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## #1 Re: Help Me ! » Any better method than guessing? » 2014-04-08 16:57:53

We have a linear Diophantine equation.
9y - 17 = 11x
y-(17/9) = x + (2x/9)
y-x-1 = (8+2x)/9
Because x and y are integers so is (8+2x)/9.
(8+2x)/9=z
8+2x = 9z
4+x=4z+(z/2)
4+x-4z=(z/2)
z=2w
we have
8+2x=18w
so we have
x=9w - 4
y=11w - 3
So we have 2 solutions x=-4 and y=-3 also x=5 and y=8
To get the other number we subtract y from 11.
so we have 83 as one solution and -38 as another solution also -83 is another solution.

## #2 Re: Help Me ! » how many possibility exist? (discrete mathematics) » 2014-04-08 15:04:42

In the example where the elements are the same, such as {1,1,1,1}, and you have X=3, the number of ways to get X=3 is simply 4C3.  Very interesting problem.

## #3 Re: Dark Discussions at Cafe Infinity » Problems from a book » 2013-10-09 15:35:52

The other problem is this, again 3 people A, B, and C.

A and B working together can prepare a shipment in 8 days.
A and C working together can prepare a shipment in 9 days.
B and C working together can prepare a shipment in 10 days.

How long would C take working alone?
The book says 22 days.  I am getting 23 and 7/31 days.

That is what I am getting also.

Yeah, I don't understand the solution in the book. They have the following equations...

(A + B)/4 = 8
(A + C)/4 = 9
(B + C)/4 = 10

So, A = 14, B = 18, and C =22.
I honestly have no idea what they're doing.

## #4 Re: This is Cool » Generating Functions » 2013-10-08 13:57:08

This is back a few pages..
Now supposing 2 die had faces of 2,4,6,8,10,12 and 2 other die had faces of 1,3,5,7,9,11.
All four die are thrown once. What is the probability of them summing to 20 or more?

Is it 545/648?

## #5 Dark Discussions at Cafe Infinity » Problems from a book » 2013-09-27 16:16:56

Fruityloop
Replies: 6

I have a book which I think has the wrong answers to 2 problems.

In the first problem there are 3 people, we'll call them A, B, and C.

A can make 5 scarves while C makes 2.
B makes 4 scarves while A makes 3.

A's scarf takes 5 times as much cloth as B's scarf.
Three of B's scarves take as much cloth as 5 of C's scarves.

C's scarves are 4 times as warm as B's scarves.
A's scarves are 3 times the C's scarves.

Who's the best overall?
Interestingly, I have 2 books which have this exact same problem, one book is from 1885 and the other book
is from 2013.  I believe that the book from 1885 has the correct answer, it claims that the best overall is C.
The book from 2013 claims the answer is A.  The book from 2013 makes the mistake (which is pointed out
by the older book) of adding the numbers once the proportions are established, when they should be multiplied instead.

The other problem is this, again 3 people A, B, and C.

A and B working together can prepare a shipment in 8 days.
A and C working together can prepare a shipment in 9 days.
B and C working together can prepare a shipment in 10 days.

How long would C take working alone?
The book says 22 days.  I am getting 23 and 7/31 days.

## #6 Re: Help Me ! » Prisms and Pyramids » 2013-09-21 15:52:20

16 x 1.5^2 != 24^2, 16^2 x 1.5^2 = 24^2.
Try moving the 1.5^2 to the right side of the equation and see what you can do.

## #7 Re: Dark Discussions at Cafe Infinity » average number of years since new record. » 2013-09-19 16:18:43

Thanks for the responses.  I was assuming an unchanging climate in my problem.
I agree that with the ever-increasing temperatures, it is becoming more and more likely to set new higher temperature records.
My idea was to figure out what would be an average number of years since a new temperature record was set and then
take an average for all 365 days from a weather station and do a comparison.  When I have time maybe I will do that.

## #8 Re: Help Me ! » Prisms and Pyramids » 2013-09-18 14:06:54

It seems you get two equations with two variables.
One equation for the lateral area has two variables, say x and y.
One equation for the volume is another equation with x and y.

5. the base is an isosceles triangle with a height of  8 and a base of 3 (sides of 6)
I'm a little unclear with this.

