Math and Logic Puzzles

[Espeonage]

Agh! Sorry for being so late. I was out. Anyhoo, lets start with a classic, just cuz I'm too tired to think...

You are traveling by foot to a shop in the City of Truth, where everybody tells the truth. You reach a fork in the road. You know the City of Truth is one way, but you don't know which. You also know that the City of Lies, where everybody lies, is on the opposite side of the road as the City of Truth. There are two children standing at the fork. You can tell that one of them is from the City of Truth, and one is from the City of Lies, but you cant tell them apart. You can ask them one question. What would be the best question to ask?

Sorry for unoriginality and possibly awkward wording, but I'm already late enough as is

Ask one what the other would say if I asked then which way to the city of truth. You then go the other way.

[Espeonage]

He missed the bit where you have one question to determine what way to go.

I did algorithms and problem solving at uni last year. We spent three weeks on knights and knaves.

[Espeonage]

TS should win that. Espe asked a question to both, which is 2 questions. For TS's, all you need to do is ask one of them.

How did I ask a question to both?

[Espeonage]

I doesn't have one. I can't remember many of the ones I did accurately enough.

I make up an impromtu knights and knaves that is similar but different.

You are at a fork in the road. This time there is only one person there. You do not know if they always tell the truth or if they always lie. What do you aske them to find which way to go?

Considering its sister problem was the last one this should be easy.

[Espeonage]

Nope

[Espeonage]

Which way will somebody from the other city say leads to the City of Truth?

Right question.

[Espeonage]

yeah

[Espeonage]

I've done this puzzle before. I can't remember how to do it.

there is a formula that can be applied. And there is also another puzzle to do with now many pirates you can have before the first pirate won't survive.

[Espeonage]

Exactly 288.

I just did it and got 384.

I'd like to see the answer.

[Espeonage]

OK, so let's arbitrarily assume that the first row is 1-2-3-4. (We will compensate for this later.) The next row is one of four possibilities. Two of them have two sub-possibilities each, and the other two of them have four. (2*2)+(2*4)=(2*6)=12. The compensation factor is 4!, or 24. 12*24=288. Q.E.D.

Ah.

Still don't know where I went wrong though.

I did it.

4! for first box.
Top left of the next box has two possible values and same with the one underneath. So 4!x4
This is the exact same for the next box. So 4!x16.
Last box is already determined.

4! x 16 = 384

[Espeonage]

Okay. I'm going to answer the sudoku question now, but in a slightly different way.

Lets start with the upper-left 2x2 grid. There are 24 distinct possibilities for it (4!). Now, the lower-right grid also has 24 distinct possibilities. HOWEVER, some of these possibilities will make the remaining two grids impossible to fill in. Lets consider a sample upper-left 2x2 grid filling in of:

1 2 X X
3 4 X X
X X A B
X X C D

Now, if A and B are either 1 and 3 or 2 and 4, there will be a square in the lower-left grid which can't be filled in. So A and B are either 1 and 2 or 3 and 4 or 1 and 4 or 2 and 3. The same is true of C and D.
Similarly, {A, C} and {B, D} can't be {1, 2} or {3, 4}. So from this, we get only a few possible combinations for ABCD from the original 4!:

if A=1, B = 2 or 4 and C = 3 or 4. So this allows:
1 2
3 4,
1 4
3 2,
1 2
4 3. So 3 possibilities. Similarly, there are 3 possibilities if A is 2, 3, or 4. Therefore, there are 3*4=12 possibilities for the lower-right grid. Whatever the lower-right grid is, as it winds up, the upper-right and lower-left grids will have already been determined.

So I get 12*24 = 288, which agrees with personalidiot.

OK. Get it now. Thanks.

[Espeonage]

Call me stupid if I'm wrong but couldn't you line the bottoms of both jars up with blacks and then put whites only at the top of the jars thus meaning that if you pick from the very top you will always pick white?

[Espeonage]

Oh. SKrew you.

I bet you chose this username for that sole purpose, didn't you?

[Espeonage]

3.

.5 + .25 + .1 + 3x .5

[Espeonage]

From the guy at the end staying silent he knows that it is impossible for the two in front to both have black hats.

The second guy in line knows this and sees either a white hat in front or a black hat in front. If he sees a black hat in front he would know he had a white hat because of what the first guy didn't say. The guy at front knows this also and concludes that for the second guy in line to have not said anything he must have a white hat.

[Espeonage]

Yay. let me think of a puzzle.

[Espeonage]

_____________
| o o o o o o o o |
|_____________|
O O

Which direction is the bus going? WHY?

Need to answer the second to get it.

[Espeonage]

Aww gay. Spaces didn't work. You get the point.

[Espeonage]

So you're saying we should smell out the problem?

[Espeonage]

The nest one is the time at which you can hear Sudo touching himself?