gavinksong

I apologize, plasmid. It seems that you did not overlook what I thought you did. However, your final answer is still wrong.

Can you give a counterexample, or would it be too much of a blatant hint if you did? I can't give you a counterexample, but I can suggest that you look at the diagram that you drew in your first post again. The puzzle is just designed to cause people to make the same mistake you made. The correct solution isn't as crazy as bonanova made it seem, though I suppose it may be confusing at first glance. Don't let him scare you.

This explanation might not be the greatest but...

We cannot assume that there are any practical limits to the number of apples.

I think you guys need a little something.

Sneaky, but wrong. All Alice would have to do is put more than one or two apples in every room, and your strategy would fall flat.

What did you mean when you said the "die was rounded on its four corners just tangent to top and bottom edges"? Does that mean that the base of the resulting cylinder is the same size as the largest circle that can be drawn within a face of the original die?

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

An admirable attempt, plasmid - but incorrect Your explanation was very rigorous, but you overlooked something critical.

Are you sure about that, bonanova? If his mean distance is zero, that would mean that he is 100% likely to end up at the origin every time, since distance cannot be negative. As for your question, I octodad is limited to moving around only on the xy-plane, but he may move in any direction (not just horizontally or vertically). Cheers

Nice try. This is what k-man got as well. It's the obvious answer, but not the best one.

Plasmid, You may imagine the cake to stick back together after returning a slice. If you make another slice and flip it, the entire slice is rotated en bloc. You may end up rotating the cake by over 360 degrees. I hope that answers all of your questions.

This is a variant of BMAD's recent post on the probability of choosing a natural number with a 1 in the digits. From the set of all natural numbers... 1) What is the probability of choosing a natural number whose digits sum to an even number? 2) What is the probability of choosing a natural number whose digits sum to a number divisible by N?

Haha, it's this thing again. Probability and infinity. This isn't the answer I had in mind, but it did make me stop and think for a moment. The real strategy actually tends to work in real life. In fact, it is one that a normal human being would use almost instinctively.

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: Ohh, I see what you were getting at now, Karthick.

Nice try, but not quite. This problem is a bit more interesting than it looks at first glance.

In the popular board game, Risk, players try to occupy as much territory as possible by moving around their armies and attacking territory owned by other players. These battles are settled through a series of dice rolls. The attacker rolls up to three dice, while the defender only rolls two. First, the highest values rolled by each player are compared. If the attacker rolled a higher value, the defender loses a unit. Otherwise, the attacker loses a unit. Then, the second highest values are compared in the same manner and one of the players loses a unit. This goes on until one of the armies becomes depleted. If the attacker has fewer than three units, he may only roll the same number of dice as the number of units. Likewise, if the defender has fewer the two units, he may only roll one die. 1) What is the probability that a dice roll results in a draw? Okay, so now pretend there is a battle where the attacker has n units, and the defender has m units. 2) What is the probability that the attacker wins the exchange? 3) What is the average/best/worst case running time of the entire exchange?

(This puzzle is based off of a student-created problem on an exam for an undergraduate Intro to Discrete Mathematics course at UC Berkeley) Octodad has trouble walking straight. When he takes a step, he moves one yard in any random direction. He decides to try and practice moving around on an xy-coordinate plane. If he starts at the origin, what will be the mean and standard deviation of his distance from the origin after taking a large number of steps?

(This puzzle is taken from a blog called By Way Of Contradiction) Imagine the following two player game. Alice secretly fills 3 rooms with apples. She has an infinite supply of apples and infinitely large rooms, so each room can have any non-negative integer number of apples. She must put a different number of apples in each room. Bob will then open the doors to the rooms in any order he chooses. After opening each door and counting the apples, but before he opens the next door, Bob must accept or reject that room. Bob must accept exactly two rooms and reject exactly one room. Bob loves apples, but hates regret. Bob wins the game if the total number of apples in the two rooms he accepts is a large as possible. Equivalently, Bob wins if the single room he rejects has the fewest apples. Alice wins if Bob loses. Which of the two players has the advantage in this game?

(This puzzle is from a blog called By Way Of Contradiction.) Imagine you have a circular cake, that is frosted on the top. You cut a d degree slice out of it, and then put it back, but rotated so that it is upside down. Now, d degrees of the cake have frosting on the bottom, while 360 minus d degrees have frosting on the top. Rotate the cake d degrees, take the next slice, and put it upside down. Now, assuming the d is less than 180, 2d degrees of the cake will have frosting on the bottom. If d is 60 degrees, then after you repeat this procedure, flipping a single slice and rotating 6 times, all the frosting will be on the bottom. If you repeat the procedure 12 times, all of the frosting will be back on the top of the cake. For what values of d does the cake eventually get back to having all the frosting on the top?

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...?