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# Mathematics Brain Teasers. ## Recommended Posts  4 hours ago, Golden Duck said:

It took me a few months to figure out; ie, the entire season.

It's the question on limits I was alluding to earlier.

You are going to have to enlighten me GD.

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#### Posted Images  1 hour ago, danydandan said:

You are going to have to enlighten me GD.

Me too! This is not an area of maths that has ever appealed to me nor is it something that I have considered before.

##### Share on other sites  9 hours ago, danydandan said:

You are going to have to enlighten me GD.

7 hours ago, Ozymandias said:

Me too! This is not an area of maths that has ever appealed to me nor is it something that I have considered before.

The Binomial Probability Distribution gives us the probability of observing y success from n trials.

The Negative Binomial Probability Distribution gives the probability of observing y trials before serving r successes.

The Geometric Probability Distribution is a special case, of the Negative Binomial Distribution Function, where r = 1.  The combinations rule equals 1 in this case.

P(n) = p(1-p)^(n-1); where p is the probability

The sum of P(n) is S(n) = 1 - (1 - p)^n

The expected value, mean, or average is μ = 1/p.

The median is when S(n) equals 0.5; ie, when (1 - p)^n = 0.5.  Therefore the median is v = log_1-p(0.5) = -1/[log2(1-p)]

Dividing the median by the mean starts to converge on a number as p gets smaller.

As p approaches 0, what is the limit of

-1/[log2(1-p)] ÷ 1/p

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##### Share on other sites  3 hours ago, Golden Duck said:

The Binomial Probability Distribution gives us the probability of observing y success from n trials.

The Negative Binomial Probability Distribution gives the probability of observing y trials before serving r successes.

The Geometric Probability Distribution is a special case, of the Negative Binomial Distribution Function, where r = 1.  The combinations rule equals 1 in this case.

P(n) = p(1-p)^(n-1); where p is the probability

The sum of P(n) is S(n) = 1 - (1 - p)^n

The expected value, mean, or average is μ = 1/p.

The median is when S(n) equals 0.5; ie, when (1 - p)^n = 0.5.  Therefore the median is v = log_1-p(0.5) = -1/[log2(1-p)]

Dividing the median by the mean starts to converge on a number as p gets smaller.

As p approaches 0, what is the limit of

-1/[log2(1-p)] ÷ 1/p

Fecking hell. Haven't seen that in a few years.

I actually knew that. Good question Golden Duck.

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##### Share on other sites  @Golden Duck, since we could not get your last question and you enlightened us with the answer.

Would you please set a new question, if you'd be so kind?

##### Share on other sites  1 hour ago, danydandan said:

@Golden Duck, since we could not get your last question and you enlightened us with the answer.

Would you please set a new question, if you'd be so kind?

Spoiler

ln(2)

I feel I got lucky finding the answer using a spreadsheet. I had to hit the textbooks to help work out the proof.

I'm lacking a bit of inspiration at the moment.

Here's one I remember seeing online.

Two trains are 200km apart when they set off towards each other at 50km/h. At the same a bird sets of flying to and fro between the trains, at a speed of 60km/h, until it is squashed between the two trains when they meet.

What is the total distance the bird flies?

##### Share on other sites  1 hour ago, Golden Duck said:

Hide contents

ln(2)

I feel I got lucky finding the answer using a spreadsheet. I had to hit the textbooks to help work out the proof.

I'm lacking a bit of inspiration at the moment.

Here's one I remember seeing online.

Two trains are 200km apart when they set off towards each other at 50km/h. At the same a bird sets of flying to and fro between the trains, at a speed of 60km/h, until it is squashed between the two trains when they meet.

What is the total distance the bird flies?

120 km.

The trains' closing speed is 100km/hr and the gap between the trains is 200km. They will hit each other after travelling for 2 hours.

During that 2 hours the bird flying at 60km/hr will have travelled 120km.

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##### Share on other sites  1 hour ago, Ozymandias said:

120 km.

The trains' closing speed is 100km/hr and the gap between the trains is 200km. They will hit each other after travelling for 2 hours.

During that 2 hours the bird flying at 60km/hr will have travelled 120km.

Too easy!

##### Share on other sites  9 hours ago, Golden Duck said:

Reveal hidden contents

ln(2)

I feel I got lucky finding the answer using a spreadsheet. I had to hit the textbooks to help work out the proof.

I'm lacking a bit of inspiration at the moment.

Here's one I remember seeing online.

Two trains are 200km apart when they set off towards each other at 50km/h. At the same a bird sets of flying to and fro between the trains, at a speed of 60km/h, until it is squashed between the two trains when they meet.

What is the total distance the bird flies?

120.

Edited by danydandan
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##### Share on other sites  What do the following numbers represent?

....  11,  101,  1000,  1101,  .....

##### Share on other sites  5 hours ago, Ozymandias said:

What do the following numbers represent?

....  11,  101,  1000,  1101,  .....

Probably not going to be correct.

But.....3,5,8,13? Binary to Decimal Base. Or is that just the low hanging fruit? And I believe that 3,5,8,13 is apart of the very famous Fibonacci Sequence.

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##### Share on other sites  38 minutes ago, danydandan said:

Probably not going to be correct.

But.....3,5,8,13? Binary to Decimal Base. Or is that just the low hanging fruit? And I believe that 3,5,8,13 is apart of the very famous Fibonacci Sequence.

That's it, Dan. You got it in one!

You're turn to set a question.

But you haven't given or laid out a solition method.

Edited by Ozymandias
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##### Share on other sites  Using only the number eight (8), as many times as you'd like, write an equation to equal one thousand (1000).