## #9 Re: Dark Discussions at Cafe Infinity » average number of years since new record. » 2013-09-18 13:53:28

Each year has an equal probability of being the record-holder.  There are N years.  So for each year the probability of it being the record-holder is 1/N.  Let's say you have N balls in a bag, they are numbered 1 to N.  You reach into the bag and pull out a ball at random and without looking at the number on the ball, you write a random number between 1 and 1500 on the opposite side of the ball.  You do this for all of the N balls.  Which numbered ball will have the highest number?  It is easy to see that each ball has an equal chance of having the highest number.

## #10 Re: Dark Discussions at Cafe Infinity » average number of years since new record. » 2013-09-18 09:33:28

I feel like such a ding-dong.  I think I know the answer now.
Let's say that there are records going back N years.  The probability of each year being the one that holds the record is 1/N.
So the average number of years since the latest record (which is the record overall of course) is

I generated 65 random numbers between 1 and 1300 inclusive and did this 50 times.
The average number of numbers since the all-time high number was 30.28 which is close to 33 which is what we would expect.
Let's say you have temperature records going back 130 years, the average number of years since the all-time high
record would be 65.5.  To me this seems counter-intuitive because it seems like later years would be more likely to hold
the all-time high records.

## #11 Re: Dark Discussions at Cafe Infinity » average number of years since new record. » 2013-09-11 14:58:57

Thank you for your response Bobbym.
This is kind of a strange problem because we are dealing with something varying around an average that is unknown.
True, the setting of 50% is somewhat arbitrary, but the first year that there is a record we assume that the rainfall (or temperature)
record is just as likely to be above as below the average, since we don't know what the average is.

because each of the following 4 years has 50% chance of breaking it or not.

I'm comparing the following 4 years only to the first year.
There are 2 possible sequences for the first 2 years, with only one sequence having the 2nd year setting a new record.  So we have a probability of 1/2 for the 2nd year setting a new record.  Similar reasoning shows that the probability of the 3rd year setting a new record is 1/3 and so on...

## #12 Dark Discussions at Cafe Infinity » average number of years since new record. » 2013-09-10 15:43:48

Fruityloop
Replies: 9

I was thinking of a math problem that I've never seen before.
Let's say you have rainfall (or temperature) records at a certain location going back 80 years.
On average, how many years will it have been since the last record has been set?
I tried doing the math for 5 years and I'm not certain if it's correct or not.

My reasoning is as follows...
The probability of the first year record still standing after 5 years is
because each of the following 4 years has 50% chance of breaking it or not.
The probability of the second year setting a new record is
and the probability of it still standing after the following 3 years is

And so on...
Is this correct or not?

## #13 Dark Discussions at Cafe Infinity » 12 year-old genius » 2013-09-05 16:48:32

Fruityloop
Replies: 17

This 12 year-old is asked to prove the convergence of the following...

Just looking at this.. since sin and cos vary between -1 and 1 there doesn't seem to be any way for this series to converge.
Any thoughts?

## #14 Re: Help Me ! » Probability! » 2013-09-05 16:21:59

I think the answer is 13/14.
(5C4*3C0 + 5C3*3C1 + 5C2*3C2)/8C4 = 13/14.

## #16 Re: Dark Discussions at Cafe Infinity » Polar bear dies from starvation. » 2013-08-09 16:19:41

I agree with you bobbym.  I think humans will be extinct by 2050.  Just do a google search for the record heat that is going on around the world.  People don't realize what is coming.  We have feedback mechanisms that are coming into play now.  Maybe it will finally dawn on people that we are in serious trouble.  But then again, maybe not.
Take a look...
http://au.ibtimes.com/articles/497854/2 … gXAqSTn_cs
We have an explosion in the world population causing an explosion in CO2 emission.

## #17 Dark Discussions at Cafe Infinity » Polar bear dies from starvation. » 2013-08-08 16:22:41

Fruityloop
Replies: 5

http://www.huffingtonpost.com/2013/08/0 … 20236.html

I want everybody in the world to see these pictures.
This is heartbreaking.
We are going to see more and more of this in the future.
There seems to be no intention on the part of mankind to reduce
either our numbers or our emission of CO2.
What is especially scary is that the methane that is locked up in ice beneath the ocean is
starting to come out.  Look up methane hydrates.
The future doesn't look good.

## #18 Dark Discussions at Cafe Infinity » Difference between cents and dollars » 2013-02-07 14:56:28

Fruityloop
Replies: 4

Holy smokes!

http://verizonmath.blogspot.com/2007/08 … tomer.html

Unbelievable.

