Under Water

The diving bell that Slylock Fox and Max Mouse are using has snapped its chain. A rescue team can reach Slylock and Max within 20 minutes, but there are only 15 minutes of air left. What will Slylock do?

He will use the lug wrench to remove the sunken car’s tires and release their air into the diving bell.

Handshakes at a Party

100 guests go to a party and some of them shake hands with each other. Show that there are two guests who handshake the same number of people.

Each of the people at the party has shaken hands between 0 and 99 times. However, if someone has shaken hands 0 times (with nobody), it is impossible that another one has shaken hands 99 times (with everybody). Therefore there are at most 98 different options for the number of handshakes at the party, and thus two of the guests have shaken hands the same number of times.

Old Man’s Killer

An eighty-year-old person was living alone in his house. On a sunny Friday morning, while delivering some letters in the mailbox, the postman felt suspicious and decided to look through the window of the house. That’s when he saw the old man lying dead on the floor in a puddle of blood. When the police arrived, they found three bottles of milk, the Tuesday’s newspaper, and some gifts in front of the house. The lock of the door was broken, and they already suspected a person. Whom and why?

SOLUTION

The police suspected the newspaper boy. Since there were 3 bottles of milk, the old man must have been dead for a few days already. However, the last newspaper left in front of his house is from Tuesday. Probably the newspaper boy, knowing that the old man is dead, decided that there is no point to go to his door and deliver newspapers anymore.

King Arthur and the Knights

King Arthur and his eleven honorable knights must sit on a round-table. In how many ways can you arrange the group, if no honorable knight can sit between two older honorable knights?

The answer is 1024 ways, up to rotation around the table. To see this, notice that the youngest honorable knight must sit right next to King Arthur – there are two possible places for him. Then, the second youngest knight must sit right next to this group of two. Once again, there are two possible places for him. Continuing like this, we see that for all honorable knights, except for the oldest one, there are two possible spots on the table. Multiplying two to the power of ten out, we get 1024.

The Pasta Puzzle

You have 10 strings of pasta left on your plate. You randomly start tying up their ends, until there are no loose ends anymore. What is the average number of loops which are created?

The expected (average) number of loops at the end of the procedure is equal to the expected number of loops created after the first tying, plus the expected number of loops created after the second tying, etc. After each tying, the number of non-loop strings decreases by 1, and then the probabilities to create a new loop are 1/19, 1/17, 1/15, etc. Therefore, the answer is the sum 1/19 + 1/17 + 1/15 + … + 1/3 + 1/1 ~ 2.1.