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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.

8 logicians,Calvin, Azhaghu, Ishan, Nihar, Brian, Sandeep, Tanishq and Prasun are being chased by the deadly medieval monster, Amphisbaena.
###### Inspired from this game.

###### This question is part of the set Best of Me.

Before them is a bridge, their only hope for survival. The bridge can only hold at most 2 persons at a time. Since it is pitch dark they have to carry a lamp, which has to be walked back and forth the two ends.Each person walks at a different speed. A pair must walk together at the speed of the slower person.

They have to cross the bridge in the minimum possible time, or else they will be engulfed in the pangs of death. What is the minimum time that the logicians would have planned out (in minutes)?

**Details and Assumptions**:

-Crossing time: Calvin-2 minutes, Azhaghu- 3 minutes, Ishan-5 minutes, Nihar-7 minutes, Brian-11 minutes, Sandeep- 13 minutes, Tanishq- 17 minutes, Prasun- 19 minutes.

- Strategies such as throwing the lamp across the bridge etc are not allowed.

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Suyeon is thinking of 2 (not necessarily distinct) positive integers, \(x\) and \(y\), each of which is greater than 1. She tells Calvin the product \(P = xy\) and Aaron the sum \(S = x + y.\) The equally intelligent Calvin and Aaron then engage in a short discussion as follows:

Calvin: "I cannot determine \(S\) at this point."

Aaron: "All right then, here's a hint; \(S\) does not exceed \(20\), and if that's all you need to know to uniquely determine \(S\) then I will know what \(P\) is."

Calvin: "I am now able to uniquely determine \(S\)."

Find \(S + P\).

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Joe is playing a game involving the above board. He places 14 pegs on the board, leaving a single space empty. He then jumps a peg with an adjacent peg. The peg that has been jumped is removed.

For example, if Joe jumped the bottom-left yellow peg with the bottom-left red peg, then he would need to remove the yellow peg and move the red one to the space above the blue, on the third row.

The game ends when no more moves are possible.

What is the maximum amount of pegs that can be left when the game is over?

**Note**: I am not asking what the maximum possible number of pegs without any possible moves is--you must be able to get to the position by playing the game.

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There is a diamond hidden on an \(N\times N\) grid at location \((x_D,y_D)\), where \(x_D\) and \(y_D\) are integers.

Every guess, you suggest a pair of coordinates (\(x_G\), \(y_G\)). And, if you get it wrong you are given a hint as to where to go to continue looking. You are told either NW, N, NE, E, SE, S, SW, or W:

- W implies \(x_D < x_G\) and \(y_D = y_G\)
- NW implies \(x_D < x_G\) and \(y_D > y_G\)
- etc.

If you can get the diamond with 10 guesses or less (i.e. at most 9 wrong guesses and one right one), you get to keep the diamond. What is the largest \(N\) for which you can guarantee success?

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You are asked to guess an integer between 1 and \(N\) inclusive.

Each time you make a guess, you are told either:

**(a)** you are too high,

**(b)** you are too low, or

**(c)** you got it!

You can guess as many times as you like, but are only allowed to guess too high 10 times and too low 3 times. That is, the \(4^\text{th}\) time you make a guess and are too low, or the \(11^\text{th}\) time you make a guess and are too high, you lose the game.

What is the maximum \(N\) for which you are guaranteed to be able to accomplish this?

**Clarification**: For example, if you were allowed to guess too high once and too low once, you could guarantee to guess the right answer if \(N=5\), but not for \(N>5\). So, in this case, the answer would be 5.

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