taeto

2 hours ago, michel123456 said:

The point is to keep i^2=-1

From my understanding if one keeps your ix⋅iy=i(xy )

for x=i and y=i you get i(xy) = i^2

You call ix imaginary when x=i? 

If was thinking that if ix is an imaginary number, then x is a real number? I am mistaken? What is the range of values that you allow for x, all imaginary numbers, all complex numbers?

33 minutes ago, michel123456 said:

From my understanding if one keeps your ix⋅iy=i(xy )

for x=i and y=i you get i(xy) = i^2

where i(xy) is a notation meaning the multiplication of ix with iy under the i system.

We were discussing imaginary numbers, weren't we? So the imaginary numbers ix and iy have imaginary parts x and y that are real numbers. It does not make sense to have imaginary real parts x and y.

16 minutes ago, studiot said:

You have not created the extension field gained by adding the special properties available in adding one special symbol with the properties of being the square root of negative 1.

I did not say that it is an extension field. It is an isomorphic field. And yes, it is kind of pointless, except the point is that it is the only way to create the symmetry that Michel postulated.

17 minutes ago, studiot said:

That doesn't work since the product of two imaginary numbers is not an imaginary number.

You can define the product of any two elements of any set to be whatever you like. 

50 minutes ago, studiot said:

What on earth does the field set look like?

It is just the same as \(\mathbb{R},\) except instead of \(x\) you have \(ix.\)

Addition is \(ix+iy = i(x+y)\) and the zero element is \(i0,\) etc. The point is that multiplication is "redefined" so that \(ix\cdot iy = i(xy)\) is the rule. It does not say that \(i^2 = -1\) or not, instead it just ignores the \(i.\)

55 minutes ago, studiot said:

How can ix be 'only a name' ?

The way I see it, it means that "i" is a letter and x is a real number. Michel can correct if he sees it differently.

30 minutes ago, michel123456 said:

You are a genius! That's it.

I agree with this.

30 minutes ago, michel123456 said:

ix is not a product, it is a name.

We agree here as well, it is only a name. But just to be clear: the point is that at least the imaginary numbers ought to form a self-respecting real vector space, so that when a is a real number and ix is an imaginary number, then the product a(ix) produces the imaginary number i(ax).

10 hours ago, michel123456 said:

What is the difference?

With the integers the square is always a positive. With i the square is always negative. It is a symmetric situation. I don't understand why I should have to treat the imaginary numbers differently from the real. If real have + and - signs, why not the imaginary ones?

It is asymmetric in that the product of two real numbers is a real number, whereas the product of two imaginary numbers is not an imaginary number.

You could define a new product on the imaginary numbers so that (ix)(iy)=ixy for real numbers x and y. Then the symmetry is restored, and the imaginary numbers form a field isomorphic to the field of real numbers. But I suspect that when you think of multiplying a real number with an imaginary number, then you think of multiplication as it is defined for complex numbers? So if you want to create symmetry, then that is when you end up with a conflict.

1 hour ago, michel123456 said:

And then I am asking what is the sign of 3i ? Because 3 is a Real number and multiplication of positives give positive but i is imaginary with the multiplication of positives give negative. There is a conflict IMHO.

There is a conflict with your thinking that \(i\) is "positive"?

Integers, rationals and real numbers have "signs". Complex numbers do not, unless you define a sign as an extra feature. You want to define the "sign" of \( 1- \sqrt{2}i\)? 

Actually when you build the foundations of complex numbers, you just start from, well, \(-1\) has two square roots, so let \(i\) be one of them, then the other one is \(-i\), right? Clearly there is nothing inherently positive or negative, since either one of the two square roots could have chosen to be the "positive" one. Which is a little different from what happens with integers.

Since my fringe field happens to be mathematics, I can add that the ICM conferences, where the Fields medals are presented, are not a hair better. They do review the submissions that go into the top sections. But you can submit any old gibberish into the main sections and get approval from the "reviewers".  It is the same principle.They scrape a lot of money in from conference fees, and no way they will reject even the worst crackpottery.

