Challenging Math Problems (Or not)

[RandomPerson]

One of? If there's more than one I'm fine with that. Actually, if there is just one I'm fine with that too.

There are tons, I've only seen about 5 or so really good ones, including the official solution.

[HappySciencePie]

Oh, just tell us one of the answers; this is making my brain hurt! xD

[RandomPerson]

Shouldn't have been that hard, I thought you guys would get closer than this:
"Where X is your number, is '1 / (x - 2)' positive"
1 -> No
2 -> I don't know
3 -> Yes

2 is I don't know because the equation will be undefined, negative infinity and positive infinity.
I've heard arguments that undefined could be considered clearly not positive by a mean guard, though I think it is still a pretty clear good answer, just not the best.
Its the most popular unofficial answer I've seen, though others are more foolproof.

[HappySciencePie]

Ohh. That's a good one!
I guess we were thinking the wrong way, then. Actually, I wasn't really thinking at all.

[ophiophagus]

Another similar solution: Where X is the number, does 3/(x-1) yield a whole number result?

1- I don't know
2-yes
3-no

[paleonaps]

Three lawyers rent a hotel room for the night. When they get to the hotel they pay the $30 fee, then go up to their room. Soon the bellhop brings up their bags and gives the lawysers back $5 because the hotel was having a special discount that weekend. So the three lawyers decide to each keep one of the $5 dollars and to give the bellhop a $2 tip. However, when they sat down to tally up their expenses for the weekend the could not explain the following details:

Each one of them had originally paid $10 (towards the initial $30), then each got back $1 which meant that they each paid $9. Then they gave the bellhop a $2 tip. HOWEVER, 3 • $9 + $2 = $29

The lawyers couldn't figure out what happened to the other dollar. After all, the three paid out $30 but could only account for $29.

Can you determine what happened?

[RandomPerson]

Another similar solution: Where X is the number, does 3/(x-1) yield a whole number result?

1- I don't know
2-yes
3-no

Yeah it works, same type of idea.
The official solution to the question was:
"you could say to the guard “I have the number 1 or 2 in mind. Is the number that you have in mind larger than the number I have in mind?” If the guard answers “yes” then you know that they are thinking of number 3. If the guard says “I don’t know” then he is thinking of number 2. If the guard says “no” then he is thinking of number 1."

An unofficial solution that I also thought was good was:
"There are three children in my family including me, I am male, and I have a sister. Is your number greater than the number of male children in my family?"

[paleonaps]

I think that (the sibling one) may be the only foolproof one, because the guard could say no for two as well as one.

[blue cobra]

Three lawyers rent a hotel room for the night. When they get to the hotel they pay the $30 fee, then go up to their room. Soon the bellhop brings up their bags and gives the lawysers back $5 because the hotel was having a special discount that weekend. So the three lawyers decide to each keep one of the $5 dollars and to give the bellhop a $2 tip. However, when they sat down to tally up their expenses for the weekend the could not explain the following details:

Each one of them had originally paid $10 (towards the initial $30), then each got back $1 which meant that they each paid $9. Then they gave the bellhop a $2 tip. HOWEVER, 3 • $9 + $2 = $29

The lawyers couldn't figure out what happened to the other dollar. After all, the three paid out $30 but could only account for $29.

Can you determine what happened?

The money spent equals the bill plus the tip. Originally, the money spent was $30, which equals the bill plus the tip, $30+$0. When the bill changed to $25 and how much they spent changed to $27, they are still equal, since the money spent, $27, still equals the bill plus the tip, $25+$2. You are comparing the new price paid to the old cost required. Tsk tsk.

[paleonaps]

Three lawyers rent a hotel room for the night. When they get to the hotel they pay the $30 fee, then go up to their room. Soon the bellhop brings up their bags and gives the lawysers back $5 because the hotel was having a special discount that weekend. So the three lawyers decide to each keep one of the $5 dollars and to give the bellhop a $2 tip. However, when they sat down to tally up their expenses for the weekend the could not explain the following details:

Each one of them had originally paid $10 (towards the initial $30), then each got back $1 which meant that they each paid $9. Then they gave the bellhop a $2 tip. HOWEVER, 3 • $9 + $2 = $29

The lawyers couldn't figure out what happened to the other dollar. After all, the three paid out $30 but could only account for $29.

Can you determine what happened?

The money spent equals the bill plus the tip. Originally, the money spent was $30, which equals the bill plus the tip, $30+$0. When the bill changed to $25 and how much they spent changed to $27, they are still equal, since the money spent, $27, still equals the bill plus the tip, $25+$2. You are comparing the new price paid to the old cost required. Tsk tsk.

I didn't make this up! But you did get it.