Math puzzles!

"Since each face has a one in six chance of coming up, when three dice are tossed there are therefore 3 in 6 chances that the number I bet on will appear... and if my number comes up more than once... I win even more! So it must be a good game to play."

ex: you bet $1 on 6.
Triple sixes occur. you get your dollar back and win 3 dollars on top of that.
double sixes occur. you get your dollar back and win 2 dollars on top of that.
single six occurs. you get your dollar back and win 1 dollar on top of that.
no sixes occur. You lose your dollar.


Is it? Your job is to work out the true odds of this game!

Going back to this one, cause it bothered me that nobody solved it this way...
For each correct die, you win $1.
so 1/6*$1 + 1/6*$1 + 1/6*$1 = $0.50 per game
But: If none of the dice are right, you lose $1
so 5/6*5/6*5/6*$-1 = 125/216*$-1 = $-0.5787 per game
Add them up: $-0.0787 per game, or $0.9213 return on $1 investment.

Going back to this one, cause it bothered me that nobody solved it this way...
For each correct die, you win $1.
so 1/6*$1 + 1/6*$1 + 1/6*$1 = $0.50 per game
But: If none of the dice are right, you lose $1
so 5/6*5/6*5/6*$-1 = 125/216*$-1 = $-0.5787 per game
Add them up: $-0.0787 per game, or $0.9213 return on $1 investment.

Quite the nice solution to that problem

http://www.wolframalpha.com/input/?i=ave...+to+sun%29

Well, that was more exact than I thought it would be.

http://www.wolframalpha.com/input/?i=ave...+to+sun%29

Well, that was more exact than I thought it would be.

My thoughts exactly.

Now noah, how 'bout you give us a nice approximation :3

 Spoiler

Not creative at all.. but I love the Gaussian Integral. (:

Here, howsabout I give you guys a nice example....

It's one of my favorites:

pi: ln(640320^3 + 744)/sqrt(163)

My image of this borked D:


That is approximately pi.. and is accurate to roughly 30 places I think. :3

 Spoiler

This has gotten me very intrigued. How does that work out? I'm a little sad that I can't figure out what's wrong with the logic.

This has gotten me very intrigued. How does that work out? I'm a little sad that I can't figure out what's wrong with the logic.

You can calculate the difference in the perimeters at each step... that sequence doesn't converge to zero. It's just the sequence: 4-pi, 4-pi, 4-pi, 4-pi..... so the error is always 4-pi even after you "repeat to infinity". Likewise the process from the 4th to the 5th panel is technically impossible. It'll never be smooth enough using that method. So you can never say that they are equal, even in the limit. It's not like the n-side polygon inside of a circle trick.... where you let n go to infinity and you get the circle. That one DOES smooth out enough.... but it's really hard to explain without getting into slightly complicated math.

You can calculate the difference in the perimeters at each step... that sequence doesn't converge to zero. It's just the sequence: 4-pi, 4-pi, 4-pi, 4-pi..... so the error is always 4-pi even after you "repeat to infinity". Likewise the process from the 4th to the 5th panel is technically impossible. It'll never be smooth enough using that method. So you can never say that they are equal, even in the limit. It's not like the n-side polygon inside of a circle trick.... where you let n go to infinity and you get the circle. That one DOES smooth out enough.... but it's really hard to explain without getting into slightly complicated math.

which is why this troll was so beautiful, to the point it had to be "solved" by majors at math, so that everyone would stop flipping their shits.

which is why this troll was so beautiful, to the point it had to be "solved" by majors at math, so that everyone would stop flipping their pomegranates.

The "smoothness" is the only tough part to explain. The sequence is pretty easy to explain and suffices to disprove it...

but it's still such a wonderful troll x3



L> MOAR approximations to pi

14) And in the mean time, let's have a bit more fun with pi. My friend (let's call him Mike) has a very unique social security number... 123-45-6789. Quite fun. I told him his social security number is actually found in the decimal expansion of pi. In fact, I can prove it... can you?

Assumptions/facts we need:
a)pi goes on forever
b)pi is normal... never repeats and is, for all intents and purposes, truly random.

The "smoothness" is the only tough part to explain. The sequence is pretty easy to explain and suffices to disprove it...

I don't think it's sufficient, you can't assume from the start what the perimeter is, because it's what you're trying to solve. If pi = 4, then the perimeter of the circle is 2*pi*r = 4.

This assumption is IMO also a problem for the limits of sin/cos, they're defined based on a value of pi, so of course they'll spit out pi if you take sin^-1(1) in a program that knows sin(pi)=1.

please don't tell me the troll math took over the thread.

If pi is infinite and is truly random and doesn't repeat, then statistically any finite string of numbers will appear since there are an infinite number of possibilities. If it's infinite, everything will happen sooner or later.

If pi is infinite and is truly random and doesn't repeat, then statistically any finite string of numbers will appear since there are an infinite number of possibilities. If it's infinite, everything will happen sooner or later.

Good explanation, kinda.... but not a proof :3

If pi is infinite and is truly random and doesn't repeat, then statistically any finite string of numbers will appear since there are an infinite number of possibilities. If it's infinite, everything will happen sooner or later.

Good explanation, kinda.... but not a proof :3

Not 100% sure if this is proof but here it goes,
Given a random and infinite string of numbers N. Let's divide N into an infinite number of 9-digit numbers, let's call them Xi, where i from from 1 to infinite.
We now have an infinite number of 9-digit number strings that are all random. Since each string is random, and we have an infinite number of them we have a probability of 1 of at least one being 123456789. As a matter of fact we have an infinite number of 123456789 strings, but that's another story.

Not 100% sure if this is proof but here it goes,
Given a random and infinite string of numbers N. Let's divide N into an infinite number of 9-digit numbers, let's call them Xi, where i from from 1 to infinite.
We now have an infinite number of 9-digit number strings that are all random. Since each string is random, and we have an infinite number of them we have a probability of 1 of at least one being 123456789. As a matter of fact we have an infinite number of 123456789 strings, but that's another story.

You're getting at it... but you have to explain WHY we have a probability of 1... not just "because there's an infinite amount of them"


Here's how (using your notation):

The probability of any Xi being 123456789 is 1/(1,000,000,000). So the probability that it isn't 123456789 is 999,999,999/1,000,000,000.

The probability that they ALL aren't Xi is:

limit x->inf (999,999,999/1,000,000,000)^x

Which, since the fraction is less than 1, is equal to zero.

Thus the probability that at least 1 of them is 123456789 is 1.


You pretty much had it.