[unreality]

some problems I came up with... they may be easy, they may be hard, you decide:

(1) You have exactly 2.3 oz of exactly 2.3 pH coca~cola in a large container... exactly how much water must be added to dilute the coke's the pH to 7?

(2) You have a straight line- two points are picked randomly on the line and that is where the line will be broken- now you have three lines. What is the probability that you will be able to make a triangle out of these lines?

(2-EXTRA) You don't have to do these if you don't want to, but they are some variations: What if you picked one random point, broke the line there, then picked another random point on the larger of two pieces to break it again? What about if you picked the second point on the smaller piece? What if it had a 1/2 chance for each piece after the first break to pick a point randomly on?

[itachi-san]

for 1)

the pH of the water. Most water found nowadays is less than 7 pH, so using a typical water like Poland Spring or something, you will never be able to get the total pH of the solution up to 7. Water can be made to have a pH higher than 7 though, which is actually very healthy for us. Water can be consumed at a very high pH like 10. So we would need to know the pH of the water used in this process before answering.

for 2)

a line is defined as being infinite in both directions. Therefore by splitting it into 3 sections we would only be left with 1 finite segment and 2 rays, thus incapable of making a triangle.

[bonanova]

some problems I came up with... they may be easy, they may be hard, you decide:

(1) You have exactly 2.3 oz of exactly 2.3 pH coca~cola in a large container... exactly how much water must be added to dilute the coke's the pH to 7?

(2) You have a straight line- two points are picked randomly on the line and that is where the line will be broken- now you have three lines. What is the probability that you will be able to make a triangle out of these lines?

(2-EXTRA) You don't have to do these if you don't want to, but they are some variations: What if you picked one random point, broke the line there, then picked another random point on the larger of two pieces to break it again? What about if you picked the second point on the smaller piece? What if it had a 1/2 chance for each piece after the first break to pick a point randomly on?

1. Always less than 7.

2. I'll do the math later - intuitive guess is .5

2extra: [1] .5 [2] 0 [3] .5

[bonanova]

for 1)

the pH of the water. Most water found nowadays is less than 7 pH, so using a typical water like Poland Spring or something, you will never be able to get the total pH of the solution up to 7. Water can be made to have a pH higher than 7 though, which is actually very healthy for us. Water can be consumed at a very high pH like 10. So we would need to know the pH of the water used in this process before answering.

for 2)

a line is defined as being infinite in both directions. Therefore by splitting it into 3 sections we would only be left with 1 finite segment and 2 rays, thus incapable of making a triangle.

itachi-san,

My first thought is that if your assumption in [2] is correct, it's always possible.

the two infinite rays attached to the one finite piece will always meet at infinity.

[Guest]

For all intents and purposes, water is always pH 7. Actual is something ridiculously close to 7, maybe about 6.98 or 6.99 (insignificant).

My stoichiometry is terrible, and I'm probably overlooking something, but calculations on first one look like...

3400 L = 114,920 oz.

(I really hope Coke is not pH 2.3 - some pretty strong stuff!)

[itachi-san]

For all intents and purposes, water is always pH 7. Actual is something ridiculously close to 7, maybe about 6.98 or 6.99 (insignificant).

My stoichiometry is terrible, and I'm probably overlooking something, but calculations on first one look like...

3400 L = 114,920 oz.

(I really hope Coke is not pH 2.3 - some pretty strong stuff!)

This is not true. Most water you will find: your tap, bottled, purified will all be acidic and all have different pH. Water can be treated with Hydrogen ions to achieve a proper 9-10 pH which is ideal for consumption. In a pure environment, mountain spring water would be alkaline.

[Guest]

I don't think treating water with hydrogen ions will make it more basic

And sure, I'll stipulate, drinking water that is a little on the alkaline side is better, it only gets that way through treatment. Naturally occurring water you'll find is at pH 7. LAB GRADE water used for any peer reviewed study is at pH 7. I'll venture a guess that less than 1% of the world's water is outside of the pH 6.7-7.3 range.

I say solve the puzzle and pretend water is at pH 7, otherwise you open up possibility of MANY different pH values, and all of those are due to some sort of special treatment (remember - h20 <--> h+ + OH- (that equilibrium has equal acidic/basic components)).

