Sean Moran must be new here...
1 

Right or Left? 
The Famous Hole in a Sphere
Actually, I checked out the picture and you might be right, but the poster only said the length of the cylinder, not radius or circumference or anything, so the answer would not be known unless the radius were so small as to not make the difference matter much. Also, using my method you still get the same amount, so apparently the source didn't think about it. I still keep my comment about volume, though. context 
The Famous Hole in a Sphere
Volume is the measure of how much space there is in an area, not how much stuff is in there. The unit of volume is length cubed, which implies this as well. Therefore, a solid sphere has the same amount of volume as the same area of vacuum or air. As for your second point, the poster said the cylinder was drilled straight through the middle, and by definition a line straight through the middle of a circle or sphere is the diameter. Also, using my method, I got the same answer as the poster got. context 
The Famous Hole in a Sphere
You could just do: The diameter d of the sphere is 6 in (because the cylinder is 6 in long & it goes through the middle of the sphere, i.e. the part that has the "true" diameter). Therefore, the radius r is 3 in (r = d / 2 = 6 / 2 = 3). Then, using the equation V = 4/3pi(r^3) we get 36pi. Also note the cylindrical hole doesn't take any volume away, so the volume before the hole is taken out is the same as the volume after. context 
3 Coins
If that's all true, which makes sense, so I'm not going to dispute it, then for the 2nd case it's the same for heads or tails as if you got 2 heads it's not the tails coin and if you got 2 tails it's not the heads coin so it's: Still 1/5 for the fair coin or 4/5 for the allheads or alltails coin then (1/5)(1/2)(1/2) + (4/5)(1)(1) = 1/20 + 4/5 = 1/20 + 16/20 = 17/20 context 
Tick Tock
This didn't really explain enough to complete the problem. For example, can you take more than 15 minutes to time it, as the problem says "what is the quickest way?", which implies it may take more than 15 minutes (or less). context 
Don't Mate in One
OK that makes it better, thanks. context 
Multiples of 8 plus 1
I'm not a mathematician, I just loooooove math. I also don't know any mathematicians (minus random people on the internet who I don't really know). context 
Parodoxical Statements from Life
For the last question, nein (or non, no (in Spanish), etc.). context 
Time To Fuse
You never said it takes the same amount of time to burn the same amount of fuse, which the answer requires you to "know", only that it takes a minute to burn the whole fuse; much less that the fuse burns from both ends at the same rate. context 
Betsy and Patsy
*SPOILER** They're still twins, twins are anyone born on the same day of the same year, whether there are 2 of them or 500 of them. context 
The North Pole Bear Puzzle Reloaded
He could easily be anywhere he could travel in more than 2 dimensions, and he would travel diagonally instead of straight. He would have to travel pretty far in the other direction, though. Also, it never said he had to be on Earth, he could be at the north pole of another planet or in some place in space twisted so much by gravity as to be unrecognizable to modern physics. context 
Don't Mate in One
This problem didn't explain enough to figure it out. I think it meant, "Find a move for white that results in check for black but not checkmate", but even that could have much more than 1 answer. context 
3 Painted Ladies
I would just keep laughing, and probably even laugh harder. context 
Fork in the road
Well you can kind of give them mystical quantities and say one of them ALWAYS tells the truth no matter if he/she knows it & same with the other twin lying. Although the answer to the solution's question could still be "I don't know" in that case... context 
Right or Left?
I saw her spinning clockwise. Sometimes, you can make yourself see the picture happening the opposite way, like a barber's pole, but I can't see her spinning counterclockwise no matter how hard I try. Incidentally, I'm lefthanded, and lefthanded people are supposed to use the right side of their brain more, too, I think. context 
Don't be a Square
Was there supposed to be a picture here? I don't see one. context 
Multiples of 8 plus 1
Sorry I write a lot (and talk a lot). context 
Multiples of 8 plus 1
I don't think they have any proofs for powers as a whole, which is kind of surprising, as they have proofs of like everything else and have had 1000's of years to do it. Especially if someone like me could come up with a couple in like 10 years. context 
Multiples of 8 plus 1
Well k is the lower number, so for the difference between, say, 6^2 and 7^2, k would be 6. The difference involves the sequence 1 + 2 + 3 + 4..., and according to Wikipedia, that sequence is triangular numbers. So the difference between 3^2 and 4^ would be 6k(1 + 2 + 3) + 1, or 37. Incidentally, this means for powers up to 3, the difference involves factorials, as the difference between any 2 consecutive integer powers of 1 is 1, the difference between powers of 2 involves 2 (2! or 2 * 1), and for cubes it's 6 (1 * 2 * 3). Then each next power involves adding something weird to the mix, like triangular numbers. context 
The Neighbourhood
I was wondering about that one. I just kept putting "other." context 
Multiples of 8 plus 1
Yes, there are many patterns among powers. I've been kind of trying to figure out all of them, maybe one day I will & it'll be like my magnum opus or something. As someone hinted up above for squares the difference between the squares of any 2 integers, positive or negative, is 2k + 1, where k is the lower number, so between 1^2 & 2^2 it's 2(1) + 1, between 5^2 & 6^2 it's 2(5) + 1, etc. I also found that for cubes, it's 6k(triangular number ending in k) + 1. I'm sure there are more. context 