Is the number \(10^{2002} + 8\) divisible by 9?
Is the sum of the numbers \(1 + 2 + 3 + \dots + 1999\) divisible by 1999?
Try to read the word in the first figure, using the key (see the second figure).
Before you is a lock “with a secret” (see the picture).
If you put the arrows on the desired letters, you will get the keyword and the lock will open. What is this word?
Six chess players participated in a tournament. Each two participants of the tournament played one game against each other. How many games were played? How many games did each participant play? How many points did the chess players collect all together?
Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?
The distance between Athos and Aramis, galloping along one road, is 20 leagues. In an hour Athos covers 4 leagues, and Aramis – 5 leagues.
What will the distance between them be in an hour?
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?
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?
Try to decipher this excerpt from the book “Alice Through the Looking Glass”:
“Zkhq L xvh d zrug,” Kxpswb Gxpswb vdlg, lq udwkhu d vfruqixo wrqh, “lw phdqv mxvw zkdw L fkrrvh lw wr phdq – qhlwkhu pruh qru ohvv”.
The text is encrypted using the Caesar Cipher technique where each letter is replaced with a different letter a fixed number of places down in the alphabet. Note that the capital letters have not been removed from the encryption.