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 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.

You go into your garage and find a pile of 100 batteries (all the same type).

You happen to know that half of them are good and half are bad, but you can't tell which is which.

You have a flashlight which uses two batteries, and requires both to work in order to turn on.

In the worst case scenario, with an optimal strategy, what is the minimum number of times you will need to put batteries into the flashlight before you can guarantee to get a working pair in the flashlight?

\(\)

**Details and Assumptions:**

The flashlight either turns on or it doesn't, i.e. there is no way to distinguish between one of the two working and neither one working.

Your answer should be the number of "flashlight loadings" you will need to actually get your flashlight working, not just how many you would need to identify two good batteries.

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.

There is a circle of \(n\) light bulbs with a switch next to each of them. Each switch can be flipped between two positions, thereby toggling the on/off states of three lights: its own and the two lights adjacent to it. Initially, all the lights are off.

Let the minimum number of flips needed to turn on all the \(n=12\) and \(n=13\) light bulbs be \(a\) and \(b\), respectively. Then what is the value of \(a+b\)?

You give Alice and Bob each a closed box containing a number. They both know that the two numbers are consecutive positive integers, but do not know the opponent's number. Then you give them a blank card and a pencil each, and have them play the following game:

- In each turn, they could predict what the number in their opponent's box is by writing it down on the card. Whoever can do so wins. They are only allowed to do this when they
*know*the answer. - Or, they could choose not to play the turn and exchange blank cards, indicating that they do not know the answer yet as of that turn.

Given that the two integers given to Alice and Bob are 16 and 17, respectively, who wins and after how many turns?

**Clarification**: Alice/Bob only attempts to answer if they *know* the correct answer. This is not a guessing game.

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