[A Prisoner Problem]

I had a solution here, but found a flaw in it... *REMOVED*

[Picking his sentence]

If the judge had said that there were 3 boxes whose labels matched their contents, then those boxes must have been the ones labelled 4, 5, and 10. In this scenario, box #3 would contain 0 coins. The only other possible answers the judge could give (that I can see) are 0 or 1, neither of which would have been helpful to the accused. So, the answer the judge neglected to give must have been 3, and Box #3 should be chosen.

[Number puzzle that is irking me]

47. Pattern is (*2, +5, *2, +4, *2, +3, *2...)

[An irrational "aha!" puzzle]

Let g = p / q, with p and q both positive integers and gcd(p,q) = 1. g satisfies the equation g^2 = g + 1, therefore (p/q)^2 = (p/q) + 1, and (p^2)/q = p + q. Since p and q are integers, (p^2)/q must be an integer, therefore gcd(p,q) must be at least q . This is a contradiction since q cannot be 1 (sqrt(5), a component of g, cannot be expressed as an integer).

[How many faces?]

Seven. Each shape "sacrifices" one of its faces to be joined to the other shape. (5 - 1) + (4 - 1) = 7.

[The Fork]

I was thinking the latter... but since you mention it, let's analyze. Do any possible "false offers" from a lying tribesman result in a different approach? The only ways I can think for the offer itself to be a lie is if he will answer more than just one question (which only makes life easier), or will not answer any questions (which means you just have to guess), or will choose his answer from some other set of responses besides "Yes / No" (which I think would leave you with too little information to ask the right question). Any other options I missed?

I would add to the list:

1. Answer a different number of questions
2. Or none at all
3. Replies other than yes/no
4. promise that his yes/no answer would be truthful may not be reliable.

It was option 4 actually that came to my mind.

But perhaps a liar lying about telling the truth and having to tell the truth for the false promise to be false cancels out.

e.g. Both a T and a L will say "I am telling the truth;" neither can say "I am lying."

It's probably the latter, and we are worrying about nothing.

I'd tend to agree; the liar and truth teller will both claim to offer honest assistance. The liar instead offers dishonest assistance by lying in response to the question, thus making his original offer a lie, and preserving his "alignment" as a liar.

[The Fork]

Ask which way would a member of the other tribe tell you is the correct path to your destination. Then go the other way.

This will work for some similar puzzles, but not this one. The tribesman will answer one question with "Yes" or with "No", and that is all. He will not point to the correct path. Furthermore, as this is a three-way fork, there is still a choice of two "other ways" you can go even if you eliminate one option.

[The Fork]

Should we take into account that his offer is not to be taken as reliable in the case that he is a liar?

Or should we assume a trusted person tells us this person will truthfully or falsely answer a yes/no question if possible?

I assume it's the latter, but the former would be interesting to analyze as well.

I was thinking the latter... but since you mention it, let's analyze. Do any possible "false offers" from a lying tribesman result in a different approach? The only ways I can think for the offer itself to be a lie is if he will answer more than just one question (which only makes life easier), or will not answer any questions (which means you just have to guess), or will choose his answer from some other set of responses besides "Yes / No" (which I think would leave you with too little information to ask the right question). Any other options I missed?

[The Fork]

With a tip of the hat to itsmeee's 999 puzzle, and bonanvoa's Mad Hatter puzzle...

You come to a three-way fork in the road. You know that one path leads to your destination, and that the other two paths lead to Certain Doom. You know that in the area is a tribe of truth-tellers and a tribe of liars, both of which will answer appropriately whenever they can, or remain silent whenever they cannot answer a question truthfully (or falsely). A tribesman from one of these tribes -- you know not which -- is standing at the fork, and -- as if for his own amusement -- offers to answer one single question with "Yes" or "No" to help you find your way. What question do you ask him?

[9, 99 or 999?]

(for 99) He couldn't provide any response as they (1) wouldn't be truthful or (2) wouldn't contain an equitable number of letters to digits.

