Peter recorded an example of an addition on a board, after which he replaced some digits with letters, with the same figures being replaced with the same letters, and different figures with different letters. He did it such that he was left with the sum: \(CROSS + 2011 = START\). Prove that Peter made a mistake.
Four numbers (from 1 to 9) have been used to create two numbers with four-digits each. These two numbers are the maximum and minimum numbers, respectively, possible. The sum of these two numbers is equal to 11990. What could the two numbers be?
There are 100 boxes numbered from 1 to 100. In one box there is a prize and the presenter knows where the prize is. The spectator can send the presented a pack of notes with questions that require a “yes” or “no” answer. The presenter mixes the notes in a bag and, without reading out the questions aloud, honestly answers all of them. What is the smallest number of notes you need to send to know for sure where the prize is?
At a round table, 30 people are sitting – knights and liars (knights always tell the truth, and liars always lie). It is known that each of them at that table has exactly one friend, and for each knight this friend is a liar, and for a liar this friend is a knight (friendship is always mutual). To the question “Does your friend sit next to you?” those in every other seat answered “yes”. How many of the others could also have said “Yes”?
In the equality \(TIME + TICK = SPIT\), replace the same letters with the same numbers, and different letters with different digits so that the word \(TICK\) is as small as possible (there are no zeros among the digits).
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
Four children said the following about each other.
Mary: Sarah, Nathan and George solved the problem.
Sarah: Mary, Nathan and George didn’t solve the problem.
Nathan: Mary and Sarah lied.
George: Mary, Sarah and Nathan told the truth.
How many of the children actually told the truth?
On a board there are written four three-digit numbers, totaling 2012. To write them all, only two different digits were used.
Give an example of such numbers.
Matt, Conrad and Louie ate some sweets. Their surnames are Smith, Jones and Cooper. Smith ate 2 sweets fewer than Matt, Jones – 2 sweets fewer than Conrad, and Conrad ate more than anyone. Which of them has which last name?
2012 pine cones lay under the fir-tree. Winnie the Pooh and the donkey Eeyore play a game: they take turns picking up these pine cones. Winnie-the-Pooh takes either one or four cones in each of his turns, and Eeyore – either one or three. Winnie the Pooh goes first. The player who cannot make a move loses. Which of the players can be guaranteed to win, no matter how their opponent plays?