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#426 Re: Puzzles and Games » Add 1, then post a letter. » 2013-04-20 00:44:23
Actually 1 is a triangular number.
2 P
#428 Re: Help Me ! » Sum of roots » 2013-04-16 02:17:13
Since, √432 = 12√3, therefore the pairs are 0 + (12√3)^2, (√3)^2 + (11√3)^2, (2√3)^2 + (9√3)^2, and so on. So a total of 12 ordered pairs.
Wont there actually be 13 ordered pairs (because of the 0)?
#429 Re: Dark Discussions at Cafe Infinity » 257 Questions » 2013-04-16 00:38:01
What happens when you run out of those 257 questions? 
#430 Re: Puzzles and Games » Nehushtans challenge problems » 2013-04-16 00:06:50
Splitting into cases is tedious. The short and sweet solution for #1 is
It looks like no-one is going to get #2 so I might as well post its solution as well.
New challenge problem:
#431 Re: Puzzles and Games » Nehushtans challenge problems » 2013-04-15 02:22:03
Hi gAr.
You apparently assumed , , to be positive. The result has to work for all real numbers, not just positive ones.
#432 Re: This is Cool » Julianthemath's number 2 » 2013-04-14 16:34:48
Isnt this simply a[sub]n[/sub] = 2n[sup]2[/sup] ?
#433 Re: Help Me ! » Cookies! » 2013-04-14 02:50:22
I just have to convince Agnishom and he can be a toughie.
I thought you were going to say he could be a tough cookie.
#434 Re: Help Me ! » Cookies! » 2013-04-14 02:38:53
Lets rephrase the original cookies question in a different way. Suppose the 14 cookies are lined up in a row. What is the probability that the 7th cookie from the left is a chocolate-chip one? 
#435 Re: Help Me ! » Cookies! » 2013-04-14 01:22:52
Because it is a trick question and using a computational idea on it just shows how stupid I am.
Youre not alone. I was tricked as well. 
#436 Re: Puzzles and Games » 5 Pirates Puzzles » 2013-04-14 00:24:41
Im glad the other thread was closed because it was totally pointless. I suggest this thread be closed as well.
[comment removed by administrator]
#437 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 19:02:02
You have once again changed the definition of rational for the puzzle's sake. Offering one gold in Collin's position is not a rational means of saving one's life when confronted with bloodthirsty pirates.
I have changed nothing. You are the one who is having the obsession about Colins death wish which has nothing to do with the puzzle. Why isnt it rational for Eddie to accept 1 gold when otherwise he would have no gold? Not only is it rational of Eddie, it is rational of Colin too. Colin would like to keep all 100 gold but he cant hell be outvoted and thrown overboard. However, by offering 1 gold to Eddie, he can ensure he wont be outvoted and thus save his life.
Because you have changed the definition of greed for your own sake again. The definition of greed should not change because we are in the context of a puzzle.
Change what definition? Greedy here simply means taking as much as one can get even if it means taking only 1 gold as opposed to nothing.
Again, the puzzle set the stage, and the pirates are described as bloodthirsty.
Only they are more greedy than bloodthirsty whereas you seem hellbent on thinking its the other way round.
The title was chosen to stimulate intelligent debate.
The title gives the impression that you think you are right and everybody else is wrong. That is not the way to stimulate intelligent debate! 
#438 Re: Puzzles and Games » 5 Pirates Puzzles » 2013-04-13 18:34:06
Hi.
Regarding this bit:
As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.
I would suggest adding that if a pirate would get less coins by voting against a proposal, then he would not vote against it. This is just for greater clarification of the puzzle.
#439 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 17:46:24
And likewise, why is it rational? Why does the definition of the word "rational" change because we are in the context of a puzzle?
It is rational because accepting 1 gold is better than accepting none.
Are you saying that Eddie is greedy by accepting 1 gold coin? Is that greedy?
Yes. Why shouldnt it be?
However, what is more relevant to the puzzle is Colins greed. Colin would like to keep all 100 gold to himself. But he cant hell be outvoted by Duncan and Eddie. His next best option is to keep 99 and give Eddie 1 gold. And Eddie would prefer 1 gold to no gold because hes greedy.
And, tell me, by the parameters in the puzzle, how you can say that Eddie's "greed" of accepting 1 gold coin will outweigh his bloodthirstiness?
Read this rule again:
As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.
The pirates are bloodthirsty only in so far as their greed is not satisfied. If by being bloodthirsty they get less instead of more (as would happen to Eddie if he voted against Colins proposal) then they wont be bloodthirsty, preferring to be greedy instead.
#440 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 17:03:06
Again, it says that the pirates are rational and intelligent. No specific definitions of those words were given. We all know that in real life, it would be completely irrational to offer 1 gold coin in that position and expect a greedy, bloodthirsty pirate to accept the offer.
