Math Is Fun Forum

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Loaded Dice

Show that you cannot have two loaded 6-faced die, such that, when you roll them together all scores from 2 to 12 are equally likely

Last edited by Agnishom (2017-03-23 01:57:33)

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

Hi;

Standard faced 1,2,3,4,5,6 on each side die?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

Yes. But maybe loaded, as in not fair.

In other words, show that

is not a product of

and

where

for all i and

Last edited by Agnishom (2017-03-23 14:10:32)

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

HI;

I have seen the proof using generating functions many times but never wrote it down.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

How about you write it down now, in this forum?

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

The dice expert:

http://www.madandmoonly.com/doctormatt/ … /dice1.pdf

Post #43 I think applies.

See if you can spot me as a contributor! Mr anonimnystefy would have also been a major contributor but I could not get him to release his findings for publication.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

That is a very cool pdf.

Somebody else gave me a different solution involving zeroes of polynomials though.

A contributor for what?

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

To that site/page (just in a very small way).

Somebody else gave me a different solution involving zeroes of polynomials though.

I have seen it in some book but I am afraid I can not think of it. The solution given by Matt is not adequate?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

It sure is

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

Then we are done?

Here is how to prove it computationally, you see that M knows it is impossible.

This may take a while to finish.

FindInstance[{a1*b1 == 1/11, (a2*b1 + a1*b2) == 1/
   11, (a3*b1 + a2*b2 + a1*b3) == 1/
   11 , (a4*b1 + a3*b2 + a2*b3 + a1*b4) == 1/
   11, (a5*b1 + a4*b2 + a3*b3 + a2*b4 + a1*b5) == 1/
   11, (a6*b1 + a5*b2 + a4*b3 + a3*b4 + a2*b5 + a1*b6) == 1/
   11, (a6*b2 + a5*b3 + a4*b4 + a3*b5 + a2*b6) == 1/
   11, (a6*b3 + a5*b4 + a4*b5 + a3*b6) == 1/
   11, (a6*b4 + a5*b5 + a4*b6) == 1/11, (a6*b5 + a5*b6) == 1/11, 
  a6*b6 == 1/11, 1 > a1 >= a2 >= a3 >= a4 >= a5 >= a6 >= 0, 
  1 > b1 >= b2 >= b3 >= b4 >= b5 >= b6 > 0, 
  a1 + a2 + a3 + a4 + a5 + a6 == 1, 
  b1 + b2 + b3 + b4 + b5 + b6 == 1}, {a1, a2, a3, a4, a5, a6, b1, b2, 
  b3, b4, b5, b6}, Reals]

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

Interesting. How does M know this?

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

That is a good question. Supposedly, it has rules that tell it when a set of equations has a solution. Does that constitute what math types would call a proof? I was just reading yet another article on this very topic. Some say yes and most still say no. I think in terms of probability, if Maple, Sage and M said the same thing and some math type said something else I would say that the chance that all 3 CAS are wrong is one in a billion. The chance that said math type is wrong based on my experiences with them, one in 20.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

Well, what counts as a proof is subjective. However, for a formalist, it is less subjective.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

However, for a formalist, it is less subjective.

I agree with Doron, let us banish the concept of proof or at least knock it off its high horse.

A battle is on the horizon...

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

Once in the mechanics class, our physics professor gave a very strange argument to claim that an expression is true.

I was perplexed and asked how that is a proof of anything?

He said, its a physicist's proof. A physicist understands that if this went wrong then something very weird would happen.

I asked, So you mean a verification?

He said, no, it's a legit proof. Keep in mind that a mathematician's idea of what a proof is, is different from what a physicist thinks a proof is.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

I am neither, in Bumpkinland, proofs are outlawed.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

I live in the Riemann Sphere

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

Worse than your physicist teacher is that Doron video where he accepts something is a proof with little evidence. If I said that they would cart me to the looney bin but he is a leading mathematician and a brilliant guy on top of that. We should start asking why Doron and Wolfram do not like math.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

They are already publishing their opinions in their blogs and webpages

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

But is anyone paying attention to it. Doron says a program is better than a proof. Do teachers change their next lecture or do they teach the same proof as they did last year?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

Doron says a program is better than a proof.

Do you know what Curry-Howard Isomorphism is?

Do teachers change their next lecture or do they teach the same proof as they did last year?

They do. Our Haskell prof offers a course called "Proofs and Types"

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

But is he teaching math or physics? Everyone knows that computer guys do not agree with how math is taught or how it is done.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

He is teaching Computer Science. Computer Science is the best!

By the way, proofs and programs are the same thing

Last edited by Agnishom (2017-03-27 16:00:50)

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Loaded Dice

I sort of like it too.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Loaded Dice

That makes two of us!

Have you heard of Curry-Howard isomorphism?

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.