This reminds me of the time when I was playing blackjack and they payed 2 to 1 on a suited blackjack.
I received a suited blackjack and the dealer asked me if I wanted even money.
I tried explaining that because the payoff was 2 to 1 they couldn't just give even money.
The pit-boss came over and agreed with the dealer.
Sometimes people just don't get it.

## #19 Dark Discussions at Cafe Infinity » Red light runner partially responsible for accident??? » 2013-01-28 22:58:53

Fruityloop
Replies: 3

Take a look at this...

www.statesman.com/news/news/local/jury- … age/nRgqf/

Maybe somebody can explain the logic behind this....

If I run a red light and collide with another vehicle..
If the other vehicle is below the speed limit I'm 100% responsible.
If the other vehicle is speeding I'm partially at fault.

I don't see what relevance the speed of the other vehicle has with the cause of
the accident.

This is like saying somebody is at fault for an accident because they weren't
wearing their seatbelt.

## #20 Dark Discussions at Cafe Infinity » Almost the greatest, but not quite... » 2013-01-25 13:05:00

Fruityloop
Replies: 1

We always remember the best and brightest and the ones who are almost as great are forgotten about.
Now imagine you were the poor soul who came up with

That's almost as bad as coming up with

or maybe

http://www.sciencedaily.com/releases/20 … 103931.htm

## #21 Re: Puzzles and Games » Evaluate this integral » 2012-07-24 16:15:42

I enjoyed looking at your answer Bobbym, but I don't understand how you got

from

I'm not certain if this answer is correct, but it's the same answer from Mathematica.

## #22 Puzzles and Games » Evaluate this integral » 2012-07-21 15:46:41

Fruityloop
Replies: 54

This integral comes from trying to get the surface area of y=1/x rotated about the x-axis.
evaluate the integral

Good luck! It's a toughie.

## #23 Re: Help Me ! » Nice Geometry Problem! » 2012-05-23 19:34:24

I used the idea from gurthbruins to solve this problem.
This is a very hard problem.  I had printed out the problem on a sheet of
paper and decided to try to solve it.  First, we draw a straight line connecting
points A and M.  The point where this line intersects XY we'll call point Z.
We'll make another point called G which is the mid-point of AB.  The point of
tangency for the big circle we'll call W.  The center of the hexagon is the origin.
Notice that

We need to know what the coordinates of Z are.  The x-coordinate is

so the coordinates of point Z are

So we must solve the linear equation

simultaneously with the equations
in order to get the
coordinates of X and Y. Many, many steps later one ends of with
the x and y coordinates of ponts X and Y as follows...

using
to get the distance of XY we finally end
up with after many more steps.. drum roll please.....
as distance of XY.
Then afterwards be sure to take some Advil for the headache and
a tissue to wipe the blood now oozing from your eyes.

## #24 Dark Discussions at Cafe Infinity » Proportionality... » 2012-02-15 06:19:27

Fruityloop
Replies: 1

Hello everybody!  I haven't posted in a long time.  But anyways...
I was working on this problem called 'The fighting fishes of Siam' by the late, great, Sam Loyd and it states that
there are two kinds of fishes, king fish and devil fish.  They inevitably attack each other on sight.  Three devil fish
counterbalance one king fish.  Four devil fish can kill a king fish in 3 minutes with each additional fish making the new group
proportionately quicker. So I did the following... 4*3 = 12, 5*x=12, so x=2.4 or 144 seconds for 5 fish.
But the solution in the back of the book says that each fish added to a group of x fish reduces the time taken by 1/x. So for five fish they say it will take 135 seconds. The time is reduced by 1/4.  This seems reasonable, but what if one devil fish killed a king fish in 3 minutes with each additional fish making the new group proportionately quicker?  Then two fish would take no time at all!  What about three fish then? What's going on?

## #25 Re: Help Me ! » Volume of a Ring » 2011-08-09 00:10:36

I got the same answer by a different method. I first calculated the centroid of a semi-circle of radius r centered on the y-axis.
the x-coordinate of the centroid will be zero, so we just need to calculate the y-coordinate...

which comes out to be
now we need to divide by the area so...

so the centroid is located at

The volume is equal to the area multiplied by the distance traveled by the centroid. So we have..