Lots of scam journals around like this. Usually promising, and delivering, quick "publication", though at a hefty publication fee. A good deal for crackpots who have no other way to get their crap published. But a pretty tough deal for beginning researchers should they fall into the trap.

Whatever the original question might be, it does seem like an interesting and original attempt at creating an attractive mathematical/logical puzzle.

However, it appears a failed attempt. It is really hard to come up with interpretations of the different parts of the question which make things fall into place and provide a solution. I have been thinking about it a bit, and I have not yet discovered a way to reasonably interpret the question which allows for just a single solution, let alone a unique one. 

Generally you could use integral notation, and in the continuous case, like for the normal distribution, you should do so. Integral notation covers both cases.

I don't remember how to work out the expected value of the median estimator for \(\theta.\)

For the average estimator it should be fairly straightforward. With \(T=\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i\) we get

\[ E(T) = \int_{x_1,\ldots,x_n} \frac{1}{n}(\sum_{i=1}^n x_i) \prod_{i=1}^n \frac{1}{\sqrt{2\pi}\sigma} e^{\frac{(x_i-\mu)^2}{2\sigma^2}}dx_1\cdots dx_n.\]

Actually after simplification it is just the sum of the expected values of each \(x_i\) divided by \(n\). Since \(E(x_i)=\mu,\) we get \(E(T)=\mu\). I hesitate to work out the details now, because I am not at home and only have my little notebook available.

1 hour ago, Ghideon said:

Can we see the original question? (Even if it's in Turkish it might help at this point)

Excellent points, I agree! 

17 hours ago, studiot said:

8221 has a sum of 13, a product of 32 and what I interpret as the range of 7

some other possibilities with a sum of 13 are

4333 product = 108 and range 1

9211 product 18 range 8

So why did John choose 8221?

      I will try to take a stab.

John thinks about 4333.

If Jack knows only \(r=1,\) then Jack would not have said that James cannot know the solution, because the solution might have been 2111. This rules out 4333 for both James and John. 

If Jack even knows \(R=(4,3),\)  then Jack knows that the product will be one of 108, 144, and 192. Apart from 4333, another possible solution could then be 9431, 9441, or 8432, respectively, so far as James knows.

Now there is a complication: if the actual solution were 9431, then Jack would have \(R= (9,1)\) and maybe Jack would have to admit that James might possibly deduce the solution from knowing \(p.\) The solution could be 9751 in that case. Then 7533 is the only other alternative that has the same product. Remarkably, 7533 can be ruled out: if Jack knows \(R=(7,3),\) then James would know the precise solution in the case 7773, which is the only 4-digit number with product \(3\cdot 7^3.\) The same argument rules out all other solutions with \(R=(7,3)\) and \(R=(9,1)\) as well. 

So 9431 and 9441 get eliminated. Possibilities such as 9322, 9422 and 8432  still remain though.

So far John seems unable to eliminate 4333 as a possibility. But we have not yet considered James's statement. The product 108 cannot occur in many ways: 9322, 9431 are the only two alternatives. As above, 9431 can be dismissed just from the info that Jack gave. James observes that the possible sums are 13, 16. Neither of these allows John to determine a solution, certainly this is true of the sum 13, which is shared by 8221, the (supposed) actual solution. And sum 16 has enough candidates that no exact identification is possible, it seems.

End of analysis of the case 4333. I do not see how John can rule it out as the solution if Jack knows \(R.\) But it is tricky, and I can definitely have overlooked something.

Did I make a mistake? 

Edit: I do not see how John can rule out 4333. But I do see now how both James and John can rule out 8221, as follows.

Jack knows only \(R=(8,1).\)

The solution could be 8771, so far as Jack knows. The only other number with the same product is 7742.