[itachi-san]

I don't think treating water with hydrogen ions will make it more basic

And sure, I'll stipulate, drinking water that is a little on the alkaline side is better, it only gets that way through treatment. Naturally occurring water you'll find is at pH 7. LAB GRADE water used for any peer reviewed study is at pH 7. I'll venture a guess that less than 1% of the world's water is outside of the pH 6.7-7.3 range.

I say solve the puzzle and pretend water is at pH 7, otherwise you open up possibility of MANY different pH values, and all of those are due to some sort of special treatment (remember - h20 <--> h+ + OH- (that equilibrium has equal acidic/basic components)).

Agreed. I think it's best to just say that water has a pH of 7. Just a side note though, you'd be surprised at how acidic most widely-consumed waters are

[Guest]

Agreed. I think it's best to just say that water has a pH of 7. Just a side note though, you'd be surprised at how acidic most widely-consumed waters are

Ignorance is bliss, my friend

[Guest]

(2) You have a straight line- two points are picked randomly on the line and that is where the line will be broken- now you have three lines. What is the probability that you will be able to make a triangle out of these lines?

I believe you would only be able to make a triangle if both selected points were on opposite sides of the midpoint of the original line segment, and the distance between the points was not greater than half the length of the segment. So another way of looking at it would be: What are the odds that if you randomly pick two numbers between 1 and 100, one will be less than 50, the other is greater than 50, and the difference is less than 50? At the moment, I'm not exactly sure how you solve that with probabilities.

[bonanova]

I believe you would only be able to make a triangle if both selected points were on opposite sides of the midpoint of the original line segment, and the distance between the points was not greater than half the length of the segment. So another way of looking at it would be: What are the odds that if you randomly pick two numbers between 1 and 100, one will be less than 50, the other is greater than 50, and the difference is less than 50? At the moment, I'm not exactly sure how you solve that with probabilities.

Let the line extend from [0, 100] in percent.

Let x and y be the two break points.

When x is in [0, 50) y must be in [50, x+50) - an interval whose average length is 25%.

When x is in [50, 100] y must be in [0, x-50] - an interval whose average length is 25%.

Thus for any x, the expected permissible range of y is 25% of the the interval [0, 100].

The probability that y is in that range is thus 0.25.

2- extra follows pretty easily.

These questions assume you've chosen x and ask about two possibilities for y:

(a) y is on the longer segment

(b) y is on the shorter segment

[c) y is on the longer segment with 50% probability.

The conditions for making a triangle listed above requires x and y to be on opposite sides of the halfway point.

Thus for a triangle to me made, [a] must be true - y must be on the longer segment.

So the answers follow immediately:

(a) 0.25

(b) 0.00

[c) 0.125

[Guest]

Let the line extend from [0, 100] in percent.

Let x and y be the two break points.

When x is in [0, 50) y must be in [50, x+50) - an interval whose average length is 25%.

When x is in [50, 100] y must be in [0, x-50] - an interval whose average length is 25%.

Thus for any x, the expected permissible range of y is 25% of the the interval [0, 100].

The probability that y is in that range is thus 0.25.

2- extra follows pretty easily.

These questions assume you've chosen x and ask about two possibilities for y:

(a) y is on the longer segment

(b) y is on the shorter segment

[c) y is on the longer segment with 50% probability.

The conditions for making a triangle listed above requires x and y to be on opposite sides of the halfway point.

Thus for a triangle to me made, [a] must be true - y must be on the longer segment.

So the answers follow immediately:

(a) 0.25

(b) 0.00

[c) 0.125

Answer is 1/8 or 0.125

and it is definite using the graphical probabilities.

See the image attached with this.

Now to make a perfect triangle from one line, consider the line is 1 unit long.

Now this line must be cut in such a way that one side should not be less than 0.25 unit and other should not be grater than 0.5 unit.

Because, sum of 2 sides should be greater than the third side and difference of 2 sides should be less than the third side in order to make a traingle.

So from the image the only portion that is left for any values for 3 parts lies between the coloured portion out of the full area of the graph.

So it is 1/8th of the total area.

If not able to understand then post me,I will try to reply with detailed explanation.

[bonanova]

Answer is 1/8 or 0.125

and it is definite using the graphical probabilities.

See the image attached with this.

Now to make a perfect triangle from one line, consider the line is 1 unit long.

