Brain Teasers 46 - Puzzle Prime

Zombie Attack

Section: Brain Teasers Category: Mathematics

Difficulty: Expert

Oh no, zombies are attacking your house!

Every second, a new zombie drops down on one of the 9 spots of your lawn, which is currently unoccupied. All zombies move towards your house on the left with constant speed, and each of them needs exactly 1 second to traverse a spot of the lawn. Once a zombie steps out of the lawn, it enters your house and waits there for the others (thus each zombie travels total distance between 1 and 9 spots).

Show that after some time, the total distance traversed by any 1000 consecutive zombies will be within the range of just 50 spots.

Remark: Assume your house can accommodate an unlimited amount of zombies.

SOLUTION

Since the number of zombies on the lawn never decreases, it must stabilize at some point. Therefore after some time T, there will be exactly K zombies on the lawn at all times, 1 ≤ K ≤ 9.

Consider any 1000 consecutive zombies appearing past time T and take a picture of the lawn at the moment each of them gets dropped on it – this makes a total of 1000 pictures. Since on every picture, there are exactly K zombies, we see exactly 1000K zombies on these pictures.

Now notice that almost all of the selected 1000 zombies appear on as many pictures as lawn spots they traverse. The zombies for which this is not true are just the K zombies, which appear on the last picture. We easily see that they have traveled between 1 + 2 + … + K − 1 and (10 − K) + (11 − K) + … + 8 spots more than the number of pictures they appear on.

Similarly, on the 1000 pictures we have taken, there are K−1 additional zombies, which appear multiple times (you can see all of them on the first picture). The total number of times they show up is again between 1 + 2 + … + K − 1 and (10 − K) + (11 − K) + … + 8.

Therefore we conclude that the total distance the selected 1000 zombies travel is equal to 1000K ± {[(10 − K) + (11 − K) + … + 8] − [1 + 2 + … + K − 1]} = 1000K ± (9 − K)(K − 1).Since (9 − K)(K − 1) is a number between 0 and 16, this solves the problem.

Unknown Author May 25, 2018

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Place a Coin

Section: Brain Teasers Category: Insight

Difficulty: Advanced

Two friends are playing the following game – taking turns, they place identical coins on a square table, so that no two coins touch each other. Whoever can not make a move, loses the game. Who has a winning strategy?

SOLUTION

The first player has a winning strategy. He just has to place the first coin in the center of the table and then each consecutive one symmetrically opposite to the last coin of his opponent.

Unknown Author May 25, 2018

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Stuck in a Blender

Section: Brain Teasers Category: Science

Difficulty: Advanced

You get shrunk 100 times (preserving your density), then thrown in a blender. You have 2 minutes until the blades of the blender start spinning. How are you going to escape?

SOLUTION

If you get shrunk 100 times, but your density remains the same, your muscles will become so strong relative to your size, that you will be able to jump out of the blender. The reason is that your weight will decrease 1000000 times, but your muscle cross-section just 10000 times.

Unknown Author May 8, 2018

4 Comments

The Cherries Puzzle

Section: Brain Teasers Category: Practical

Difficulty: Advanced

Get a stripe of paper, then cut two holes and two slits in it, as shown on the image below. Then try to attach two large cherries to paper, without tearing it apart.

SOLUTION

Pull the handle in the paper through the holes, attach the cherries, then pull it back up.

Jacques Ozanam May 5, 2018

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Bonded Forever

Section: Brain Teasers Category: Practical

Difficulty: Advanced

Get 2 ropes and tie them up around the hands of 2 of your friends, as shown on the picture. Then, let your friends separate from each other, without untying the knots or cutting the ropes.

SOLUTION

The solution is shown below.

Unknown Author April 28, 2018

2 Comments

Cross – Parallel

Section: Brain Teasers Category: Practical

Difficulty: Advanced

This puzzle/game is played with groups of people, in which some of the participants know the rules, and others are trying to figure them out.

All players must sit in a circle, facing each other. Then one person who is familiar with the rules starts by pointing at another and saying “cross-cross”, “cross-parallel”, “parallel-cross”, or “parallel-parallel”. After that, the person who got picked must choose another one and guess which one of the four expressions above he must say. If he gets it wrong, he gets corrected, and then the next person continues. The game ends when everyone finds out what the secret is.

