Maths Puzzles (or your Next Exam)

[yellowdingo]

Please feel free to submit a Mathematical Puzzle.

1.
g = x^(n-1) + {x^n} : g = sec^2(theta) + {tan(theta)}
p = n mod x XOR n(max) mod x : p =

 

[Mallory]

Please feel free to submit a Mathematical Puzzle.

1.

[FONT=Bodoni MT]g = x^(n-1) + {x^n}[/FONT]​

[FONT=Bodoni MT]:[/FONT]
[FONT=Bodoni MT]g = sec^2(theta) + {tan^(theta)}[/FONT]

[FONT=Bodoni MT]p = n mod x XOR n(max) mod x[/FONT]​

[FONT=Bodoni MT]:[/FONT][FONT=Bodoni MT][/FONT]
[FONT=Bodoni MT]p = [/FONT]

Just make sure you remove all the font commands first, OK?

 

[Jadzia]

what is n(max) supposed to mean?

what does tan^(theta) supposed to mean?

If you want us to solve for p, which of these variables are we allowed to express the solution in?

It looks like nonsense.

 

[yellowdingo]

Please feel free to submit a Mathematical Puzzle.

1.

[FONT=Bodoni MT]g = x^(n-1) + {x^n}[/FONT]​

[FONT=Bodoni MT]:[/FONT]
[FONT=Bodoni MT]g = sec^2(theta) + {tan^(theta)}[/FONT]

[FONT=Bodoni MT]p = n mod x XOR n(max) mod x[/FONT]​

[FONT=Bodoni MT]:[/FONT][FONT=Bodoni MT][/FONT]
[FONT=Bodoni MT]p = [/FONT]

Just make sure you remove all the font commands first, OK?

Yeah! I should have expected them to show up...serves me right for Posting from Word Documents.

 

[yellowdingo]

what is n(max) supposed to mean?

If you want us to solve for p, which of these variables are we allowed to express the solution in?

It looks like nonsense.

You need to solve for P as a Trigonometric Solution. But the idea is to submit a Maths Puzzle.

 

[All Seeing Eye]

what is n(max) supposed to mean?

what does tan^(theta) supposed to mean?

If you want us to solve for p, which of these variables are we allowed to express the solution in?

It looks like nonsense.

Yes yellowdingo, your equation makes no sense.

 

[The Borg Queen]

42.

 

[Jadzia]

Nobody has attempted mine from the end of last year...

Q1.
On one summer afternoon, a neighbor has an ipod playing on desktop speakers, and you overhear some of the tunes and notice there are songs playing from various albums.

Over a period of time, 40 tracks are heard, 3 of which are recognized to be from the same artist, and belonging to the same album. A quick google confirms this fact, and you learn that this album has a total of 15 songs.

Assuming that the ipod is on random shuffle, estimate the number of songs in its memory.


Q2.
An internet radio station is left playing all day long, with playlist of music from the 1980s. Over a period of 1 day, 200 tracks are heard, and you notice at the end of the day that 6% of the tracks you heard today, you have already heard earlier today, on this same station.

Assuming the songs played are selected randomly from the archive, and with each selection, there is no preference for one song over the other, estimate the size of the radio station's song archive.

 

[Hermiod]

Nobody has attempted mine from the end of last year...

Q1.
On one summer afternoon, a neighbor has an ipod playing on desktop speakers, and you overhear some of the tunes and notice there are songs playing from various albums.

Over a period of time, 40 tracks are heard, 3 of which are recognized to be from the same artist, and belonging to the same album. A quick google confirms this fact, and you learn that this album has a total of 15 songs.

Assuming that the ipod is on random shuffle, estimate the number of songs in its memory.

Erm, since the song shuffling is random, you can't make a reasonable estimate. The smallest iPod Shuffle sold has a 1gb capacity and supposedly holds 240 songs. However, by reducing the bitrate below the 128kpbs AAC they use to estimate this, you could store a lot more.

 

[The Borg Queen]

Q1: 200 songs.

Q2: Ugh... I used up all my brain juice on Q1.

 

[Jadzia]

Q1: 200 songs.

Q2: Ugh... I used up all my brain juice on Q1.

Yep. Q1 is correct.

Shuffle constructs a randomised playlist of all songs in the ipod's memory.

So if 3 notable songs are heard in a subset of 40 songs, then the proportion of these notable songs is 3/40 = 0.075

We can extend this playlist to cover N songs, maintaining the same proportion of 0.075 (the best estimate we can make), and we want there to be 15 notable songs, then

N x 0.075 = 15

Meaning there is estimated to be 15/0.075=200 songs in the ipod's memory.


Any takers for Q2?

 

[The Borg Queen]

Q1: 200 songs.

Q2: Ugh... I used up all my brain juice on Q1.

Yep. Q1 is correct.

Shuffle constructs a randomised playlist of all songs in the ipod's memory.

So if 3 notable songs are heard in a subset of 40 songs, then the proportion of these notable songs is 3/40 = 0.075

We can extend this playlist to cover N songs, maintaining the same proportion of 0.075 (the best estimate we can make), and we want there to be 15 notable songs, then

N x 0.075 = 15

Meaning there is estimated to be 15/0.075=200 songs in the ipod's memory.


Any takers for Q2?

Uh... okay.

I just worked it out as:

If three songs mean an album of 15, then thirty songs mean 150.

Ten songs would then be 50.

Add them together.

 

[Jono]

Q2.
An internet radio station is left playing all day long, with playlist of music from the 1980s. Over a period of 1 day, 200 tracks are heard, and you notice at the end of the day that 6% of the tracks you heard today, you have already heard earlier today, on this same station.

Assuming the songs played are selected randomly from the archive, and with each selection, there is no preference for one song over the other, estimate the size of the radio station's song archive.

Trick question, I don't listen to 80s Internet radio stations so I couldn't have heard any songs so the station has no archive because it does not exist.

 

[Showdown]

Q1. My version

There are 40 songs heard.
There are 3 songs are from one album.
There are 15 songs on one album.

20% songs from one album played.
Possibly 20% songs of ipod is played.

If 40 songs is 20% of ipod,
and 200 x 0.2 = 40,
then 200 songs are on ipod.

Q2. My version

200 tracks were heard in one day.
6% percent of the tracks (12 tracks) were reapeted.

If all songs were played because of total randomness with 12 repeated,
and 200 - 12 = 188,
then there 188 songs are in the archive.

Q3. My question

What is A+B?

 

[The Borg Queen]

6% percent of the tracks (12 tracks) were reapeted.

If all songs were played because of total randomness with 12 repeated,
and 200 - 12 = 188,

I feel like such a moron.

I'd been completely unable to think what 6% of 200 is...

 

[Jadzia]

Q2. My version

200 tracks were heard in one day.
6% percent of the tracks (12 tracks) were reapeted.

If all songs were played because of total randomness with 12 repeated,
and 200 - 12 = 188,
then there 188 songs are in the archive.

No, because this doesn't use shuffle. This is sampling with replacement, so the station doesn't need to play all of its songs before a song is given a second play.

So to start you off...
If there are N songs in the archive, then every song has an equal probability of 1/N of being selected for the next play.

You might find it helpful to ask yourself what is the probability of the next song being a repeat (contributing to that 6%), given that X songs have been played so far?