Beautiful Tapestry

A piece of a beautiful tapestry is missing. Can you figure out what its colors are?

Each 12×12 square on the tapestry represents a number between 2 and 26, such that all squares representing prime numbers are painted in single colors. The colors of the squares representing composite numbers are determined by the factors of these numbers.

The number 2 is represented by orange color. The number 3 is represented by green color. The number 4 = 2×2 is represented once again by orange (2, 2) color. The number 5 is represented by red color. The number 6 = 2×3 is represented by orange (2) and green (3) colors. The number 7 is represented by blue color. The number 8 = 2×2×2 is represented once again by orange (2, 2, 2) color. The number 9 =3×3 is represented once again by green (3, 3) color. The number 10 = 2×5 is represented by orange (2) and red (5) colors, and so on.

Does Not Divide Another

How many numbers between 1 and 100 can you pick at most, so that none of them divides another?

You can choose at most 50 numbers: 51, 52, 53, …, 100.

In order to see that you cannot choose more than 50, express each number in the form 2ⁿ×m, where m is an odd number. Since no 2 numbers can have the same m in their expressions, and there are only 50 odd numbers between 1 and 100, the statement of the problem follows.

Murdered Wife

One night, a man received a call from the Police. The Police told the man that his wife was murdered, and that he should get to the crime scene as soon as possible. The man immediately hung up the phone and drove his car for 20 minutes. As soon as he got to the crime scene, the Police arrested him, and he got convicted for murder.

How did the Police know that the man committed the crime?

The police did not tell the man where the crime scene was.