Now this line must be cut in such a way that one side should not be less than 0.25 unit and other should not be grater than 0.5 unit.

Because, sum of 2 sides should be greater than the third side and difference of 2 sides should be less than the third side in order to make a traingle.

So from the image the only portion that is left for any values for 3 parts lies between the coloured portion out of the full area of the graph.

So it is 1/8th of the total area.

If not able to understand then post me,I will try to reply with detailed explanation.

Not greater than 0.5 I would agree with.

But any one side can be arbitrarily small I think.

For example, 0.1, 0.45 and 0.45 works.

It's not clear what your square represents -- where would .1, .45 and .45 lie on the square?

Also, the shaded area looks like 3/4 x 1/4 = 3/16 of the total area of the square. How does 1/8 come from it?

[Guest]

Not greater than 0.5 I would agree with.

But any one side can be arbitrarily small I think.

For example, 0.1, 0.45 and 0.45 works.

Also, the shaded area looks like 3/4 x 1/4 = 3/16 of the total area of the square.

How do you get 1/8 from that?

OOPs !. There is a flaw in my resoning, I agree.

Will come up with the correct explanation. But I know it can be explained using this graph but not able to recall how. I did this some 4-5 years back..

[bonanova]

I could be wrong about 1/4.

I just worked it out on some scratch paper.

Thanks.

[Guest]

For all intents and purposes, water is always pH 7. Actual is something ridiculously close to 7, maybe about 6.98 or 6.99 (insignificant).

My stoichiometry is terrible, and I'm probably overlooking something, but calculations on first one look like...

3400 L = 114,920 oz.

(I really hope Coke is not pH 2.3 - some pretty strong stuff!)

Lord knows I'm not a chemist but...

isn't pH a measure of the ratio of unbalanced H or OH ions to the whole?

PURE water which I'll write as H OH is perfectly balanced and therefore pH 7 (neutral)

How could you possibly get the pH of an Acid to pH 7 by adding something that is pH 7? Wouldn't it be an Asymptotic as in you add 2.3 oz of HOH to the Coke you get something with a pH halfway between Coke and Water?

[bonanova]

Lord knows I'm not a chemist but...

isn't pH a measure of the ratio of unbalanced H or OH ions to the whole?

PURE water which I'll write as H OH is perfectly balanced and therefore pH 7 (neutral)

How could you possibly get the pH of an Acid to pH 7 by adding something that is pH 7? Wouldn't it be an Asymptotic as in you add 2.3 oz of HOH to the Coke you get something with a pH halfway between Coke and Water?

I agree.

[unreality]

For everyone arguing over the pH of the water, reread my OP. I clearly said 7 ;D

only 2 people have gotten #1 so far, I won't say who (but I'll give a hint: it's a trick question)

okay I'll say who, lol

Bonanova was the first to get it (surprise surprise )

pH 7 is IMPOSSIBLE to obtain from an acid such as coke. No matter how much water you add to dilute the coke, there will always be some coke in it, thus making the pH more acidic than 7

for #2... it's a good one ;D

CLARIFICATION FOR #2: By "line" I meant "line segment", with two ends and a finite length ;D

HINT FOR #2:

btw I havent looked at any of the given answers for #2 yet, so I don't know who's correct

HINT: I'm sure everyone knows this, but in case not, but for three line segments to make a triangle, they all have to actually be line segments (ie, bigger than a single point or nonexistant point) and the length of the largest of the line segments must be LESS than the sum of the lengths of the two smaller line segments

[Guest]

number 1 is by far a trick question the way it's written.

number 2 I would guess 50%?

[unreality]

My solution to #2:

The answer would be a teeny bit LESS than 1/2, by my figuring

The thing with the line being broken up into three smaller, random, lines is a diversion when you think about it. It doesn't matter, since the original line is of random length. So basically the riddle is about three line segments of completely random length. Call them (and their lengths) a,b,c

for a triangle to be made, the sum of the two shortest sides must be greater than the longer side

if a is the largest side, b+c>a

if b is the largest side, a+c>b

if c is the largest side, a+b>c

since they are all random lengths of the same "randomness", we can say that they have the same average, whatever that average may be (based on length of original line).

So now let's call the largest side 'a', and the other two 'b' and 'c'. b+c must be greater than a to make a triangle.

