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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.

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.*

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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.

###### This is a harder version of an earlier problem.

###### Image Credit: Flickr Mark Mullen.

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.

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There are two integers, x and y, both of them are valued between 2 and 99 (inclusive)

Mr. S only knows the sum of the two numbers, while Mr. P only knows the product of the two numbers.

Mr. S told Mr. P, "I know you won't know what the two numbers are."

Then Mr. P said "Now I know what are they."

Followed by Mr. S "I get it now."

What is the product of the two numbers?

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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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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.

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