Problem 389

[psujono]

Problem 389 (View Problem)
What is meant by "Find the variance of I"?

[TripleM]

http://en.wikipedia.org/wiki/Variance#Definition

[psujono]

So that would be the square of the standard deviation?

[mdean]

I was wondering what the code was with those letters for a minute before I figured it out:

Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

Ah well, probability is one of my weak points, but I may just have to attempt this problem anyway. Hoping I can complete it and maybe learn something from the other solutions.

[thundre]

Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

The Platonic solids. I didn't even notice!

You can make fair dice (all faces congruent and symmetric) with any even number of sides, but the Platonic solids are the only ones where the faces are all regular polygons.

[richard]

I feel like I am solving this correctly but my answer is not accepted so obviously I've done something wrong. I've spent a lot of time looking for a mistake in my implementation but I'm not having any luck. Can someone post the variance of O (the results of the 8-sided dice) as an example that I can check against?

If that's too much of a clue maybe someone can give the variance for some other scenario involving dice with different numbers of sides from the actual problem (but involving at least 3 different types of dice) For example, one 5-sided = A then A 7-sided = B then B 9-sided = C, give the variance of C).

[Marcus_Andrews]

The answer to your 5-7-9 example: 1180.0

[richard]

Thanks Marcus, I was able to track my bug down given the sample solution. I had calculated the variance across every unique roll sequence instead of based on the actual probabilities of the sequences.

[Marcus_Andrews]

For future reference: I hope this is not too much of a spoiler, but a way to sanity-check your results manually if you are having trouble is to make a brute-force program that simulates dice rolls and calculates values for the variable I over and over again. That way you can calculate the variance on the fly to help cross-check your main program.

[hk]

For future reference: I hope this is not too much of a spoiler

I certainly isn't a spoiler.
It cannot be said often enough:
If you have an idea how to bruteforce for lower limits, or simpler cases or less accurate: just make that bruteforcer and check the results of more advanced programs with that.

It's just a way of solving your problems that should become a second nature.
I also use it when designing problems and when checking other's proposals.

[haphaeu]

Hi there,
sorry, this is sort of a silly question but I just want to make sure I'm getting it right...
Since this problem involves a floating point, the answer should be given with the dot and the following 4 places? Like xyz.1234 ?
Or should I just ignore the dot and enter, say, xyz1234?
Thanks!

[hk]

Like xyz.1234.

[haphaeu]

Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

The Platonic solids. I didn't even notice!

You can make fair dice (all faces congruent and symmetric) with any even number of sides, but the Platonic solids are the only ones where the faces are all regular polygons.

wouldn't it make more sense using Hexahedron?

[thundre]

You can make fair dice (all faces congruent and symmetric) with any even number of sides, but the Platonic solids are the only ones where the faces are all regular polygons.

wouldn't it make more sense using Hexahedron?

Good point! A Hexahedron isn't Platonic because the vertexes don't all join the same number of sides. But all the faces are regular (provided all the sides are the same length).

How many other solids have regular congruent faces but aren't Platonic?

[hk]

How about http://en.wikipedia.org/wiki/Platonic_solid ?
There it says that a cube=a hexahedron=a six sided Platonic solid.

Please note that the problem statement doesn't use the terms -hedron or cube.

[thundre]

How about http://en.wikipedia.org/wiki/Platonic_solid ?
There it says that a cube=a hexahedron=a six sided Platonic solid.

Please note that the problem statement doesn't use the terms -hedron or cube.

Oops! I thought it meant two tetrahedrons joined at the face. My bad.

[square1001]

What numbers are written in each dice? I don't know even 1-indexed or 0-indexed.
Please tell me.

[Animus]

The dice are 1-indexed, e.g. the numbers on the 4 sided die run from 1 to 4.

[Junglemath]

The dice are 1-indexed, e.g. the numbers on the 4 sided die run from 1 to 4.

I had the same question as the person you replied to. The question does not specify the numbers on the dice, but only how many faces they have. I don't think it should be assumed that the numbers on the faces go from 1 to $n$. Can you add this condition explicitly to the problem statement?