bonanova

Hello coolfflame, and welcome to the Den. There is enough information to solve this. A word of advice. You will earn respect from members when you respect them. That would mean first assuming a puzzle does have an answer and trying to find it. If in the process you come to believe that it can't be solved, show that in the form of a proof. Finally, please read the READ BEFORE POSTING link at the top of the page. Name calling, even in jest, won't earn you points with the site administrator.

None of the following experiments can be carried out in practice. But that doesn't stop us from thinking about them. In one way or another each of them involves the infinite series 1/2 + 1/4 + 1/8 + ... + = 1. Each asks for a response. Which responses, if any, can be given? A lamp is switched on and remains on for 1/2 minute. Then it's switched off for 1/4 minute. Then on for 1/8 minute, and so on. After 1 minute, of course, the switching stops. Describe the condition [on or off] of the lamp. . A supercomputer is programmed to calculate a print the value of pi, one digit at at time. The first digit prints after 1/2 minute. Each succeeding digit prints in 1/2 the time, and, [to keep the supply of printer paper finite], in 1/2 the space of the previous digit. After 1 minute, of course, the printing stops. Can we at last determine pi's final digit? .A marble rests in tray A. After 1/4 minute a machine transfers the marble to tray B and rests for another 1/4 minute while a different machine transfers the marble back to tray A. During the next 1/8 minute the first machine puts the marble back to tray B and rests 1/8 minute while the second machine returns it to tray A. After 1 minute the halved-time transferring process stops. Describe the location [tray A or B] of the marble. .A constant-speed runner completes 1/2 of his journey in 1/4 minute; then he rests for 1/4 minute. In the next 1/8 minute he completes 1/4 of his journey, resting thereafter for 1/8 minute. And so on. Describe the position of the runner after 1 minute. .A certain toy ball returns to 1/2 its original height 1/2 minute after being dropped. It falls from that point and returns to 1/4 the original height 1/4 minute later, and to 1/8 the height 1/8 minute after that. And so on. Describe the motion of the ball 1 minute after the initial drop. .The ball in question 5 is initially red. It changes color, alternating between red and blue, each time it strikes the ground. Describe its color after 1 minute. .

Last night at Morty's, Alex asked his buddies: What is the probability that a random chord drawn through a circle has length greater than the side of an inscribed equilateral triangle? After scribbling a moment on the classy new, M - monogrammed napkins, each of them had an answer: Ian quickly announced, it's 1/4. Jamie was more optimistic with his result of 1/3. Davie smiled and said, If ya do it truly randomly, it'll happen 1/2 the time. Which, if any, of them is right? This is one puzzle where ambiguity in the OP is not only OK, it's necessary. Give reasons, and please use spoilers.

No, it doesn't, and no one else does, either. But that's not a solution. Try harder.

Apologies for the ambiguous wording... - bn

Yes, the process is often the most satisfying part of solving a puzzle. And you might start out solving the 6-ball and 10-ball racks.

Alex slides a cold one down the bar to superp!

Exactly. But, you know Alex....

I meant for the comments added mistakenly to DeeGee's last post to be a reply of my own. Moderators get confused sometimes.

What's with Alex? Jamie was watching his friend brood over a square of paper, writing some numbers, erasing them, trying other numbers and erasing again ... He had just racked for a game of 8-ball with Ian, Davie explained; and he noticed the 8 ball was out of position. He also noted something about the placement of the other balls in the rack. While he worked out their positions, and before he could fix the rack, Ian broke. Now he's trying to reconstruct the layout. And what was so special? Jamie wondered. Looking at the rack as Ian saw it, Davie continued, with the triangle pointing toward him, each ball except for the five in the back row was the positive difference of the two balls diagonally behind it. Huh? Jamie had downed a few, and his mind was not the clearest. Look - suppose there were only three balls. Then it could have been the 1-ball in front of the 2- and 3-balls. Or the 2-ball in front of the 1 and 3. Get it? But with all 15 balls it gets more complicated. I still don't get it, there must be a bunch ways that could happen with that many balls. Why is it taking him so long? He thought so, too, and there are, for smaller triangles of 3, 6 and 10 balls. But for 15, he's convinced now that there is only one [not counting the reflected] solution. With that, Alex broke his silence, passed them both a square of paper and offered to buy drinks if either could find the fifteen-ball "layout with a difference" before he did. Davie and Jamie began to scribble ... And now here's your paper. The clock's ticking! Enjoy

You can do pretty much the same math using distances and Pythagoras [x2 + y2 = z2], or angles and trig identities [sin2 + cos2 = 1]. In this case the angle a had an identifiable significance - the angle of paths 1, 3 and 5 - so I went with angles. Your x is my 1/sin(a).

Perhaps, but twenty prisoners [as specified in the OP] in a line before the executioner are well able to hear the replies. If there is crowd noise, or if there are millions of prisoners [not provided in the OP] these impediments would be overcome by the 10MW PA system and giant projection displays [also not provided in the OP ]. The OP could have been more crisp on what's allowed, but the consensus seems to be that "not communicating" other than to say RED or BLACK precludes stamping, shouting, whining, singing, whispering, altering pace, or in other ways offering information additional to the color guess.

This is not a duplicate post, and I don't mean to divert discussion, but, a while ago, I suggested a which was never fully solved. When discussion here is finished, you might want to take a look.

According to folklore, marital fortunes were once told by a simple method. A girl would hold six long blades of grass in her closed hand, the blades protruding from each side. Another girl would tie the ends in pairs, at random, first on one side, then on the other. The first girl would then open her hand and inspect how the blades were connected. The possible outcomes are three small loops; one small loop and one medium loop; and one large loop. If the blades became one large loop, the girl doing the tying would be married within the year! The puzzle is in two parts: . What is the probability of obtaining one large loop?How many blades of grass must the first girl hold for the probability to fall to 1/8?. As always, enjoy!

p_m and DG have it. Scratching my head for a puzzle that will last longer than 8 minutes ...

It's trivial to determine the maximum number of unit squares that can be arranged in such a way that each square shares a portion of an edge with each of the others. So let's up the ante. What is the maximum number of unit cubes that can be arranged in such a way that each cube shares a portion of a surface with each of the others. A common surface need not be an entire face, but it must not be merely a line or a point.

I think this refers to the increase of a sphere's area as its radius increases. If sea level were to rise, say, by 1 inch, that would add 1 inch to the sea's spherical radius, and the sea's spherical surface area would increase accordingly. So if a given increase of sea water produced the first inch rise of sea level, a slightly larger increase of sea water would be needed to produce the next 1-inch rise. And so on. Ah... re-reading, now I think this refers to the increase in sea area, which began in the OP as 3/4 the area of a 4000-mile-radius sphere, but increases as land masses are inundated, giving up their area to the sea. Both apply. But while the first is easier to calculate; the second is probably a larger effect.

Hi Admiral, and welcome to the Den. This answer is temptingly straightforward, and many have offered it previously in this thread. And it would work, if all that was required is to know the color of your hat. Unfortunately, you're required to say your own color, correctly, or you die. So you see this solution does not guarantee survival for any of the prisoners.

Kudos to tbrophy for the initial analysis and DeeGee who brought it home!