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# easy puzzles..

## Question

my friends at school shared these 'puzzles' during our break to avoid boredom. I dunno where they got them so I also don't know if some of these are already posted in this site.

ESCAPE of MARTHA

A wicked wizard banished Martha to a room and the only exits available are two doors (no windows, no chimneys, no exit except those two doors). One door leads to her freedom, another leads to her doom; However, she doesn't know which is which. In the middle of the room are two goblins. One always tell the truth and the other always lie, and again, Martha doesn't know which goblin tells the truth or which goblin lies. What question must she ask in order to know which of the two doors is the door to freedom if she is only permitted to ask ONE question? The question may be asked from one goblin or both or them, as log as it is only a SINGLE question. If she exceeds one question, she will be imprisoned forever. Let us suppose that the goblins are fairly intelligent and they do not side on anyone. (I don't know why the wizard is so mad at Martha but that is not our problem. Good Luck Martha!

THE WATER PROBLEM

You are a prisoner in King Noia's contest. There are two glasses of water, and a big pail. To save yourself, you must solve this problem. How would you place the water in the big pail, if the requirements are:

1. you must know which water came from which glass

2. you are not allowed to use any sort of divider

3. only PURE water must be present in the pail

LOCKED with LOCKERS

In a school with 1000 OPENED lockers arranged in a ordered and single line, a teacher made an experiment with 1000 of her students. She calls the first student and asked him to go to all lockers which are multiple of one and close the lockers which are opened OR open the lockers which are closed. She then calls the 2nd student and asked her to go to the lockers which are located at multiples of two and do the same thing. In short, a student goes to the ALL lockers which are MULTIPLE to his/her number and close them if they are opened OR open them if they are closed (example: student #55 goes to all lockers found at locations which are multiple of 55, and closes them if they opened OR opens them if they are closed). Which lockers are opened and which lockers are closed at the end of this so-called 'experiment'?

(Let us suppose that all students have infinite stamina to survive the long process)

## 5 answers to this question

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I dunno where they got them so I also don't know if some of these are already posted in this site.

only ESCAPE of MARTHA is on this site - check Honestants and Swindlecants III.

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ESCAPE of MARTHA

If I were to ask you which door leads to freedom, which door would you point to? Believe the answer.

THE WATER PROBLEM --

Pour the first glass into the pail. Freeze it. Pour the second glass into the pail

LOCKED with LOCKERS

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900 961.

The others are open

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regarding Bonova's solutions:

Problem is as soon as you poor the 2nd glass into the pail, it unfreezes some of the water when it touches, mixing it. I'd suggest freezing both glasses, then place each frozen ice block in the pail, but separated so they don't touch

notice that each of the numbers is a perfect square? anyone care to explain why?

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lockers are opened or closed by students whose numbers are factors of the locker numbers.

Factors come in pairs.

So generally, the lockers are touched by an even number of students and end up open.

The exceptions are lockers whose numbers are squares.

Because squares have a repeated factor, those lockers are touched by an odd number of students and end up closed.

Example - a locker whose number is not a square.

Locker #8 is touched by students [1, 8] and [2, 4] - an even number of students, and ends up open.

Example - a locker whose number is a square.

Locker #9 is touched by students [1, 9] and [3, 3] - an odd number of students, and ends up closed.

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LOCKED with LOCKERS

Thanks bonanova, I agree. If I may throw my own explanation into the ring (as an alternative, not to compete with yours!).

First, understand that this problem is equivalent to:

There are 1000 open lockers numbered 1 to 1000. For each number from 1 to 1000 you take that many stones and try to arrange them into rectangles (squares included of course). For every rectangle you can make of unique width and height, change the state of the corresponding locker.

Realising that, it's obvious why only the square numbered lockers end up closed. For every rectangle you can make (e.g. for the 12th locker, a 2x6 rectangle) you can also make the opposite rectangle (a 6x2 rectangle) - you end up changing the corresponding lockers state twice, so the two actions cancel each other out and the locker stays open. The only exception is when the rectangle happens to be a square (e.g., for the 16th locker, a 4x4 square), which is it's own opposite and can't be counted twice, so the locker gets closed. And of course, you can only make a square when you are dealing with a square number of stones, so the lockers 1, 4, 9, 16 and so on all end up closed while the rest stay open as they were from the start.

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