Why is it irrational IN A PUZZLE to have a pirate accept 1 gold because otherwise he would have nothing at all? If this situation offends your sensibility, you can always assume that later on, the pirate having the most gold gets murdered in his sleep, the other pirates start squabbling over the gold, etc. Do what you like with your imagination outside the puzzle. However, when you are in the puzzle itself, you go by the rules of the puzzle. You dont go making up your own.
#441 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 16:41:04
Is has been my contention from the start that case 3 doesn't matter, because the puzzle starts and ends at case 5.
As I said, you have to understand the three-pirate case in order to figure out the five-pirate case.
I don't agree that it is rational or intelligent to believe that an offer of 1 coin is suitable for guaranteeing that you will stay alive.
That has nothing to do with the rules of the puzzle. Its just an opinion.
Again, parameters of the puzzle.
Which you dont seem to be able to understand.
#442 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 16:28:35
Again, it is irrational to assume that a greedy, bloodthirsty pirate would accept 1 gold coin.
It is also irrational to assume that you would live with such a pittance.
This is NOT stated in the puzzle. This is something you made up yourself. When you play a game, you follow the rules of the game. You DO NOT make up your own rules! 
Okay, look at it this way. Colin reasons in the following line.
1. Suppose he keeps all gold and gives nothing to Duncan and Eddie. Then both will vote no and he will die. So this is no good.
2. Suppose he takes 99 gold and gives 1 gold to one of the the other two. Should he give to Duncan or Eddie? Suppose Duncan. Duncan will vote no, obviously. How will Eddie vote? Eddie gets nothing here. If Colin dies and Duncan proposes, he will get nothing too. Either way he will get nothing. Therefore he will vote no because of the following rule:
if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against
So this is no good to Colin either.
3. Suppose Colin takes 99 gold and gives 1 gold to Eddie. Duncan will vote no as before, but what matters is Eddies vote. If Colin dies Eddie will get no gold so Eddie has to agree to Colins proposal and accept 1 gold rather than no gold. So this works and is the scenario for the three-pirate case.
Again, the puzzle starts with 5 pirates, not 3. Case 3 is irrelevant.
Case 5 does NOT depend on case 3. Case 5 comes first.
In order to understand the five-pirate case, you have to understand the three-pirate case first.
#443 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 16:01:40
The parameters of the puzzle are not only to maximize your profits, but to stay alive as well.
Exactly. Colin KNOWS that he will NOT die because he knows that Eddie will NOT vote against his proposal. That is why he gives Eddie 1 gold. (If Eddie does not like it and gets rid of Colin he will get nothing; therefore Eddie votes to keep Colin alive and pocket the 1 gold.)
#444 Re: Puzzles and Games » 5 Pirates Solution is wrong - here is the real solution » 2013-04-13 15:20:53
anonimnystefy wrote:Let's talk about the 3 case first, because that is where the disagreement begins.
If you were Eddie and I was Collin, and I proposed to you the 99:0:1 deal, would you accept (knowing that you will get nothing if I die)?
No I wouldn't, because as Eddie, I know I am in control and can do much better for myself.
No, Eddie most certainly CANNOT do better. Under Colins proposal Eddie will get 1 gold. If Eddie does not accept, then he will end up with nothing because Colin will die and Duncan will propose to keep all the gold for himself. No matter what Eddie does, there is no way he can get more than 1 gold. Since 1 gold is better than no gold, Eddie is forced to vote for Colins split.
#445 Re: Help Me ! » vector » 2013-04-13 07:58:45
Then the angle between
and is and taking the dot product givesThus you can take any x, y satisfying the last equation and z such that
for example, , , .#446 Re: Help Me ! » Long division help :( » 2013-04-13 07:10:59
Are you sure you want 11 divided by 1012 rather than 1012 divided by 11?
#447 Re: This is Cool » Sum of squares and primes » 2013-04-13 07:05:49
You might this theorem useful: A prime p can be written as the sum of two integer squares if and only if p ≡ 1 (mod 4).
#448 Re: Computer Math » The two trains » 2013-04-13 06:54:21
Do you agree with the algebra answer of 10.8?
10.8 seconds? Yeah, Id say so.
Look at the problem this way. Each train is travelling at 100 km/hour relative to the other and has 300 metres to travel. So the time taken is .
#449 Re: Help Me ! » boundaries of an integration » 2013-04-13 06:43:55
The RHS means: integrate
from 0 to , integrate from 0 to , , integrate from 0 to , and finally integrate from 0 to 1. The limits of each integral come from the way the domain of integration is defined.#450 Re: Help Me ! » can anyone do this? » 2013-04-13 06:27:28
Similarly
.