At both the beginning and the end, the two cups have exactly the same amount of liquid in them. Therefore, the amount of tea that ends up in the coffee cup must be exactly equal to the amount of coffee that ends up in the tea cup.

0, 1, 2, 720!, 24!!!

The rule is *n* followed by *n* factorial signs.

1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98

The differences between pairs of numbers are all the numbers that are not in the sequence!

1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345

R.A.T.S. means Reverse Add Then Sort. 55778 + 87755 = 143533, and then sorting the digits gives the next number.

No. A domino must cover one black and one white square, and on the mutilated chess board there are 32 black squares and 30 white squares (or vice versa).

Put 1000 black balls and 999 white balls into one basket, and the remaining white ball into the other basket. Now your chance of survival is nearly 3/4.

Firstly, the disjointness is a red herring, since if you can find
*any* two sets that sum to the same value then you can always
remove the elements that they have in common to get two disjoint sets
that sum to the same value.

Secondly, there are 20 elements in the set X, so the number of subsets is 2 raised to the power 20, which is more than 1,000,000. But the sum of all the elements in X is 639576, so each subset must sum to less than this. Therefore there must be two subsets that sum to the same value.

A neat application of the pigeon-hole principle. There are more subsets than possible sums, so two subsets must sum to the same value.

Sorry, no neat solution to this one. But the unique solution happens to be:

SUM PRODUCT X Y 17 52 4 13

You say to one oracle: "if I was to ask the other oracle if the left path went to heaven, would he say yes?" If the reply is yes then the right path goes to heaven, and if the reply is no then the left path goes to heaven. The proof is by cases: first examine what the oracle would say if he was the truth teller and the other one lied, and then the other way around.

The second problem is a bit more complicated: the hint is to eliminate the random truth teller with the first question.

Yes, you should change. There is a 2/3 chance that the other door contains the prize.

To see this, imagine that there are not three doors, but 100. You pick one, and then the gameshow host opens 98 wrong doors. You can then choose to stick with your original door, or switch to the one remaining. Here it's plain that your door still only has 1/100 chance of being right, so the other door must have 99/100 chance of being right.

Look carefully at the picture. The long side of the large triangle (the one that contains the four shapes) is not a straight line! (It can't be, it's the join of two triangles: one with gradient 2/5 and the other with gradient 3/8.) In the top picture this bend adds area to the large triangle, and in the bottom picture it subtracts area. The amount of area that it adds or subtracts: exactly 1/2 a square.