bonanova

ROTFL I'm with ya ... I don't know who Karl Childers is...

For any non-zero, finite number, say 13, for sake of discussion, then usually: [1] 0/13 = 0 [2] 13/0 = infinity [3] 0/0 = undefined [that is, could be 242, for example] [4] infinity/infinity = undefined [could be 0.327, for example] [5] 0 x infinity = undefined [could be 69, for example] But it turns out that both zero and infinity have different flavors, called orders. There are 1st-order zeros, and 1st-order infinities, as well as 2nd, 3rd-order, etc. of both. In these cases, the higher order predominates: [3a] 0[2]/0[1] = 0; 0[1]/0[2] = infinity [4a] infinity[2]/infinity[1] = infinity; infinity[1]/infinity[2] = 0. [5a] 0[2] x infinity[1] = 0; 0[1] x infinity[2] = infinity. where 0[n] means n-th order 0 and infinity[p] means p-th order infinity. You can see what that means by changing one of these zeros a wee bit, then inching your way back. Let x be a small number - say 1/100. [6] x/x=1. no problem, since x isn't zero. Let x inch its way to zero, the ratio stays at 1. [7] 2x/x = 2. no problem, since x isn't zero [yet]. Let x -> 0, the ratio stays 2. That's why 0/0 is usually thought of as undefined: you could have used any number instead of 2. But look at [8] x^2/x = x. now the ratio is x. Let x ->0, and the ratio [always = x] ->0 as well. What happened? both numerator and denominator -> 0, so the ratio goes to 0/0. How come it's not undefined? How come in this case we know it's zero? The reason is x^2 becomes a 2nd-order zero: 0^2 so to speak; it's stronger than just 0. Same thing about the infinity you get by dividing by x^2 instead of by x as x->0. There you get a stronger infinity - a 2nd-order infinity - infinity^2 so to speak. There is a rule called L'Hopital's rule that gives you these answers for [6] - [8]. Read more about that here: http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule Basically you step away from the divide-by-zero situation and evaluate the division using a legal small number, then see what happens when it approaches zero, and that gives you a defined answer, for that particular case. L'Hopital's rule can make some paradoxical problems intuitive. Absent some expression like [2x/x] ->2, or [x^2/x] ->0, or [x/x^2] -> infinity, you're just left with undefined and you don't try to prove too much with expressions that contain infinities in the numerator or zeros in the denominator.

Interesting. 0 does not equal 0. But let's keep this a secret, OK? Word gets out, imagine the recall of all those text books. Would be devastating.

The inscriptions on Gold and Silver agree, so they must have the same truth value. By conditions of the problem, Lead has the opposite truth value. Gold and Silver False means [1] Ring is in Silver [2] Lead is true -> Ring is in Lead. Contradiction Gold and Silver True means [1] Ring is in Gold or Lead [2] Lead is false -> Ring is not in Lead. Ring is in gold box.

That's it. Here's my method: [care to share yours?]

That's it.

Considering only positive integers, 7 is unambiguously specified by the phrase the smallest number not specifiable using fewer than two syllables. What is the smallest number not specifiable using fewer than twenty-three syllables?

I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. It's not a small number, but it's not really big, either. When I looked for a smaller number with this property I couldn't find one. Can you find it?

A prime, N, is a number with no factors other than [1,N]. Can the arithmetic mean of two consecutive prime numbers be itself a prime number?

You can pose this puzzle with 10 bulbs instead of with 100. But then it's too easy to grind it out; boring. You can also pose it with 1000 bulbs. But life is short, and people with any sense walk away, and miss the fun. Michlips appears to be in that [sensible <!-- s;) --><!-- s;) --> ] group. The whole idea is to find the short cut; and grinding it out partially can light the way. ergo, 100 bulbs is only moderately complex and sucks you in to the search. KamZhiYhi took the first step and realized bulb #1 would be on - partial credit. unreality went farther, saw the pattern, and found the answer. Here's another way to see the solution.

In a room are 100 light bulbs, numbered 1 to 100. Outside the room are 100 switches, numbered 1 to 100. Each switch controls its same-numbered bulb. Initially, all the switches are off. 100 people, numbered 1 to 100, are asked to flip some or all of the switches. "Flip" means "change its state" - if it's on, turn it off; if it's off, turn it on. Person 1 must flip every switch: 1, 2, 3, ..., 97, 98, 99, 100. Person 2 must flip every 2nd switch: 2, 4, 6, ..., 96, 98, 100. Person 3 must flip every 3rd switch: 3, 6, 9, ..., 93, 96, 99. ------ Person 15 must flip every 15th switch: 15, 30, 45, 60, 75, 90. ------ Person 99 must flip every 99th switch: 99. Person 100 must flip every 100th switch: 100. When each person has flipped his/her assigned switches, s/he leaves. Finally, you enter the room to check the bulbs. Assuming the 100 people did as they were instructed: [1] are any bulbs lit? [2] if not, why not? [3] if so, which one?