You can use any operator, function or whatever but you can only use the number eight (8).

##### Share on other sites  1000  =  8 × 125

1000  =  8  ×  8  ×  15.675

1000  =  (8 × 8)(8 + 8 - 0.375)

1000  =  (8 × 8)(8 + 8 - 3/8)

1000  =  (8 × 8)[8 + 8 - (1/8) - (1/8) - (1/8)]

1000  =  (8 × 8)[8 + 8 - (8/64) - (8/64) - (8/64)]

1000  =  (8 × 8){8 + 8 - [8/8(8)] - [8/8(8)] - [8/8(8)]}

Alternatively (and prompted by Golden Duck's following post #269):

8 [(8 × 8) + (8 × 8) - 8/8 - 8/8 - 8/8]

Edited by Ozymandias
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##### Share on other sites  Is this cheating?

```>>> s = '8'
>>> for i in range(int(8*8 + 8*8 - 8/8 - 8/8 - 8/8 - 8/8)): s = s + " + 8"

>>> s
'8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8'
>>> eval(_)
1000
>>> ```

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##### Share on other sites  25 minutes ago, Golden Duck said:

Is this cheating?

```
>>> s = '8'
>>> for i in range(int(8*8 + 8*8 - 8/8 - 8/8 - 8/8 - 8/8)): s = s + " + 8"

>>> s
'8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8'
>>> eval(_)
1000
>>> ```

Why do you think it might be cheating? Is it because you are using a computer?

Edited by Ozymandias
##### Share on other sites  20 minutes ago, Ozymandias said:

Why do you think it might be cheating? Is it because you are using a computer?

Perhaps because I'm using alphabetical characters.

Thinking about this question a bit deeper... is the benefit of this type of question to cement in understanding of various identity laws?

##### Share on other sites  A man walks up to three boys playing in a vacant lot. All three boys have a dirt smudge on their forehead.

The man says "I'll give a dollar to the first boy that can tell me if he has a smudge on his forehead."

The boys think about it, then finally one raises his hand and says "I have a smudge on my forehead."

The man paid him.

How did the boy know?

Harte

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##### Share on other sites  Both answers are correct, @Ozymandias and @Golden Duck. I did say just use the number eight and any operator, function or whatever.

A very simple solution is the following, hopefully that question was a good brain teaser.

888+88+8+8+8=1000. Yes I use eight eights to get the result, I thought that was eloquent.

Take it away Ozy it's your turn you posted first.

Edited by danydandan
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##### Share on other sites  10 hours ago, Harte said:

A man walks up to three boys playing in a vacant lot. All three boys have a dirt smudge on their forehead.

The man says "I'll give a dollar to the first boy that can tell me if he has a smudge on his forehead."

The boys think about it, then finally one raises his hand and says "I have a smudge on my forehead."

The man paid him.

How did the boy know?

Harte

Instead of setting a new problem, and if Harte doesn't mind, I choose to let his teaser (quoted above) to stand for mine.

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##### Share on other sites  1 hour ago, Ozymandias said:

Instead of setting a new problem, and if Harte doesn't mind, I choose to let his teaser (quoted above) to stand for mine.

I didn't mean to post out of turn. I just remembered that little logic problem so I posted it.

Harte

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##### Share on other sites  11 hours ago, Harte said:

A man walks up to three boys playing in a vacant lot. All three boys have a dirt smudge on their forehead.

The man says "I'll give a dollar to the first boy that can tell me if he has a smudge on his forehead."

The boys think about it, then finally one raises his hand and says "I have a smudge on my forehead."

The man paid him.

How did the boy know?

Harte

1

Each boy has the knowledge of what he sees - two boys with smudges; and, the words of the man - "I'll give a dollar to the first boy that can tell me if he has a smudge on his forehead."

If you can infer "...first boy..." means there is more than must mean there is more than one boy with a smudge, then one boy must, based on the uncertainty of the others, can rule out the group of boys is made up of two smudges and one clean.  Therefore, it means all boys must be smudged.

Again I'm banking on "...first boy..." cannot mean only one.

##### Share on other sites  My take is this:

None of the boys can know that they have a smudge on their own forehead, only that the others do. That is why none can claim they have a smudge with certainty.

The best bet, in your ignorance, is to take a punt and claim that you do before the other two, because 'if you're not in you can't win'!

Relying on the man's use of 'first boy' is logically unsound because he could be saying that if only two boys had smudges.

Edited by Ozymandias
##### Share on other sites  40 minutes ago, Ozymandias said:

My take is this:

None of the boys can know that they have a smudge on their own forehead, only that the others do. That is why none can claim they have a smudge with certainty.

The best bet, in your ignorance, is to take a punt and claim that you do before the other two, because 'if you're not in you can't win'!

Relying on the man's use of 'first boy' is logically unsound because he could be saying that if only two boys had smudges.

Corey sees two smudges.

Corey knows Billy and Andy each see at least one smudge. If "first boy" means more that one and if there were only two smudges, then Corey also knows that Billy and Andy should know they have a smudge.  Corey deduces that both Billy and Andy, because of their uncertainty, see two smudges meaning Corey himself must be smudged.

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##### Share on other sites  21 hours ago, Harte said:

A man walks up to three boys playing in a vacant lot. All three boys have a dirt smudge on their forehead.

The man says "I'll give a dollar to the first boy that can tell me if he has a smudge on his forehead."

The boys think about it, then finally one raises his hand and says "I have a smudge on my forehead."

The man paid him.

How did the boy know?

Harte

He assumed he did because the others didn't raise their hands.

Edited by danydandan
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