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, note 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.

The Missing Digit

The number 229 has 9 digits, all different. Which digit is missing?

Bonus: Is the number 9991 prime?

Let the missing digit be m. Every number and the sum of its digits give the same remainder when divided by 9. The number 229 = 32 * 644 gives remainder 5 when divided by 9, and therefore 9 divides (0 + 1 + 2 + … + 9) – 5 – m = 40 – m. Thus, the missing digit is 4.

Bonus: 9991 = 10000 – 9 = 1002 – 32 = (100 – 3)(100 + 3) = 97 * 103. Therefore the number 9991 is not prime.