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Three friends, A, B and C, want to find out what their average salary is without disclosing their own salaries to the others. How can they do it using only verbal communication?
A tells B some number, then B adds his salary to it and tells the result to C, then C adds his salary and tells the result to A. Now A subtracts the number he told B in the beginning, adds his own salary and divides by 3. Repeat the same procedure with B and C starting first.
Five hundred begins it,
five hundred ends it,
five in the middle is seen.
The first of all letters,
the first of all numbers
take up their stations between.
Put them all together
and bring before you
the name of an eminent king.
The answer is DAVID. In Roman numerals D=500, V=5, I=1.
Six identical glasses are placed in a row on the table – first three filled with water, and then three empty ones. Can you move just one glass, so that empty and full glasses alternate?
Take the second full glass, pour all the water into the second empty glass, and then put it back in its place.
Nine pirates have captured a treasure chest. In order to protect it, they decide to lock it using multiple locks and distribute several keys for each of these locks among them, so that the chest can be opened only by a majority of the pirates. What is the minimum number of keys each of the pirates should get?
First, we show that for every four pirates, there exists a lock which cannot be opened only by them and can be opened by everyone else. We choose an arbitrary group of four pirates. If they can open every lock, then they can access the treasure without the need of a majority. If any of the remaining pirates cannot open that lock, then he, together with the initial group of four still cannot access the treasure. Thus, the claim is proved and to each group of four pirates we can assign a unique lock. These are \binom{9}{4}=126 locks in total. Finally, every pirate should get keys for \binom{8}{4}=70 of these locks, one for each group of four additional pirates he can be a group of.
Three friends, Adam, Bob, and Charlie are buried in the sand up to their necks, all facing West. Charlie can see both Adam and Bom, Bom can see only Adam, and Adam cannot see anyone. Black and white hats are placed on their heads. The three friends are told that there is at least one hat from each color, and then they are asked whether anyone can guess the color of their own hat.
After a few minutes, one of them answers. Who is that?
If Adam and Bob had hats with identical colors, then Charlie would immediately be able to deduce that his hat has the opposite color. Charlie doesn’t do that, so Adam and Bob are able to figure out that their hats have opposite colors. Since Bob is the one who can see the color of Adam’s hat, he is the one that answers the question.
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”.
Rearrange the pieces, so that they form the shadow of the letter “E”, as shown below.
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?
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?”
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.
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.
Sarah lives in the UK. Which of these statements is least likely:
The answer is 3. This is because statements 1. and 2. are more general than statement 3. and therefore more likely to be true.
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?
Bobby can mate Garry in four moves:
1. d4 d5 2. Qd3 Qd6 3. Qh3 Qh6 4. Qxc8
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