We eliminate 7742 by observing that if Jack knows only \(R=(7,2),\) then James would know the exact solution if it is 7772.

Since 7742 is not a possible solution, and 8771 is the only other 4-digit number with product 392, also 8771 cannot be the solution.

We deduce that there is no solution with \(R=(8,1).\)

15 hours ago, Tor Fredrik said:

Above they use the expected value of T where T is the estimator for example mean or median as in the example in the beginning of the question. But since they find the expected value

of T must not they then use the pdf that corresponds to T? Which in the example above would be gamma and normal respectively.

I do not understand this part of your question. The point is that \(t\) can be any estimator for \(\theta,\) providing the expected value of \(t\) is actually equal to \(\theta.\) You are not assuming any particular pdf for \(T,\) except that which is given by the pdf \(f\) for the individual outcomes \(x_1,\ldots,x_n.\)      

It appears that the original text was formulated in Turkish language.

Depending on the quality of the translation, it may not be safe to rely on any assumptions about the meaning of "range" or whether Jack indicates that he knows at least one of the digits.

The phrase "Jack should be just found it" could make one suspicious.

18 minutes ago, studiot said:

Jack does not say that John can't deduce the solution, only James.

That is correct: "no matter what the product \( p\) is, James will not be able to deduce the solution". Nothing about John there.

18 minutes ago, studiot said:

Why can p not equal 0 ?

If 8221 is the assumed solution, then \(p\) is not \(0\).

I gather that we do not agree that the rule is that Jack does not tell James and John that the value of \(r\) is 7, but he does tell them that he can inform them that no matter what the product \(p\) is, James will not be able to deduce the solution, even if James knows the value of \(p\)?

23 minutes ago, studiot said:

Note also at the end that no one says Jack would be able to deduce the solution

The last statement of the problem description is "John says: I just found it and Jack should be just found it as well."

It does not contradict what you are saying?

What is the function \(f\) which has only one argument but has the same name as the density function \(f\) that has two arguments? 

You are missing my point.

Let us just assume that the 4-digit number is 8221. 

We agree that Jack does not know that this is the solution, but somehow he knows that the difference between highest and lowest digit is \(r=7\), right?

We agree that the rule is that Jack does not tell James and John that the value of \(r\) is 7, but he does tell them that he can inform them that no matter what the product \(p\) is, James will not be able to deduce the solution, even if James knows the value of \(p\)?

James now knows that \(r\) is not \(0,\) because if \(r=0,\) then James will be able to deduce from \(p=1\) or \(p=6561\) what the exact solution is, namely 1111 or 9999, respectively, do we agree? 

James also knows that \(r\) is not \(1\), because if \(r=1,\) then James will deduce the solution from \(p=2,\) namely 2111.

And so on, until \(r=7,\) which it would be for 8221. But again, Jack cannot say that James will not deduce the solution from knowing the value of \(p.\) Since if \(p\) is equal to \(8,\) then James knows immediately that the answer is 8111.

Please tell me what you are missing here, or what it is that seems wrong to you.

Yes, 8221 is a very interesting answer. It does not lie in any of the extremes, as when having a 9 at the start or a 0 at the end, or mostly the same digits, etc.

The problem is to figure out with reverse engineering which question this might be the correct answer for. 

I will attempt by pure improvisation from my own pet theory, that "range" means the pair of the smallest and the largest digits.

In this case the range would be \(R = (8,1). \) So Jack knows this.

The information from Jack that James will not be able to deduce the answer from knowing the product \(p=32\) does not help James or John a lot, unless there are lots of ranges for which the solution is obvious once you know the product. It is basically only ranges like \( (1,1),(5,5),\ldots,(9,9)\) and a few more for which this is possible.  

Knowing that the product is \(p=2^5\) however will help James along quite a bit. It leaves only \(8411\), \(8221\),\(4421\), and \(4222\).