For b+c to be equal to or less than a (which would mean a triangle cannot be made) the mean of b and c must be equal to or less than 1/2 of the average random length. So it seems to be 1/2

But note the "equal to OR less" for it not to be a triangle, while to be a triangle, it must be "more".

In other words, if there's a tie, it goes to the non-triangle side. You can't have a triangle with lengths of 10, 7 and 3- it would just be a line with a length of 10.

With ties going to the non-triangle side, the chance of getting a triangle from three random lines is slightly less than 1/2

[Guest]

(I really hope Coke is not pH 2.3 - some pretty strong stuff!)

Actually it is around pH 2.3. Here's a link to some measurements:

http://www.selah.k12.wa.us/SOAR/SciProj200...hs/MarshalN.pdf

You can google something like "pH level soda" to get more information. I stopped drinking carbonated beverages 3 years ago and now I cannot even have a sip as it "burns" too much. Strong stuff.

Edited April 14, 2008 by KarenClark

[unreality]

yep

btw I posted my #2 answer on page 2 if anyone's interested ;D

[bonanova]

My solution to #2:

The answer would be a teeny bit LESS than 1/2, by my figuring

The thing with the line being broken up into three smaller, random, lines is a diversion when you think about it. It doesn't matter, since the original line is of random length. So basically the riddle is about three line segments of completely random length. Call them (and their lengths) a,b,c

for a triangle to be made, the sum of the two shortest sides must be greater than the longer side

if a is the largest side, b+c>a

if b is the largest side, a+c>b

if c is the largest side, a+b>c

since they are all random lengths of the same "randomness", we can say that they have the same average, whatever that average may be (based on length of original line).

So now let's call the largest side 'a', and the other two 'b' and 'c'. b+c must be greater than a to make a triangle.

For b+c to be equal to or less than a (which would mean a triangle cannot be made) the mean of b and c must be equal to or less than 1/2 of the average random length. So it seems to be 1/2

But note the "equal to OR less" for it not to be a triangle, while to be a triangle, it must be "more".

In other words, if there's a tie, it goes to the non-triangle side. You can't have a triangle with lengths of 10, 7 and 3- it would just be a line with a length of 10.

With ties going to the non-triangle side, the chance of getting a triangle from three random lines is slightly less than 1/2

(2) You have a straight line- two points are picked randomly on the line

and that is where the line will be broken- now you have three lines.

What is the probability that you will be able to make a triangle out of these lines?

Let's construct the three sides according to the OP:

Pick two points - call them x and y on the line [segment].

They define three line segments.

To make a triangle, the length of none of the line segments can be greater than 50% of the length of the original line segment.

That is, x and y can't be more than 50% apart, and x and y can't be on the same side of the midpoint of the original line segment.

When x is in [0, 50), y must be in [50, x+50).

When x is in [50, 100] y must be in [0, x-50]. [edit] y must be in [x-50, 50]. [/edit]

We can visualize this.

The graph shows possible values of x: between 0% and 100% along the original line segment.

For each value of x, the values of y that will permit a triangle to be formed are shown in grey.

The entire square defined by x in [0, 100], y in [0, 100] contains the possible random x,y pairs.

The grey area is 1/4 of that area.

The probability of being able to form a triangle from the line segments defined by two randomly chosen points is therefore 1/4.

[Guest]

My solution to #2:

The answer would be a teeny bit LESS than 1/2, by my figuring

From spoiler ...

The thing with the line being broken up into three smaller, random, lines is a diversion when you think about it. It doesn't matter, since the original line is of random length.

I think that's where you made a mistake. It does matter, because choosing dividing points on a single segment enforces a relationship between the three segments that does not exist when you choose all three lengths randomly. To put it another way, when you choose one segment length, you've just limited the remaining possibilities to choose from for the lengths of the other two. That's why bonanova's method solution works.

[bonanova]

It's the bonanova - Duh Puck solution actually.

Bona took the problem statement from the above link and simply plotted the result.

[Guest]

It's the bonanova - Duh Puck solution actually.

Bona took the problem statement from the above link and simply plotted the result.

The solution is right and interesting. Only a slight correction. In the lower right square the upper triangle should be grey not the lower one.

Also no one has yet rated this puzzle, which to me is 5 star.

Edited April 15, 2008 by imran