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Sequence 1, 11, 21, 1112

What follows next in the following sequence:

1, 11, 21, 1112, 3112, 211213, 312213, ?

Every term in the sequence describes how many 1s, 2s, 3s, etc. were there in the previous term. “11” means that in the previous term there was one 1. “21” means that in the previous term there were two 1s. “1112” means that in the previous term there were one 1 and one 2. “3112” means that in the previous term there were three 1s and one 2. “211213” means that in the previous term there were two 1s, one 2 and one 3. “312213” means that in the previous term there were three 1s, two 2s and one 3. Therefore, the next number in the sequence should be 212223.

The Fragile Bridge

You have to cross a long bridge which supports weight up to 180 pounds. However, you weigh 175 pounds and also carry with yourself 3 golden eggs, each of which weighs 2 pounds. How can you get to the other side?

 P. S. If you leave an egg unattended, someone can steal it.

Simply juggle the eggs while crossing the bridge.

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Optical Illusions

If you count carefully the number of people before the tiles scramble and after that, you will see that one person disappears. Can you explain how this is possible?

Similarly, in this picture it looks like after changing the places of the tiles in the diagram, their total area decreases by one. Can you explain this?

If you look carefully, you will notice that every person in the picture of 12 people is slightly taller than the corresponding person in the picture of 13 people. Basically, we can cut little pieces from 12 different people without making noticeable changes and arrange them into a new person.

For the second question, none of the shapes before and after the scrambling is really a triangle. One of them is a bit curved in at the hypotenuse and the other one is a bit curved out. This is barely noticeable, because the red and the blue triangle have very similar proportions of their sides – 5/2 ~ 7/3.

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3 Men, 1 Woman

Warning: this puzzle involves mature themes that are inappropriate for younger audiences. If you are not an adult, please skip this puzzle.

3 men must have sex with 1 woman, but they have only 2 condoms. Each of the 4 people has some unique STD which they don’t want to transfer to the rest. What can they do?

They can start by putting the two condoms on top of each other and letting the first man use them. After that, the second man can take the inner condom out and use just the outer condom. Finally, the third man can take the removed inner condom, turn it inside out, and place it back inside the outer condom. Then he, he can use the two condoms simultaneously.

Ants on a Stick

On the ground there is a stick and 10 ants standing on top of it. All ants have the same constant speed and each of them can travel along the entire stick in exactly 1 minute (if it is left alone). The ants start moving simultaneously straightforward, either towards the left or the right end of the stick. When two ants collide with each other, they both turn around and continue moving in the opposite directions. How much time at most would it take until all ants fall off the stick?

Imagine the ants are just dots moving along the stick. Now it looks looks like all dots keep moving in their initially chosen directions and just occasionally pass by each other. Therefore it will take no more than a minute until they fall off the stick. If any of them starts at one end of the stick and moves towards the other end, then the time it will take for it to fall off will be exactly 1 minute.

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Prisoners and a Bulb

There are 100 prisoners in solitary cells. There is a central living room with one light bulb in it, which can be either on or off initially. No prisoner can see the light bulb from his or her own cell. Every day, the warden picks a prisoner at random and that prisoner visits the living room. While there, the prisoner can toggle the light bulb if he wishes to do so. Also, at any time, every prisoner has the option of asserting that all 100 prisoners have already been in the living room. If this assertion is false, all 100 prisoners will be executed. If it is correct, all prisoners will be set free.

The prisoners are allowed to get together one night in the courtyard and come up with a plan. What plan should they agree on, so that eventually someone will make a correct assertion and will set everyone free, assuming the warden will bring each of them an infinite number of times to the central living room?

First, the prisoners should elect one of them to be a leader and the rest – followers. The first two times a follower visits the living room and sees that the light bulb is turned off, he should turn it on; after that he shouldn’t touch it anymore. Every time the leader visits the living room and sees that the light bulb is turned on, he should turn it off. After the leader turns off the lightbulb 198 times, this will mean that all followers have already visited the room. Then he can make the assertion and set everyone free.

Merlin and Hermes: Mysterious Lines

Two adventurers, Merlin and Hermes, approached a large iron door built into a cliff face.”Well…”, said Hermes, “What do we do now?”. Merlin produced an old, large piece of crumpled paper from his pocket. “Hrm…”, Merlin mumbled. “It says here that we must speak the six letter keyword to open the door and enter the secret chamber, but I don’t remember seeing any signs as to what that keyword might be…”

After a bit of searching, Hermes notices something etched into the ground. “Come over here!”, he yelled, pointing frantically. And sure enough, barely visible and obscured by dust, was a series of lines of different colors etched into the ground:

“Ah”, Merlin said, “So that is the keyword.” Hermes was lost and confused. After staring at it for another thirty seconds, he grumbled “What keyword!? All I see is a bunch of lines!”. Merlin simply responded, “You’re just looking at it the wrong way. It’s obvious!”

Isn’t it?

The signs are engraved letters on the ground and Merlin and Hermes are looking at them from above (the italic “looking at it the wrong way” is a hint). The darker a part from some sign is, the farther from the ground it is. The only letters which could correspond to this description are U – N – L – I – N – K. Therefore the keyword is “UNLINK”.

Source:

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