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# Logical Reasoning

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.

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 \(S + P\).

"This test" refers to the following set of three questions. In particular, "questions in this test" can count itself.

- How many questions in this test (including this) have a different answer from this question?
- How many questions in this test (including this) have the same answer as this question?
- What is the square of the answer to this question?

Concatenate the answers to the three questions in order. For example, if the answers are \(0, 1, 2\) for Questions 1, 2, 3 respectively, then submit \(012 = 12\).

*This problem is created by Nikolai Beluhov.*

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 \(a+b\)?

Suppose there are 10 balls identical in appearance, where 8 of them each have a mass of \(x\) grams, and each of the other two \((x + \delta)\) grams. Now, \(\delta \, (\gt 0)\) is so small that the difference can't be detected using your own hands, and can only be detected by the precise balance scale you have been provided with. You can put any number of balls in each pan of the scale.

What is the minimum number of weighings you will need to make to guarantee the identification of both of the heavier balls?

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