James now knows that John will be looking at a sum \(s \in \{10,11,13,14\}.\) Indeed, John is looking at \(s=13.\) Now 9400 is one of the possibilities that John has to think about. It is consistent with Jack saying that he does not know all digits by knowing that the range is \(R=(9,0).\) And also with Jack saying that James will not be able to figure out the answer even knowing that the product is \(p=0.\) The only thing that 9400 is not consistent with is when John says that he knows that Jack also can deduce the answer. To Jack, the two possible solutions 9400 and 9220 would look exactly the same, judging from the discussion between James and John. This excludes 9400 as solution. 

Edit: that was garbage, please ignore. The real question is whether John can eliminate 9400 as a solution, and right now I do not see how he can. 

There is a list of additional possible solutions with sum \(s=13\) for John to consider, 9310, 9220, 8500, 8410, 8311, 8221, 7600 etc. Surely each, except 8221, can be excluded on similar grounds, no?

25 minutes ago, studiot said:

8221 has a sum of 13, a product of 32 and what I interpret as the range of 7

You really have to give up on your idea of "range".

If Jack only knows that the difference \(a-d\) is equal to \(7,\) he most definitely cannot ascertain that James will not figure out the precise solution once James knows the product.

Yes, exactly, the first and last digit would be the information that Jack has. Which he does not communicate to anyone else.

But if the range were to mean their difference, then Jack effectively tells everyone that the first digit is 9 and the last is 0 anyway. This way there is less information conveyed.

Not to be overly pedantic, but for those who want to google, the name is correctly spelled Martin Gardner.

And is it possible to post an exact version of the question, as a .jpeg image or similar?

Usually this style of puzzle is standard and has step-by-step arguments and solutions posted online. This particular one is interesting in its combination of number theory and logic. And I have not found it anywhere online after some googling attempts. I would be quite interested to use a simplified version in a test for students of a number theory course, or maybe a course on abstract algebra. Preferably not in an exam, since even if simplified it may still feel too difficult.

18 hours ago, studiot said:

Going back to Jack and the range.

So I do not know all the digits implies he knows at least one of them.

So when Jack says:" I dont know all the digits but James doesn't either." And if Jack is actually implying that he knows at least one of them.

Then what does "James doesn't either" mean exactly?

I suppose that \(x/(x+1)\) is the simplest expression that starts at \(0\) for \(x=0\) and increases to \(1\) in the limit as \(x\to \infty.\) 

It is not completely natural though, in that the \(+1\) may appear a little arbitrary, since you can replace \(x+1\) by \(x+c\) for any choice of \(c > 0\) and have the same effect. Also the function is not defined for \(x=-1\) resp. \(x=-c.\) 

Actually I meant the hyperbolic tangent \(\tanh \) in place of the \(x/(x+1)\) bit, but mistyped out of habit. It is nice throughout the reals and behaves similarly from \(0\) and up. It has no "arbitrary" looking parameters either. 

The "range" has to mean something different from \(r=a-d.\) If the value of \(r\) is all that Jack knows, then his statement that James cannot know the answer leads to elimination of all the possibilities \(r=0,1,\ldots,8\) in turn: If the solution happens to be \(a=r+1\) and \(b=c=d=1,\) then James can actually figure it out just from knowing the product \(p=r+1.\) That leaves only the possibility \(r=9,\) so \(a=9\) and \(d=0.\) Then James cannot be right in stating that John will not be able to know the answer, e.g. if the sum is \(s=27.\) (Another way to argue is to consider that James has no more information than John already knows, so it cannot be true that John can find the solution after hearing what James has to say.)

Hence "range" should mean the same as the pair \( R:=(a,d); \) the hyphen between biggest and smallest in the problem description is a hyphen, not a minus. 

59 minutes ago, Ghideon said:

My guess is:

  Reveal hidden contents

4111

 

So you assume that Jack knows that the range is 3.

How does Jack think about the possibility that James knows that the product \(a\cdot b\cdot c \cdot d\) is equal to \(250\)?