[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 621

 

There is another option, without "recursion" ("either... or..." is a domestic version of XOR):

Either you're a liar or you have a telly.

Very reminiscent of "Either you're a bastard or you've got a telly!".

 

I have a couple of thoughts about synthesizing Boolean functions. I myself am not familiar with all sorts of intricacies like DNF, QNF and so on, so I analyze just as an amateur. Let's take the TV problem as an example.

Let A = You are a liar.

X = You have a telly.

We need to make a function f(A,X) which has a single two properties:

f(~A, X) = ~f(A, X). [Liar inverts the value of a Boolean function]

f(A, ~X) = ~f(A, X). [For the same type of person, the answers at different X's must be different]

Explanation of the first property: since the liar inverts the inverted value, the answers are the same regardless of the type of person.

I know such functions based on xor: f = A xor X and their derivatives. Hence the answer.

(There are only 16 functions of two variables, so the enumeration is finite.)

Now to the dumb sentinel problem:

Now A = "yyy=true", B = "You are a liar", X = "This road is right".

f(~A, B, X) = ~f(A, B, X).

f(A, ~B, X) = ~f(A, B, X).

Suppose this function can be composed by the superposition of two - f1 and f2. Let's put parameters A and B into function f1(), and the result of f1 and X into the second f2().

Then if y1 = f1(A, B), then f(A, B, X) = f2(y1, X).

On the other hand, we have these properties of f1():

f1(~A, B) = ~f1(A, B) [liar inverts answer].

f1(A, ~B) = ~f1(A, B) [this is no longer necessarily a liar, but the answer is still inverted if the underlying value system (yu/yu) is expressed differently].

We know such a function: it is again (A xor B). Now for the function f2():

f2(~y, X) = ~f2(y,X).

f2(y, ~X) = ~f2(y,X).

Explanations of why these properties are so were given in the previous TV problem. Again the function (y xor X).

It turns out simply: f() = (A xor B) xor X = A xor B xor X. Let's check (A = "yyy=true", B = "You're a liar", X = "This road is right"):


yyy=true, Liar, True: true xor true xor true = true = yyyy. Inverts to "woo".

yyy=true, True, True: true xor true xor false xor true = false = woo. "woo".

yyy=False, Liar, True: false xor true xor true = false = yyy. Inverts to "woo".

yyy=False, True, True: false xor false xor true = true = woo. "woo".


yyy=true, Liar, Incorrect: true xor true xor true = false = woo. Inverts to "yoo".

yyy=true, True, Incorrect: true xor true xor false xor false = true = yyy. "yyyy".

yyy=False, Liar, Incorrect: false xor true xor false xor false = true = yyy. Inverts to "yoo".

yyy=False, True, Incorrect: false xor false xor false xor false = false = yyy. "yyyy".


That's it. Master class on amateur analysis is over :)

Correct judgement (either...or...or is a household exclusionary XOR): Either "yyyy" is true, or you are a liar, or this way is right.

Or stricter, so that there is no variation: ("yyy" is true) XOR (You are a liar) XOR (This road is right).

We managed to say it in 15 words.

 

Right.

In my solution I also started with XOR between three expressions, but the output of formula(representation through AND and OR as for binary XOR ) for "three"-dimensional XOR got messed up:(

 
This answer has been in front of me all along (following the pattern of the TV problem), but for some reason I didn't dare to check it.
 

Challenge. Complexity 10.

Calculate the probability that on N tosses of a coin tails will fall Y times more often than eagles.

(Write a summary formula to calculate the probability. )

 

What's 10? There's barely 2 here. And that's only because people are bad friends with terver. Oh, why are neophytes tormenting Bernoulli's ashes so much these days...

Lizavetto, look at the solution here, we just recently solved a very similar one for Dima .

Besides, that's not a very... uh... practical. Failure rate is better to specify in an interval (here's how Dima, for example: "at least 30 successes out of 120 trials"). You can, of course, calculate in your case, but it will be a very small probability, and also not very applicable in practice.

 

Mathemat:

Besides, you didn't set a very... uh... practical. It's better to specify the fallout fraction in some interval (like Dima, for example: "at least 30 successes out of 120 tests"). You can, of course, calculate in your case, but it will be a very small probability, and also not very applicable in practice.

It's OK. It seems to me that the formulation is practical. That is, the function from one variable N needs to be plotted. At the same time a chart can be plotted.

By the way - I thought this was a "non-practical" branch... Have I made a mistake? :))

// Ah yes. Also Y is variable... Yeah, well, then the graph will be three-dimensional. As long as it's not "four-dimensional", otherwise you won't be able to see it. ;)

 
Mathemat:


Or stricter, so that there is no misunderstanding:(XOR (You're a liar) XOR (This road is the right one).

That'sabout 15 words long.

I'm afraid that's not going to work with a dumb guard. (
 
jelizavettka:

Mathemat:

Besides, you didn't set a very... uh... practical. It's better to specify the fallout fraction in some interval (like Dima, for example: "at least 30 successes out of 120 tests"). You can, of course, calculate in your case, but it will be a very small probability, and also not very applicable in practice.

jelizavettka, what is meant is that it would be more practical to ask, for example, "per N coin tosses, tails will fall Y or more times more often than heads". To solve the problem of "exactly that many times more" is ..... well, like comparing two numbers double ...
 
jelizavettka:

The challenge.

And the problem itself is reduced in one step to the well-known variant - if X is the number of eagle's falls, then according to the condition we get X + X*Y = N, where X = N/(1+Y), after that we can write the usual formula for binomial distribution (or approximate it by Gaussian, if N is very big).