If my number is 99, and you ask whether I could provide an answer with as many letters as digits in my number, my truthful response would be "Yes" because I

could provide such an answer (the fact that said answer is actually "No" doesn't matter).

[Mad Hatter question]

I would re-label the doors 1, 2, and 3, and then ask, "What answer can you give that has as many letters as the number on the correct door?". If the Hatter says "Yes" I choose door 3 / C; if he says "No" I choose door 2 / B, if he vanishes I choose door 1 / A.

[9, 99 or 999?]

Will your response contain the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

But as phrased, if the number were 999 a response of either 'yes' or 'no' is equally valid. However, clearly, we need a question that can give 3 different types of responses, a yes, a no, or a lack of response...

I would ask: what answer has as many letters as digits in the number you are thinking of? If "Yes" the number is 999, if "No" it is 99; silence implies it is 9.

Could you provide a truthful response which contains the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

If he says "Yes", the number might be either 999 or 99, because the truthful response he

could have provided might have been either "Yes" or "No".

[9, 99 or 999?]

Will your response contain the same number of letters as the number's digits?

yes - 999

no - 9

(no answer) - 99

But as phrased, if the number were 999 a response of either 'yes' or 'no' is equally valid. However, clearly, we need a question that can give 3 different types of responses, a yes, a no, or a lack of response...

I would ask: what answer has as many letters as digits in the number you are thinking of? If "Yes" the number is 999, if "No" it is 99; silence implies it is 9.

[four of a kind]

Assume that a four of a kind is defined as only needing to match the numeric (or letter?) value or in other words that suits do not matter.

In that case, like Sp said. And the number of decks does not matter.

well this is embarrassing but for some reason i am getting a higher number, what am i missing?

There are 13 types of card in each suit. If you draw 39 cards and still do not have four of a kind, then you must have exactly 3 of each type. The 40th card will match with one of the sets of 3 you already have, making four of a kind.

[A series]

Each number is (2*n)^3 -1.

(A) 513 is incorrect.

(B) 511 should be in its place.

© 1727 and 2743.

[The Liar, the Truthteller, and the Trickster]

Depending on the behavior of the "alternator"

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "alternator" is in liar mode or truth-teller mode when queried, he must point in the correct direction.

Depending on the behavior of the "trickster",

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "trickster" is in liar mode or truth-teller mode when queried, he must also point in the correct direction.

OP says the trickster randomly lies or tells the truth. That kind of "trickster" can be modeled.

The responder who randomly says Yes or No cannot be modeled.

You might sharpen the question by asking: If I were to ask "Is THIS the road [pointing to one of the roads] that leads to my destination?", how would you respond? This constrains the reply to be Yes or No. As stated, the person is free to reply in a non-responsive manner: "I am not in a mood to answer," or "The sky is an unusual shade of blue today, isn't it?"

The question can certainly be sharpened. But be careful: in your phrasing, you lost one of my own "sharpenings":

The trickster is given by the OP to alternate between lying and truth-telling, so I felt it was important to specify that my hypothetical question was in place of the question I am currently asking, thus preserving the trickster's "state". Otherwise, as given, the answer could be interpreted to depend upon the timing of when the question was asked. The trickster might read the question as "If my next question to you was '

xyz...'."
The case of the random responder is more interesting... I have not solved that yet.

Also, note that the questions were not constrained by the OP to the yes-no variety, so I think my original answer makes sense so far as it goes.

[Numbered Foreheads Concluded]

Hint:

This is not a tedious problem... not much harder than the original. There is a simple way to solve it; a pattern to be observed. An exhaustive description of how each logician answered in his turn would not be very long...

[The Liar, the Truthteller, and the Trickster]

Depending on the behavior of the "trickster",

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "trickster" is in liar mode or truth-teller mode when queried, he must also point in the correct direction.

