bonanova

I did the calculus afterward, and however you do the calculation, it's difficult. The remaining volume is a volume of revolution with requires finding the cross sectional area and spinning it thru 360 degrees. It's no easier to compute the cylindrical volume removed, cuz there are spherical "caps" on the cylinder which don't have formulas that I could find. The logical way is easier, and I'll post it in a day or so if you don't get it... have fun.

Yes.

Clarifications: [1] the hole is a circular cylinder of empty space whose axis passes through the center of the sphere - just as a drill would make if you aimed the center of the drill at the center of the sphere and made sure you drilled all the way through. [2] the length of the hole [6 inches] is the height of the cylinder that forms the inside surface once the hole is drilled. picture the inside surface as viewed from inside the hole and measure the length of that surface in the direction of the axis of the drill. in this sense, you could for example drill a 6-inch hole through the earth. the diameter of the hole would be huge, and you'd just have a tiny remnant of the earth left. but if you could set it on a table [a big table] it would be 6 inches high. you of course could not drill a 6-inch hole through a sphere whose diameter was less than 6 inches. it was actually this fact that led me to the logical answer and made me a genius for a couple of minutes.

He says, "I am lying." Which doesn't have a truth value.

Maybe this has already been posted. A friend asked me this a while back, and I answered her in less than a minute. She said I was a genius. But I said there were two ways to arrive at the answer, and I simply chose the easier way. A 6-inch [long] hole is drilled through [the center of] a sphere. What is the volume of the remaining portion of the sphere? The hard way involves calculus. The easy way uses logic.

Riddari, You're correct on both comments [physical attributes and spoiler for mistake.] Thanks ... !

I'm missing something. What rule is being broken?

A variation on the Lone Ranger theme: A man left home one day and began to run. After he'd gone a ways, he turned left and kept running. A bit later he turned left again and ran a little faster. Later on, he turned left a final time and ran even faster than before. In the distance he saw two masked men. Who were they?

I first heard it posed as ... if a hen and a half can lay an egg and a half in a day and a half, how many hens would it take to lay 18 eggs in 3 days? It's tempting to figure a hen can lay an egg a day so it would take 6 hens. But it takes a hen and a half to lay an egg a day, so it takes 9 .

A man has two girlfriends, reachable only by train. One lives to the North, the other to the South. Being a man, he is incapable of deciding which to marry. Since a train leaves his station each hour to the North, and another leaves each hour to the South, he decides to leave his amorous visits to chance. Let Fate decide. Being a clever person, he creates a device that sounds an alarm at a random time of the day. Each day, promptly after the alarm sounds, he takes the 5-minute walk to his station and boards the next train to arrive: Northbound or Southbound. After a year has passed, he finds he has visited one of the girlfriends [turns out it was the one to the South] more than 300 times, and so he marries her. Assuming his random time-of-day device was working properly, how could this have happened?

What goes up the chimney down, but not down the chimney up? He used his umbrella.

Overhead from the back seat... She: I am perfect. He: _______? Clue: How do get to Carnegie hall?

2498?

your math is off -- or your typist made a typo.... Two typists [together] can type at a pace of 1 page/minute. and therefore, also, they can type 18 pages in 18 minutes

water?

My name is Joe; my wife is Mary. Our daughter is Amber. Mary's mother is Barbera. Who is Barbera's daughter? Clue: it's Mary. Who is my daughter? Clue: it's Amber. Who is her mother? Clue: it's Mary. Listen as Joe speaks: Barbera's daughter is my daughter's mother. The speaker could be Barbera's son-in-law.

"You will give me the gold coin." is close. They can give you the gold coin [thus making the statement true and requiring them to do so.] But they can also give you nothing [thus making the statement false and requiring them not to give you anything.] You have to add something to change the second outcome. "You will give me the gold coin or you will give me nothing." Now they can't give you nothing because that makes the statement true.

My head says any lying day; my heart says Monday. What does the unicorn have to do with this?

Use the shorthand notation IH IS IN GH GS GN for the 6 cases of [innocent Guilty] [Honestant Swindlecant Normal]. Craft statements that distinguish you from the perpetrator [P] for 4 different scenarios: [1] You are an IS. The court knows P is a GS. They don't know whether you are H or S. [2] You are a GS. The court knows P is a GS. They don't know whether you are H or S. [3] You are an IH. The court knows P is a GH. They don't know whether you are H or S. [4] You are IH IS or IN [innocent, unspecified]. The court knows P is GH or GS [Guilty, but not Normal]. [1] Distinguish yourself [an IS] from P [a GS]. Create a statement that anyone but a GS can make. It must be true for a GS [can't] or any H [can] and false for IS [can]. i.e. the statement must be false iff the speaker is IS. "I am not an Innocent Swindlecant." What the court can conclude is this: If you were a GS [and fit P's known profile] this statement would be true and you couldn't make it. If you were an IS it would be false, and that's ok. If you were any flavor of H the statement would be true and you could make it; but no H is a GS. Thus, only an innocent person can make this statement. P cannot, and anyone else can. You could be anybody but P. Go home. [2] Distinguish yourself [a GS] from P [a GS]. You can't. You're cooked. Too bad. There is no statement that distinguishes a GS from a GS. Please report to the gas chamber. [3] Distinguish yourself [an IH] from P [a GH]. Create a statement that anybody but a GH could make. It must be true for an IH, false for a GH and false for any flavor of S. i.e. the statement must be true iff the speaker is IH "I am an innocent honestant." You're free to go. [4] Distinguish yourself [you are IH IN or IS] from P [a GH or GS]. Create a statement that any I can make, but neither a GH nor a GS can make. Cases and Truth requirements. IH - statement must be T IN - doesn't matter: N's can say T or F as they choose. IS - must be F GH - must be F GN - doesn't matter: such people do not exist. P is known not to be a GN. GS - must be T Taking the cases that do matter, the statement must be true iff the speaker is IH or GS. "I am either an innocent honestant or a guilty swindlecant" What the court can conclude is this: If you were an IH the statement would be true and you can make it. If you were an IN you could say anything you like, so you could say this. If you were an IS, the statement would be false, and you could make it. So you could be any flavor of Innocent. Further, If you were a GH the statement would be false and you could not make it. If you were a GS the statement would be true and you could not make it. You can't be a GN: P is known not to be N. GN's don't exist. So you could not be any flavor of Guilty All charges are dropped. You're free to go.

