The answer is 50%. The reason is that no matter what birth plan the society comes up with, the chance for having a boy or a girl during every birth is 50/50.

Every term in the sequence describes how many 1s, 2s, 3s, etc. were there in the previous term. “11” means that in the previous term there was one 1. “21” means that in the previous term there were two 1s. “1112” means that in the previous term there were one 1 and one 2. “3112” means that in the previous term there were three 1s and one 2. “211213” means that in the previous term there were two 1s, one 2 and one 3. “312213” means that in the previous term there were three 1s, two 2s and one 3. Therefore, the next number in the sequence should be 212223.

This is the so called “Monty Hall” problem. The answer is that in order to maximize your chances of winning $1.000.000, you should switch your box. The reason is that if initially you picked a box with a penny, then after switching you will get a box with $1.000.000. If initially you picked a box with $1.000.000, then after switching you will get a box with a penny. Since in the beginning the chance to get a penny is 2/3, then after switching your chance to get $1.000.000 is also 2/3. If you stay with your current box, then your chance to get $1.000.000 will be just 1/3.

If you look carefully, you will notice that every person in the picture of 12 people is slightly taller than the corresponding person in the picture of 13 people. Basically, we can cut little pieces from 12 different people without making noticeable changes and arrange them into a new person.
For the second question, none of the shapes before and after the scrambling is really a triangle. One of them is a bit curved in at the hypotenuse and the other one is a bit curved out. This is barely noticeable, because the red and the blue triangle have very similar proportions of their sides – 5/2 ~ 7/3.

3 men must have sex with 1 woman, but they have only 2 condoms. Each of the 4 people has some unique STD which they don’t want to transfer to the rest. What can they do?

Since we can not see the passenger doors, they should be on the other side of the bus, and therefore the bus should be on the other side of the street. Here we assume the bus is from a right-hand traffic country.

Yes, you can, as shown on the picture below.

It will remain the same. The amount of water that the ice cube displaces is equal to its mass. Since the mass does not change and the density of water is equal to 1, the extra water after melting will be the same amount as the displaced water before that.

Separate the coins into one group of X coins and into another group of 50-X coins, then flip every coin in the first group. If in the first group there were Y heads up initially, then after flipping there would be X-Y – exactly the number of heads up in the second group.

The patient should keep taking one pill from each bottle until there are 4 pills remaining – 1 in the first bottle and 3 on the table. On the 29th day he splits the pills in halves and takes one half from each pill. On the 30th day he takes the remaining halves of the pills.