To play this puzzle game with your friends, at least one of you must know the secret rules and solution, which are explained below. Just keep in mind that whoever learns the rules, will lose the enjoyment of figuring them out by himself.

SOLUTION

When you point at somebody, you say “cross-cross”, if yours and the other person’s legs are crossed, “cross-parallel”, if yours are crossed and the other person’s are parallel, etc.

Unknown Author April 27, 2018

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Murder or Suicide

Section: Brain Teasers Category: Detective

Difficulty: Advanced

Examine the picture and determine whether the woman was killed or she committed a suicide.

SOLUTION

This was a murder, which was supposed to look like a suicide. These are some of the reasons:

1. The dead person has not finished their cigarette yet, which is a normal thing to do when someone commits a suicide.
2. The person is left-handed, judging by the position of the pen and the lamp, but is holding the gun in their right hand.
3. It was supposed to look like the person was writing a death note before killing themselves, but it is night and the lamp is not plugged in.

Unknown Author April 25, 2018

17 Comments

Nervous Breakdown

Section: Brain Teasers Category: Lateral

Difficulty: Advanced

Once there was a recluse who always stayed in his home. The only time anyone visited him was when his food and supplies were delivered, but nobody came inside. One winter night there was a big storm and the recluse had a nervous breakdown. He went to his room, turned off all the lights and got in his bed. The next morning he found out that he caused the deaths of several hundred people. How?

SOLUTION

The recluse was living in a lighthouse. When he turned off the lights, he caused several ships to get wrecked in the cliffs and many people die.

Unknown Author April 23, 2018

2 Comments

Beat Strata

Section: Brain Teasers Category: Insight

Difficulty: Hard

Strata is a beautiful award-winning game with mesmerizing sound and unique puzzle concept. It contains hundreds of levels with common rules and final goal. Below I present you these rules and ask you to find a universal algorithm, which will allow you to solve easily every single level of the game.

The rules are simple – you begin with an nxn board, some squares of which are colored in arbitrary colors. Then you start placing stripes of whatever color you choose over entire rows and columns of the board. Your task is after placing all available 2n stripes, the color of every (colored) square to match the color of the stripe which has been placed second over it (on top).

Can you find a simple algorithm, which results in solving any level of the game, no matter the starting position? You can watch AppSpy’s video below for better understanding of the rules.

SOLUTION

Imagine the reverse Strata puzzle – the color of every square must match the color of the first stripe which is placed upon it. Clearly, there must be a line in the grid such that all colored squares in it have the same color. Take all such lines in the grid and place on them stripes of appropriate colors. Then erase the colors from all squares covered by the stripes and repeat the procedure until you place all 2n stripes. It is easy to see that if the reverse Strata puzzle has a solution, then we will find it using this strategy. Finally, in order to solve the original Strata puzzle, just place the stripes in reverse order.

Puzzle Prime April 13, 2018

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The Rotating Square

Section: Brain Teasers Category: Insight

Difficulty: Hard

On the table in front of you there is a square with 4 coins placed on its vertices. You are blindfolded and are given the task to turn all of the coins with either heads up or tails up. Every time you turn few of the coins however, the square rotates arbitrarily on the table. Find a strategy, such that no matter the starting arrangement of the coins and no matter how the square rotates after every flip of coins, eventually you will turn all of the coins with the same face up.

SOLUTION

First assume that there is even number of tails and even number of heads on the table – 2 of each kind. Flip 2 opposite coins. If after that not all coins have the same face up, the coins’ faces along the square’s corners show T-T-H-H. Now flip 2 adjacent coins. If after that not all coins have the same face up, the coins’ faces along the square’s corners show T-H-T-H. Now flip again 2 opposite coins and you are done.

Next assume that there were intially odd number of tails and odd number of heads on the table. Then after applying the moves described above, flip one of the coins upside down. Now there is even number of heads and even number of tails on the table, so you can repeat the same procedure and accomplish the task.

Unknown Author March 28, 2018

2 Comments