karthickgururaj

One question: is it allowed to use infinite operations on the cake? I mean, let us say given a 'd' (any real number), if I can show that we can approach arbitrarily close to the initial cake "state" (i.e, all the frosting will be on top, except for an arbitrarily small section) - would that count? Or, must it always become the cake with all frosting on top in finite amount of steps?

I agree that my earlier attempt had a bit of "hand-waving" So let me try to make it slightly more formal, perhaps that will help in revealing the error.. Slightly longish post It is good to have a notation to refer the "state" of the cake. Starting from 0 deg, say the cake top is frosted for x0 deg, then cake bottom is frosted for x1 deg, etc, until 360 deg is reached. Let us denote it by: Cake(F => (0, x0); F' => (x0, x1); F => (x1, x2); ... ; (F or F') => (x, 360)) (F stands for frosted side up, F' for frosted side down). or even more briefly (though redundant, we retain 0 and 360 for clarity): C(F => 0; x0; x1; ..., xn; 360) The initial state of the cake is: C(F => 0; 360), which we will refer to as 'S0' A cake that is half-frosted can be represented as: C(F => 0; 180; 360). Note that C(F => 0; 90; 270; 360) is also half-frosted, but is considered different from C(F => 0; 180; 360) Next we define a transformation on the cake t(c, d), that changes the cake state c to a new state c' by 1. First cutting d degrees of cake (from 0 deg) 2. Then flipping the cut part over 3. And then rotating the cake by d degrees clockwise (so there is a "cut" at 0 degrees) Note: We assume 0 < d < 360 here (we'll remove that assumption later..) So, as an example t(S0, 60) = C(F' => 0; 60; 360) t(C(F' => 0; 60; 360), 60) = C(F' => 0; 120; 360) We also define: t(t...t(t(c, d), d))..) to be tn(c, d) - i.e, repeated applications of the transformation 'n' times. So, as an example t6(S0, 60) = C(F' => 0; 360) The puzzle can then be worded as: For what values of 'd', there exists an 'n' such that: tn(S0, d) = S0 Ok, with the notation taken care of,

Your analysis isn't wrong; doing it like that won't work. Try another approach. Go on. Very interesting. I'm also now wondering about the puzzle itself.. What should Bob do: have a strategy that ensures he rejects the room with least number of apples,

I didn't read plasmid's solution, but came up with the same end result (seems that keeps happening to me)..

Why would you say that?

The number of apples in rooms 1 and 2 are described by the same words. What if Bob had gone to Room 2 first? Or is that the flaw? No.. the flaw is that we can't directly deal with infinities, as you yourself showed in a different context A better way to approach would be to assume an upper limit on the number of apples that may be kept in a room, and then take limits to infinity... wouldn't that answer the OP? So, in short: I don't know the answer to the puzzle

Agree.

"infinite supply of apples and infinitely large rooms, so each room can have any non-negative integer number of apples": Ting..ting..ting.. This is setting off a few warning bells for me For instance, how about the following rationale why Bob must always reject the first room:,

Can you show how you got this? Off the top of my head, I feel like the case with five circles would be a counterexample...? Why would five circle case be a counter example? My reasoning is thus:

Thanks all for the interesting discussion. I understand the folly in my own solution (at the least!). I'm summarizing some of the interesting posts for a reader who might come across this thread in future. Might be worthwhile to also check the initial set of solutions that attempted to solve the problem "directly" on the infinite plane (including my own). The solutions varied from 1, 0.75, 0.64.. Also, you can check a related discussion: Didn't discuss the approach below much further, but same result as bonanova's (closest to simulation result): Bonanova's solution: If the OP has a solution, it is the same as for a circle of finite radius. Simulation results when trying over a square and a circle. Not the same result as what was predicted above, but close: Counter example to Bonanova's solution, which may be one of the reasons why the simulation result is not the same as predicted: One more counter example to Bonanova's solution:

Also, even though it doesn't solve the OP,

I agree partly, gavinksong. However, because some area will be wasted between circles (even if they are packed optimally, solution I proposed places a lower bound on the value of the circumscribed radius.

Interesting. Think it is as below..

I read it.... but it is not correct... because in this case ,for the 3rd question, the 1st person may be A,,, or may not...hmmm...then what? If the 1st person is not A, it is known within first two questions. I have annotated my solution with how the possibilities change with each question, see below (ABC means the possibility that, "A is sitting in chair 1, B in chair 2, C in chair 3").

Just a suggestion: since we do not really know if rand(..) is going to yield well-distributed points, it may make more sense to just just step x and y linearly over small increments.

Edit: It might be that the shape from which the points were selected impacts the probability that they form an obtuse triangle. And if that's the case, it's probably impossible to say what to expect if the three points are chosen from an infinite plane without boundaries -- were they drawn from an an infinitely large square with probability 72.5% or an infinitely large circle with probability 75%? The shape will definitely affect the probability! But in this question a circle is chosen, please see my response to bonanova's response (the third response, on the first page): I agree we need to assume the points to be contained in a finite circle, and then apply the limit as the radius approaches infinity. The question can be considered suitable re-worded. [spoiler=Why circle, and not any other shape?.. ] ..because a circle is "fair" Fairness was implicit in the question. It is like asking: a point is chosen at random on x-axis, what is the probability that it is positive? "Intuitively", it looks to be 0.5. But we can get to that result only if we assume a line segment centered at origin, and then apply the limit as the segment length approaches infinity. A line segment centered at origin is "fair" to both positive and negative directions. I solved the puzzle in a slightly different way (however..), without starting with a bounding circle. I'll post that solution later, not sure if it is entirely correct.

While the final answer doesn't change..

If the OP has a solution, it is the same as for a circle of finite radius. Very clever!

So obviously (at least) one of the approaches is wrong (if not both). Which one and why?

What if the 1st person is left handed and he is B? for the 1st question, he can not raise his right hand because he is not A, neither to raise his left hand which means (yes),and that is not true. There is no contradiction here. If first person is left handed and he is B, he has to answer the question 'will I raise my right hand for the question "are you A?"?'. Answer is 'Yes, right hand will be raised if the question was "Are you A"'.. Since B is left handed, he will finally raise his left hand to signify "Yes". I think my approach is essentially the same.

Wolfgang, have you seen my attempt at this? Is that correct?

I agree completely. Which is why, to be sure, we need to solve it for a bounded circle.