multiply both sides by -i2, which is 1. x = -i

Don't blink or you'll miss it. 1. sqrt[-1] = sqrt[-1] 2. sqrt[-1/1] = sqrt[1/-1] 3. sqrt[-1]/sqrt[1] = sqrt[1]/sqrt[-1] 4. sqrt[-1] x sqrt[-1] = sqrt[1] x sqrt[1] 5. -1 = 1 [/code] You know the drill; prove me wrong. [/font]

I've been thinking about your answer. Yes there are two [fundamentally] different ways to make a square. I hadn't seen that. The wording of the OP lumps them together, as you rightly conclude. We're looking for ways that a slice makes different polygons. Sorry not to reply sooner.