In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?
From a set of weights with masses 1, 2, ..., 101 g, a weight of 19 grams was lost. Can the remaining 100 weights be divided into two piles of 50 weights each in such a way that the masses of both piles are the same?
Jack and Ben had a bicycle on which they went to a neighborhood village. They rode it in turns, but whenever one rode, the other walked and did not run. They managed to arrive in the village at the same time and almost twice as fast than if they had both walked. How did they do it?
Three tourists must move from one bank of the river to another. At their disposal is an old boat, which can withstand a load of only 100 kg. The weight of one of the tourists is 45 kg, the second – 50 kg, the third – 80 kg. How should they act to move to the other side?
Alice the fox and Basilio the cat are counterfeiters. Basilio makes coins heavier than real ones, and Alice makes lighter ones. Pinocchio has 15 identical in appearance coins, but one coin is fake. How can Pinocchio determine who made the false coin – Basilio the cat or Alice the fox – with only 2 weighings?
It is known that “copper” coins that are worth 1, 2, 3, 5 pence weigh 1, 2, 3, 5 g respectively. Among the four “copper” coins (one for each denomination), there is one defective coin, differing in weight from the normal ones. How can the defective coin be determined using scales without weights?
How can we divide 24 kg of nails into two parts of 9 kg and 15 kg with the help of scales without weights?
Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.
17 squares are marked on an \(8\times 8\) chessboard. In chess a knight can move horizontally or vertically, one space then two or two spaces then one – eg: two down and one across, or one down and two across. Prove that it is always possible to pick two of these squares so that a knight would need no less than three moves to get from one to the other.
Petya and Misha play such a game. Petya takes in each hand a coin: one – 10 pence, and the other – 15. After that, the contents of the left hand are multiplied by 4, 10, 12 or 26, and the contents of the right hand – by 7, 13, 21 or 35. Then Petya adds the two results and tells Misha the result. Can Misha, knowing this result, determine which hand – the right or left – contains the 10 pence coin?