Author: I.V. Izmestyev
Postman Pat did not want to give away the parcel. So, Matt suggested that he play the following game: every move, Pat writes in a line from left to right the letters M and P, randomly alternating them, until he has a line made up of 11 letters. Matt, after each of Pat’s moves, if he wants, swaps any two letters. If in the end it turns out that the recorded word is a palindrome (that is, it is the same if read from left to right and right to left), then Pat gives Matt the parcel. Can Matt play in such a way as to get the parcel?
Authors: Folklore
There are 13 pupils in the school of witchcraft and wizardry. Before the Clairvoyance exam, the teacher put them at a round table and asked to guess who would receive the clairvoyant’s diploma. The students said nothing about themselves and two of their neighbours, but they wrote the following about all of the others: “None of these ten will get the diploma!" Of course, all of those who passed the exam guessed correctly, and all of the other students were mistaken. How many wizards received a diploma?
In an attempt create diversity the government of the planet hired \(100\) truth tellers and \(100\) liars. Each of them has at least one friend. Once exactly \(100\) members said: “All my friends are honest” and exactly \(100\) members said: “All my friends are liars.” What is the smallest possible number of pairs of friends, one of whom is honest and the other a liar?
Specify any solution of the puzzle: \(2014 + YES =BEAR\).
Of five coins, two are fake. One of the counterfeit coins is lighter than the real one, and the other is heavier than the real one by as much as the lighter one is lighter than the real coin.
Explain how in the three weighings, you can find both fake coins using scales without weights.
A traveller met five inhabitants of the planet of liars and truth tellers. To his question: “How many truth tellers are there among you?” the first replied: “None!", and another two answered: “One.” What did the final two say?
The sheikh spread out his treasures in nine sacks: 1 kg in the first bag, 2 kg in the second bag, 3 kg in the third bag, and so on, and 9 kg in the ninth bag. The insidious official stole a part of the treasure from one bag. How can the sheikh work out from which bag the official stole the treasure from using two weighings?
Author: D.V. Baranov
Vlad and Peter are playing the following game. On the board two numbers written are: \(1/2009\) and \(1/2008\). At each turn, Vlad calls any number \(x\), and Peter increases one of the numbers on the board (whichever he wants) by \(x\). Vlad wins if at some point one of the numbers on the board becomes equal to 1. Will Vlad win, no matter how Peter acts?
In the entry \({*} + {*} + {*} + {*} + {*} + {*} + {*} + {*} = {*}{*}\) replace the asterisks with different digits so that the equality is correct.
A disk contains 2013 files of 1 MB, 2 MB, 3 MB, ..., 2012 MB, 2013 MB. Can I distribute them in three folders so that each folder has the same number of files and all three folders have the same size (in MB)?