## Scratch My Head

Tear one off and scratch my head,

What once was red is black instead.

What is it?

**SOLUTION**

The answer is MATCHSTICK.

A collection of Riddle, Math, Chess, Detective, Lateral, Insight, Science, Practical, and Deduction Puzzles, carefully curated by Puzzle Prime.

Tear one off and scratch my head,

What once was red is black instead.

What is it?

The answer is MATCHSTICK.

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.

I am big on Saturday and Sunday.

I am small on Tuesday, Wednesday, and Thursday.

I am not on Monday and Friday.

What am I?

The answer is the letter S.

White to play and mate Black in 20 moves.

The moves are as follows:

- Qb7+ Rc5c6
- Qa5+ Rc4c5
- Qb3+ Rd4c4
- Qd2+ Re4d4
- Qf3+ Re5e4
- Qg5+ e5
- Qf7+ Rd6e6
- Qd8+ Rc6d6
- Qb7+ Rc5c6
- Qa5+ Rc4c5
- Qb3+ Rd4c4
- Qd2+ Re4d4
- Qf3+ e4
- Qg5+ Re6e5
- Qf7+ Rd6e6
- Qd8+ Rc6d6
- Qb7+ Rc5c6
- Qa5+ Rc4c5
- Qb3+ Rd4c4
- Qd2x

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.

What does the liar do when he dies?

He lies still.

Find all 10-digit numbers with the following property:

- the first digit shows the number of 0s in the number
- the second digit shows the number of 1s in the number
- the third digit shows the number of 2s in the number, and so on

Let the number be **ABCDEFGHIJ**. The number of all digits is 10:

A + B + C + D + E + F + G + H + I + J = 10

Therefore, the sum of all digits is 10. Then:

0×A + 1×B + 2×C + 3×D + 4×E + 5×F + 6×G + 7×H + 8×I + 9×J = 10

We see that F, G, H, I, J < 2. If H = 1, I = 1, or J = 1, then the number contains at least 7 identical digits, clearly 0s. We find A > 6 and E = F = G = 0. It is easy to see that this does not lead to solutions, and then H = I = J = 0.

If G = 1, we get E = F = 0. There is a 6 in the number, so it must be A. We get 6BCD001000, and easily find the solution **6210001000**.

If G = 0, F can be 0 or 1. If F = 1, then there must be a 5 in the number, so it must be A. We get 5BCDE10000. We don’t find any solutions.

If F = 0, then the number has at least five 0s, and therefore A > 4. However, since F = G = H = I = J = 0, the number does not have any digits larger than 4, and we get a contradiction.

Thus, the only solution is **6210001000**.

There are 5 points in a square 1×1. Show that 2 of the points are within distance 0.75.

Split the unit square into 4 small squares with side lengths 0.5. At least one of these squares will contain 2 of the points. Since the diagonals of the small squares have lengths less than 0.75, these 2 points must be within such distance.

There are 5 fish in a pond. What is the probability that you can split the pond into 2 halves using a diameter, so that all fish end up in one half?

Let us generalize the problem to N fish in a pond. We can assume that all fish are on the boundary of the pond, which is a circle, and we need to find the probability that all of them are contained within a semi-circle.

For every fish **Fᵢ**, consider the semi-circle **Cᵢ** whose left end-point is at **Fᵢ**. The probability that all fish belong to **Cᵢ** is equal to **1/2ᴺ⁻¹**. Since it is impossible to have 2 fish **Fᵢ** and **Fⱼ**, such that the semi-sircles **Cᵢ** and **Cⱼ** contain all fish, we see that the probability that all fish belong to **Cᵢ** for some **i** is equal to **N/2ᴺ⁻¹**.

When N = 5, we get that the answer is 5/16.

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