What is the secret in the pattern of this stained glass?
The image is a superposition of a blue shape and a yellow shape. The places where they coincide are colored in green (blue + yellow = green). The blue shape is consisting of horizontal stripes with lengths 3, 1, 4, 1, 5, 9, 2, 6, 5, representing the number pi, and the yellow shape is consisting of vertical stripes with lengths 4, 6, 6, 9, 2, 0, 1, 6, 1, representing the Feigenbaum constant.
A man must mail a precious necklace to his wife, but anything sent through the mail will be stolen unless it is sent in a padlocked box. A box can bear any number of padlocks, but neither of the spouses has the key to a lock owned by the other. How can the husband mail the necklace safely to his wife?
The man can put a lock on the box and send it to his wife. Then she can put her own lock and send it back. Once the man receives the box, he can remove his lock and send the box once again to his wife. When she gets it, she can finally unlock the box using her own key.
You are playing a game of Bridge. Which probability is greater – that you and your partner do not have any spades, or that you and your partner have all the spades in the game?
You and your partner do not have any spades if and only if your opponents have all the spades. Therefore the probabilities are equal.
A Sheikh dies, leaving behind three sons, 17 camels, and the following order:
1. The oldest son shall inherit one in two camels.
2. The middle son shall inherit one in three camels.
3. The youngest son shall inherit one in nine camels.
Now the three sons do not know what to do. They ask an old friend of the family for advice, and he finds a solution. What does the friend propose?
The friend lent them one camel to make the camels 18. The first son took 9 camels, the second son took 6 camels, the third son took 2 camels. Then the friend took back his camel.
There are five pirates, one monkey, and lots of coconuts on an island. The pirates are supposed to share the coconuts on the next day, but while everybody is sleeping, the first pirate gives 1 coconut to the monkey, splits the remaining coconuts into 5 equal piles, and secretly keeps one of the piles for himself. Later, the second pirate does the same, then the third one, the fourth one, and the fifth one. On the morning, the pirates wake up and split all the remaining coconuts in five, leaving one last coconut for the monkey.
What is a possible number for the number of coconuts on the island?
Notice that if we find a certain number of coconuts which works, then we can add 56 and get a new one. Now imagine the pirates start with -4 coconuts, i.e. they have a total loan of 4 coconuts. Every time a pirate wakes up, he gives 1 coconut to the monkey, which makes the total loan 5 coconuts. Then the pirate keeps a loan of 1 coconut for himself and leaves -4 coconuts. Now we just add 56 coconuts to -4 to make the number positive and get 56 – 4 coconuts as a possible answer.
When do the hour and the minute hands of the clock lie on top of each other, and also between the numbers 1 and 2?
One such time is around 1:05 – at 1 o’clock and 5.454… minutes. This can be easily found by solving the linear equation x = 12(x – 5) => x=60/11.
Another time you possibly didn’t think about is 12:00 o’clock. At that time, the hour and minute hands are on top of each other and just between the two digits of the number 12.
There are four days which start with the letter “T” – Tuesday, Thursday, and which two others?
“Today” and “tomorrow”.
John gets off work at random times between 3 and 5 PM. His mother lives uptown, his girlfriend lives downtown. John takes the first subway that comes in either direction and eats dinner with the one he is delivered to. Even though John believes he has 50-50 chance to have dinner with either his mother or his girlfriend, he visited the former only 2 times out of the last 20. How come?
The subway heading downtown arrives at 3:00, 3:10, 3:20, etc, and the subway heading uptown arrives at 3:01, 3:11, 3:21, etc. Thus, the chance that John goes to his girlfriend is about 90% (depending on train delays).
It is well known how to split fairly a cake between two people – one of them cuts, the other one picks. The question is, how can you split fairly a cake between three people?
Easy: “Fairly” means that every person gets at least 1/3 of the cake.
Hard: “Fairly” means that every person has the opportunity to get at least as much
Easy (Banach-Knaster method):
The first person cuts 1/3 piece of the cake. If the second person thinks it is larger than 1/3, he can trim it to 1/3. If the third person thinks the cut (and possibly trimmed) piece is larger than 1/3, he can trim it to 1/3 and keep it. Otherwise, the second person takes the piece if he decided to trim it, or the first one, in case he did not. After that, there are two people left, and they can easily split the remaining cake between them. This approach works for any number of people.
Hard (Selfridge-Conway method):
The first person cuts the cake in 3 pieces. The second one takes the biggest piece and trims it so that it becomes as large as the second biggest piece, puts the trimmings aside. The third person picks one of the three big pieces. Then, if the trimmed piece is still available, the second person takes it, if not – he picks whichever he likes. The first person takes the last remaining big piece. Among the first two people, whoever did not pick the trimmed big piece, splits the trimmings
You have two solid cubes of lead, which have almost the same size. You cut a hole in one of them and pass the other one through it. After measuring the cubes later, it turns out that the larger cube is still heavier than the smaller one. How is this possible?
You cut a hole in the SMALLER cube, and pass the larger cube through it. “Prince Rupert’s cube” is the largest cube which can pass through a unit cube, and it is approximately 6% larger.
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