[Crossing the bridge]

First, A and B cross, taking 2 minutes. Then A returns with the flashlight in an additional 1 minute. Then C and D cross, taking an additional 10 minutes. Then B runs back with the flashlight in 2 additional minutes. Finally, A and B cross in 2 additional minutes. 2 + 1 + 10 + 2 + 2 = 17.

[The Liar, the Truthteller, and...cake!]

I would ask either one of Dee or Dum, "If I were to ask you whether the Queen likes Chocolate better than Vanilla, what would you say?" By asking the question this way, I am guaranteed a truthful response, which gives me the relative ordering of chocolate and vanilla in the Queen's preferences. For sake of argument, say she prefers Chocolate over Vanilla. Then, once I know the Queen's "middle" preference: if it is Vanilla, the order is Chocolate, Vanilla, Red Velvet. If her middle preference is Chocolate, then the order is Red Velvet, Chocolate, Vanilla. If her middle preference is Red Velvet, then the order is Chocolate, Red Velvet, Vanilla.

[Cheapskate Date]

This distance, call it

x, must satisfy (x / 8) + (x / 3) = 7. So, x = 7*(24 / 11) = ~15.2727 miles.

[Weightloss]

I guess we can assume by "round person" that your waistline approximates a circle.

We need to know the original or final size.

RG's answer is the correct one if you reduced down to a zero waist.

I'm looking at this problem assuming you started out as *gasp* 40" waistline..

R.G.'s answer is correct for any size of waist, assuming the waistline of the pants is pinched off to accentuate the difference. The difference in circumference is 5 inches, so the "slack" must be 5 inches long, but doubled over. Its length should therefore be 2.5 inches regardless of original size. No?

[The River]

1800 ft.

why?

Call the width of the river

w. Then from the moment the boats began moving to the moment the boats crossed for the first time, boat "A" traveled 700 feet and boat "B" traveled (w - 700) feet. Since they completed their respective distances in the same amount of time, and each boat is traveling with a constant speed (say Va for boat "A" and Vb for boat "B"), we have that 700 / Va = (w - 700) / Vb. By similar reasoning, from the moment the boats first crossed to the moment they crossed a second time, boat "A" completed a distance of (w - 700) + 300, while boat "B" completed a distance of 700 + (w - 300), and so we have that (w - 400) / Va = (w + 400) / Vb.

Dividing one equation by the other cancels out the terms Va and Vb, leaving: 700 / (w - 400) = (w - 700) / (w + 400). With a few algebraic manipulations, this simplifies to w*(w - 1800) = 0, from which the answer is apparent.

[Square inscribed in a triangle]

The length of the side is 168 / 29. Reasoning: the triangle atop the square is similar to the triangle it is inscribed within, and so it has the same proportions. The two "lower" triangles are both right triangles and so the Pythagorean theorem applies. Call the square's length

x, the length along the long side of the triangle to the left of the square y and the length to the right of the square z so that x + y + z = 21. Using Pythagoras, and the similarity in triangle proportions, we also have that x^2 + y^2 = (10 - (10/21)*x)^2 and similarly that x^2 + z^2 = (17 - (17/21)*x)^2. Much further algebra yields the above result.

It always amazes me when people use strategies hidden in the problem that i didn't even consider and even more makes me jealous when their strategy is more efficient than my own.

Thanks. Although witzar's answer involves even less work than mine.

And I was actually not aware of Heron's formula, so I learned something.

[Square inscribed in a triangle]

The length of the side is 168 / 29. Reasoning: the triangle atop the square is similar to the triangle it is inscribed within, and so it has the same proportions. The two "lower" triangles are both right triangles and so the Pythagorean theorem applies. Call the square's length

x, the length along the long side of the triangle to the left of the square y and the length to the right of the square z so that x + y + z = 21. Using Pythagoras, and the similarity in triangle proportions, we also have that x^2 + y^2 = (10 - (10/21)*x)^2 and similarly that x^2 + z^2 = (17 - (17/21)*x)^2. Much further algebra yields the above result.