Not true. Quit talking about my friends like that....

[1] It is truth. Everyone lies. If you recast the question as: If somebody says about himself, that he is lying, as he speaks, is it truth or lie? [2] It is a contradiction. Here's why: [a.] every declarative sentence, say "I am 66 years old." carries the implicit assertion "It is true that ... I am 66 years old." equivalently, "I speak the truth when I say that ... I am 66 years old.". [b.] To pair that assertion with the assertion "I am lying as I speak" transforms the statement into the contradiction "I speak the truth when I tell you that ... I lie as I speak." This need not be viewed as a paradox. But it is inescapably a contradiction, of the same form, for example, as the statement "It is raining heavily as I tell you that ... the skies are perfectly clear."

Maybe, maybe not. Onelook.com says ... Quick definitions (slice) # noun: a golf shot that curves to the right for a right-handed golfer # noun: a spatula for spreading paint or ink # noun: a thin flat piece cut off of some object -- maybe # noun: a share of something (Example: "A slice of the company's revenue") # noun: a serving that has been cut from a larger portion (Example: "A slice of bread") -- maybe # noun: a wound made by cutting # verb: hit a ball so that it causes a backspin # verb: cut into slices (Example: "Slice the salami, please") -- maybe # verb: hit a ball and put a spin on it so that it travels in a different direction # verb: make a clean cut through -- maybe not. This does not imply "straight." # name: A surname (very rare: popularity rank in the U.S.: #34695)

Thank you, my friend -- I have seen the light. You've taught me well. If you'll excuse me I have a few things to do... [1] Go tell sajow4 she was right. [2]Go to Morty's and flip some coins tonight. [3] Finally -and this is the big one - I'm going to buy a lotto ticket.

The detective arrived on the scene and discovered the following: George was dead. He was found face down in the street, squashed nearly flat. He must have been there some time as he was cool to the touch. In addition, when he was found, there was a bird perched on the back of his head. What was George's last name?

Sounds good. O wait. I'll be at the Yankees game.

A cup of coffee next time yer in New York for anyone who can tell me what this is all about. Hint: reading it may put you to sleep, but it's not meant as a cure for insomnia...

For all practical purposes. Here's why that makes sense. On your first fold, you're folding a single sheet On your second fold, 2 sheets On your third fold, 4 sheets ----- ----- On your seventh fold, it's 64 sheets. If you want to try for an eighth fold, it would be 128 sheets. Try folding pages 1-256 of a book [numbers on both sides remember].

Alex wins. By a lot. Here's why:

*slaps forehead* thanks ... !

Absolutely correct. [1] If no cup is removed, a particular pair of remaining cups match colors 12 of 26 times. [2] If a red cup is removed, a particular pair of remaining cups match colors 7 of 15 times. [3] If a green cup is removed, a particular pair of remaining cups match colors 5 of 11 times. [4] If we remove a cup 26 times [say cup 1] a particular pair of remaining cups match colors 12 of 26 times. Choose your interpretation of UR's 2nd question and you have your answer. In the Monty Hall Problem you are able to take guidance from the added knowledge. You can deduce that you'll win 2/3 of the time if you swap and 1/3 if you don't. If you think UR's 2nd question is like the MH problem, then let's give it a try: You pick a cup, say it's Cup 1, and I tell you it's green. Based on your knowledge that Cup 1 was green, which of your 6 options are you guided to choose? Cups 2 and 3 Cups 2 and 4 Cups 2 and 5 Cups 3 and 4 Cups 3 and 5 Cups 4 and 5

Assuming that two halves make a whole ...

Right. When two elastic bodies collide, both energy and momentum are conserved. The fly was anything but elastic, becoming and remaining a simple grease spot, so in this case only momentum is conserved. Simply put, [Mass of fly] x [velocity change of fly] = [mass of train] x [velocity change of train] Let's say for argument's sake the train is ten times more massive than the fly. [That would be one big, m......f........ fly, by the way...] That gives you ... canceling mass of fly from both sides ... velocity change of fly = 10 x velocity change of train. the fly's velocity changed from +10 to -60 = 70 mph that means the train's velocity changed by 70/10 = 7 mph. The train continues moving forward, slowing from 60 mph to 53 mph. Even a fly weighing 1/10 the weight of the train won't stop it. Here's an interesting point. A detailed analysis of what happened to the fly reveals that it never actually came to rest! The first miniscule portion of the fly to contact the train instantaneously assumed the train's speed, producing a progressive collapse and deformation of the fly into a grease spot. The rear portions of the fly were still moving forward while its forward parts were traveling backwards. As a whole, the fly never had zero velocity, and thus technically never stopped.