The answer is - YES - head for the hills and bring your pick and shovel. But I don't think anyone has given the correct analysis yet. Here's mine: First, note the statement that was made: There is treasure only if I am an honest man. Some have made the mistake of calling this logical equivalence. It's not. A only if B is logically the same as if B then A. Logical equivalence is more restrictive: A if and only if B. The truth tables differ in the case of a false premise and a true conclusion: "False implies Truth" is True for if; it's False for if and only if. Thus, we can restate simply as if B then A: If I am an honest man then there is treasure. There are two cases: the speaker is a honestant [H] or a swindlecant . [1] H - the speaker is an honest man If the speaker is honest, the premise is true [fact] and the logical implication must be true [else he would be lying]. Therefore the conclusion is true: There is treasure. [2] S - the speaker is lying. If the speaker is lying, the premise if false [fact] and the logical implication must be false, also, [else he would be telling the truth.] But, because a false premise validly implies every conclusion, such an implication is always true. A contradiction. Thus we must conclude that the speaker could not have been a swindlecant: one cannot invalidly conclude anything [tell a lie, as a swindlecant must do] starting from a false premise. Since the speaker must have been a truth-teller, there must be a treasure.

Let's identify the four types [rich poor], [honestant, swindlecant] of creatures as RH PH RS and PS. There are two statements that must be crafted: [1] A statement that only an RS can make, proving the speaker is a rich swindlecant. [2] A statement that only an RH can make, proving the speaker is a rich honestant. [1] A statement that only an RS can make. Such a statement must be false if spoken by either type of honestant, eliminating RH and PH. It must be true if spoken by a poor swindlecant, eliminating PS It must be false if spoken by a rich swindlecant, making it OK for RS. That is, the statement must be true if and only if the speaker is a poor swindlecant. The simplest such statement is: "I am a Poor Swindlecant." Only an RS can say that in character. [2] A statement that only an RH can make. Such a statement must be true if spoken by either type of swindlecant, eliminating RS and PS. It must be false if spoken by a poor honestant, eliminating PH. It must be true if spoken by a rich honestant, making it OK for RH. That is, the statement must be false if and only if the speaker is a poor honestant. The simplest such statement is: "I am not a Poor Honestant." Only an RH can say that in character.

There are 3 cases [HH, SS and HS] and in each case one or the other of the Aborigines could have answered. So the gringo could have heard any of 6 answers. As Ploper points out, 5 of these answers will be Yes. Only one answer could be No: the answer given by the Swindlecant in the case HS. So the answer is - the gringo heard "No" and that made it clear a Swindlecant was speaking and that there was one of each. A "Yes" answer gives no information about either point.

This is correct. The final bit of analysis is that we can't know what A is.

Thanks. I love these things, too, and thoroughly enjoy this site. - bn

Remember the question to answer is: What statement shall the mother make to save her child? First, let's ask: has the croc already decided what he'll do? [1] Yes. [this is suggested by "if you don't guess his fate ..."] The answer in this case is: Nothing the mother says can affect whether the child is eaten or returned. It's a fallacy to suggest there are Mother's words that will force the return of her baby. Its fate is sealed; not open to negotiation. Not a satisfying result for the puzzle solver or the mother. Poor baby. [2] No. [this is suggested by "If you guess [correctly] ... then I will ..."] In this case, what the croc will do is determined by the response of the mother. And the task is to construct a reply which will logically bind the croc to return the child. The best solution is reply that he will choose, and his choice will be either to eat or to return. This is a clever approach because it seems to cover all the bases and force the baby's return. But by assuring the baby's return, the mother seals its fate: The croc merely says, "sorry, but you're wrong. I'll prove you're wrong by eating the baby, and thus fulfill my word which is ... [read carefully]: If you don't guess his fate I'll eat him. [Only because] she secured his return by her clever answer, she wrongly guessed its fate. The croc had dinner that night. Another unsatisfactory result. Personally, I like the strategy mentioned above of threatening the croc with unbelievably heinous consequences if the baby is not returned. Always appeal to enlightened self interest.