The Letter E
Click the image to enlarge it and download it. Then print it, cut out the three pieces, and rearrange them so that you get the letter “E”.
SOLUTION
Rearrange the pieces, so that they form the shadow of the letter “E”, as shown below.
Category: Brain Teasers
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Fish Eat Fish
A hundred fish are swimming along a stream at different velocities. If one fish catches up to another fish, it eats it and continues swimming. What is the expected number of fish that will survive?
SOLUTION
Notice that the N-th fish in the stream survives if and only if it is the fastest among the first N fish. The probability of this event happening is equal to 1/N. Since the expected number of fish that survive is equal to the sum of the survival probabilities for each of them, the answer is 1+1/2+1/3+…+1/N.
For more details on the last claim, consider reading our blog post “How Many Times on Average?”
Worm in an Apple
There is a perfectly spherical apple with a radius 50mm. A worm has entered the apple, made a tunnel of length 99mm through it and left. Prove that we can slice the apple in two pieces through the center, so that one of them is untouched by the worm.
SOLUTION
Let the entering point is A, the leaving point is B and the center of the apple is C. Consider the plane P containing the points A, B and C and project the worm’s tunnel on it. Since 99 < 2×50, the convex hull of the tunnel’s projection will not contain the center C. Therefore we can find a line L through C, such that the tunnel’s projection is entirely in one of the semi-planes of P with respect to L. Now cut the apple with a slice orthogonal to P passing through the line L and you are done.
Least Likely
Sarah lives in the UK. Which of these statements is least likely:
- Sarah speaks English.
- Sarah eats baby pandas.
- Sarah speaks English and eats baby pandas.
SOLUTION
The answer is 3. This is because statements 1. and 2. are more general than statement 3. and therefore more likely to be true.
Mirror Chess
Bobby and Garry are playing chess. If Garry is playing with the black figures and agrees to mirror every move Bobby makes, what is the minimum number of moves in which Bobby can mate his opponent?
SOLUTION
Bobby can mate Garry in four moves:
1. d4 d5 2. Qd3 Qd6 3. Qh3 Qh6 4. Qxc8
Slicing Butter
If you want to split a cubic piece of butter into 27 smaller cubes, you can easily do it using just 6 slices (imagine the Rubik’s Cube). However, after every slice you make you can also rearrange the pieces – stack them in different ways on top of each other
SOLUTION
In order to separate the little cube in the center of the butter piece from the rest, you need 6 slices – that’s the number of sides it has. Therefore, you can’t accomplish the task with less than 6 cuts.
Mute Person, Blind Person
A mute person enters the pharmacy and decides to buy a toothbrush. He imitates the movements of brushing his teeth and thus successfully explains what he needs. After him a blind person enters the pharmacy, looking for toothpaste. What does he do?
SOLUTION
He simply says “I need toothpaste”. He is blind, not mute.
Halftime Sentence
After one king got his first newborn child, he was so happy that decided to reduce all prisoners’ sentences in his kingdom by half. How did he handle the prisoners who had lifetime sentences?
SOLUTION
The king alternated keeping them 1 year in prison with 1 year out of it.
Ancient Coins
Suppose I show you two ancient coins. The first one is dated 51 B.C., the second one is marked George I. Which one is counterfeit?
SOLUTION
Both of them. People who lived 51 years before Christ didn’t know about Jesus yet. When King George I was ruling, he was the first king with this name, and so he was just called George.
Mountain Hike
A man decides to climb a mountain. He starts at sunrise from the bottom of the mountain and arrives at the top at sunset. He sleeps there and on the next day he goes back the same way, descending at higher speed. Prove that there is some point of his path, on which the man will be at the same time on both days.
SOLUTION
Imagine a second man who starts climbing from the bottom of the mountain on the second day and following the first hiker’s first day movements. At some point the first and the second hiker will meet each other, and this will be the point you are looking for.