How can you find a fake coin with a balance scale? How can you use math to pretend to read minds? Solve these puzzles and build your foundational logical reasoning skills.

Agnishom is in love and decided to talk to his crush on Valentine's Day.

**Agnishom:** What's your phone number, sweetheart?

**Agnishom's Love:** Well, if you take your phone number and replace it the leftmost digit with a 1, you get my phone number.

**Agnishom:** And the 4-digit area code that comes before it?

**Agnishom's Love:** You know, the product of the four digits in the code is actually the square root of my phone number.

**Agnishom:** Hey, that's still insufficient information.

**Agnishom's Love:** Ah, but if I tell you the sum of digits of the code, then you can figure it out exactly.

What is the 4-digit code?

Note: Even though you have fewer pieces of information (e.g. you don't know how many digits the phone number has or what it is), you can still figure out the 4-digit code from the conversation.

Two logicians must find two distinct integers \(A\) and \(B\) such that they are both between 2 and 100 inclusive, and \(A\) divides \(B\). The first logician knows the sum \( A + B \) and the second logician knows the difference \(B-A\).

Then the following discussion takes place:

**Logician 1:** I don't know them.

**Logician 2:** I already knew that.

**Logician 1:** I already know that you are supposed to know that.

**Logician 2:** I think that... I know... that you were about to say that!

**Logician 1:** I still can't figure out what the two numbers are.

**Logician 2:** Oops! My bad... my previous conclusion was unwarranted. I didn't know that yet!

What are the two numbers?

Enter your answer as a decimal number \(A.B\).

\((\)For example, if \(A=23\) and \(B=92\), write \(23.92.)\)

**Note:** In this problem, the participants are not in a contest on who finds numbers first. If one of them has sufficient information to determine the numbers, he may keep this quiet. Therefore nothing may be inferred from silence. The only information to be used are the explicit declarations in the dialogue.

At the finish line, their configuration is as follows:

- 1 runner between the red pairs,

- 2 runners between the orange pairs,

- 3 runners between the yellow pairs,

- 4 runners between the green pairs,

- 5 runners between the blue pairs,

- 6 runners between the indigo pairs,

- 7 runners between the violet pairs.

If we know that the first runner wore a red shirt, what is the total number of possible configuration(s) of all the runners (from fastest to slowest)?

As an explicit example, if the runners were arranged as ROYGBIVROYGBIV, then there are 6 runners between all of the colored pairs.

I am thinking of an integer between 1 and 2015 (inclusive). You are required to guess the integer, by asking yes/no questions about the properties.

If I'm only allowed to lie exactly once, what is the minimum number of yes/no questions you need to ask, in order to guarantee that you can find my integer?

**Details and Assumptions**:

As an explicit example, suppose the integer I'm thinking of is 2015. Then you can determine my number in 4030 questions by asking whether my number is 1, 1, 2, 2, 3, 3, ... , 2015, 2015 in that order. And I will only lie on the second last question.

You are not allowed to ask questions that don't directly relate to the number. E.g "what other people think of this number."

Find the total number of distinct ways to join the six islands shown above by bridges such that

- each island can be reached from any other island via the bridges,
- 1 of the islands has 1 bridge leading from it,
- 2 of the islands each have 2 bridges leading from them, and
- 3 of the islands each have 3 bridges leading from them.

**Details and Assumptions:**

- Neither of the 2 islands on the far left can be joined directly to either of the 2 islands on the far right.
- There can be more than 1 bridge between 2 islands.
Mirror images and 180-degree rotations are not counted as distinct.

Diagonal bridges are

**not**allowed.

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