Logic
# Logical Reasoning

In a marathon, there are 14 runners. There are exactly of two runners that are wearing the same color shirt for each color of the rainbow.

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.

Find the total number of distinct ways to join the six islands below 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.
- No two bridges can intersect with each other.

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.

You are the ruler of a great empire and you have decided to throw another celebration tomorrow. However, apparently you were forgetful of the near disaster that happened in the last celebration, and decide to hatch your own plan with poison. For during this celebration, one of your most hated rivals will come, thinking that you have finally given up your fight and you held this celebration in part to surrender to your rival's superiority. You decide to pour some drops of that same deadly poison (that have no symptoms except death 10 to 20 hours later) into his glass; however, you do not want to look guilty, so to push the blame away you will insert some poison in your own drink that will make you sick after 10 to 20 hours with some worrisome but definitely non-lethal symptoms.

You have already inserted the deadly poison in one glass and your non-lethal poison in another glass out of the 1000 glasses, but because of your forgetfulness you forgot which glasses had the poisons! Certainly, you don't want another one of your beloved guests to drink a poisoned glass, or even worse, giving yourself the lethally poisoned glass.

Fortunately, you still have your supply of death-row prisoners who you are sure won't mind lending a helping hand by taste-testing the glasses. What is the least amount of prisoners needed to guarantee successful location of the lethally and non-lethally poisoned glasses?

*Note: if the lethal poison acts before the non-lethal poison, the prisoner will die without any symptoms.*

You met a magician on a train, and after a little chat, he took out $3$ squares of different sizes, as shown below. (The figure may not be drawn to scale.)

**Magician:** "These squares' side lengths are **distinct** digits (from $1$ to $9$ inclusive) whose greatest common factor is $1$."

With a flip of his hand, he transformed the squares into one whole rectangle.

**Magician:** "Now this rectangle's area is the same as the combined area of those $3$ squares. The width of the rectangle is equal to the sum of the $3$ squares’ side lengths, while the height of the rectangle is another **distinct** digit."

**You:** "How amazing! Can you tell me that height then?"

**Magician:** "No. Even if you know it, you still can’t work out the area of the rectangle."

**You:** "Can you at least tell me just one side length of the squares then?"

**Magician:** "No. Even if you know just any one square's length, you still can’t work out the area of the rectangle."

**You:** "Thanks! Now I know the area of the rectangle."

The magician became baffled after he had been advertently tricked to slip out a big clue.

What is the area of the rectangle?

Inspired by Digitalize This.