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Gerrymandering

The city has 49 districts that vote for blue, yellow, or purple as shown in the grid. Seven electoral regions are drawn up to elect a city council. Each region consists of seven districts and each region will elect one councilor. Can you gerrymander the map so that blue will win the majority?

The solution is shown below.

Tough Decisions

You are driving your car along the road in a very harsh snowy weather and reach a bus stop. On this bus stop you see that there are three people waiting – your best friend, a sick old lady and the girl of your dreams. The car unfortunately can accommodate only 2 people (including you), so you can not take all of them with you. What will be your choice?

The best solution is to let your friend drive the old lady and you stay with the girl of your dreams on the bus stop.

Potatoes and Wire

If you have two spheres, you can always make a loop with some wire, such that it fits both of them (for example you can shape it as a circle). If you have any two potatoes, can you do the same?

Yes, you can. Just take the two potato surfaces and intersect them. Their intersection will give you the shape of a wire which fits them both.

Man in the Elevator

One man who lives on the 12th floor every day takes the elevator to go to work. When he comes back in the evening, unless it is raining or there are other people in the elevator, he goes with it just to the 8th floor and then uses the stairs for the last 4 floors. Why does he do this?

The man is very short. Unless he has an umbrella or can ask other people to press the button for him, he can’t make the elevator go to the 12th floor.

Escaping the Tower

A Queen is captured in the top room of a very high tower, along with her son and daughter. Outside their window, there is a pulley with a rope around it, and two identical baskets fastened at the ends of the rope. The queens weigh 195lbs, the daughter weighs 105lbs, and the son weighs 90lbs. If any of them gets in a basket and weighs 15lbs more than the contents in the other basket, he/she will fall too fast and will die upon impact with the ground. How can the three escape the tower, using the two baskets and one 75lbs stone they have in their room?

1. The stone is sent down in a basket.
2. The son goes down, the stone goes up.
3. The daughter goes down, the son goes up.
4. The stone is thrown back down.
5. The queen goes down, the daughter and the stone go up.
6. The stone goes down.
7. The son goes down, the stone goes up.
8. The daughter goes down, the son goes up.
9. The stone is thrown back down.
10. The son goes down, the stone goes up.

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Domino and Chess

Can you cover the chess board with 31 domino pieces, such that only two opposite corners are left uncovered?

The answer is NO. Every domino piece covers exactly one black and one white square on the chess board. Since initially you start with a different number of available white and black squares on the chess board, it is impossible to cover it with domino pieces.

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Sequence 1, 11, 21, 1112

What follows next in the following sequence:

1, 11, 21, 1112, 3112, 211213, 312213, ?

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.

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The Monty Hall Show

You are in Monty Hall’s TV show where in the final round the host gives you the option to open one of three boxes and to receive the reward inside. Two of the boxes contain just a penny, while the third box contains $1.000.000. In order to make the game more exciting, after you pick your choice, the rules require the host to open one of the two remaining boxes, such that it contains a penny inside. After that he asks you whether you want to keep your chosen box or to switch it with the third remaining one. What should you do?

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.