Two players in turn paint the sides of an n-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what n can the second player win, no matter how the first player plays?
On a table there are 2002 cards with the numbers 1, 2, 3, …, 2002. Two players take one card in turn. After all the cards are taken, the winner is the one who has a greater last digit of the sum of the numbers on the cards taken. Find out which of the players can always win regardless of the opponent’s strategy, and also explain how he should go about playing.
Janine and Zahara each thought of a natural number and said them to Alex. Alex wrote the sum of the thought of numbers onto one sheet of paper, and on the other – their product, after which one of the sheets was hidden, and the other $($ on it was written the number of 2002 $)$ was shown to Janine and Zahara. Seeing this number, Janine said that she did not know what number Zahara had thought of. Hearing this, Zahara said that she did not know what number Janine had thought of. What was the number which Zahara had thought of?
In Conrad’s collection there are four royal gold five-pound coins. Conrad was told that some two of them were fake. Conrad wants to check $($ prove or disprove $)$ that among the coins there are exactly two fake ones. Will he be able to do this with the help of two weighings on weighing scales without weights? $($ Counterfeit coins are the same in weight, real ones are also the same in weight, but false ones are lighter than real ones. $)$
Two people are playing. The first player writes out numbers from left to right, randomly alternating between 0 and 1, until there are 1999 numbers in total. Each time after the first one writes out the next digit, the second changes two numbers from the already written row $($ when only one digit is written, the second misses its move $)$. Is the second player always able to ensure that, after his last move, the arrangement of the numbers is symmetrical relative to the middle number?
A game takes place on a squared 9 × 9 piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number K, and the number of rows and columns in which there are more zeros than crosses is denoted by the number H $($ 18 rows in total $)$. The difference B = K – H is considered the winnings of the player who goes first. Find a value of B such that
1$)$ the first player can secure a win of no less than B, no matter how the second player played;
2$)$ the second player can always make it so that the first player will receive no more than B, no matter how he plays.
A cube with side length of 20 is divided into 8000 unit cubes, and on each cube a number is written. It is known that in each column of 20 cubes parallel to the edge of the cube, the sum of the numbers is equal to 1 $($ the columns in all three directions are considered $)$. On some cubes a number 10 is written. Through this cube there are three layers of 1 × 20 × 20 cubes, parallel to the faces of the cube. Find the sum of all the numbers outside of these layers.
Two play tic-tac-toe on a 10 × 10 board according to the following rules. First they fill the whole board with noughts and crosses, putting them in turn $($ the first player puts crosses, its partner – noughts $)$. Then two numbers are counted: K is the number of five consecutively standing crosses and H is the number of five consecutively standing zeros. $($ Five, standing horizontally, vertically and parallel to the diagonal are counted, if there are six crosses in a row, this gives two fives, if there are seven, then three, etc.$)$. The number K-H is considered to be the winnings of the first player $($ the losses of the second $)$.
a$)$ Does the first player have a winning strategy?
b$)$ Does the second player have a winning strategy?
On a plane there is a square, and invisible ink is dotted at a point P. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does P lie $($ if P lies on the line, then he says that P lies on the line $)$.
What is the smallest number of such questions you need to ask to find out if the point P is inside the square?
A 1 × 10 strip is divided into unit squares. The numbers 1, 2, …, 10 are written into squares. First, the number 1 is written in one square, then the number 2 is written into one of the neighboring squares, then the number 3 is written into one of the neighboring squares of those already occupied, and so on $($ the choice of the first square is made arbitrarily and the choice of the neighbor at each step $)$. In how many ways can this be done?
During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men $($ at least 80% $)$ – with a girl who was at the same time more beautiful and more intelligent. Could this happen? $($ There was an equal number of boys and girls at the ball.$)$
There is a chocolate bar with five longitudinal and eight transverse grooves, along which it can be broken $($ in total into 9 * 6 = 54 squares $)$. Two players take part, in turns. A player in his turn breaks off the chocolate bar a strip of width 1 and eats it. Another player who plays in his turn does the same with the part that is left, etc. The one who breaks a strip of width 2 into two strips of width 1 eats one of them, and the other is eaten by his partner. Prove that the first player can act in such a way that he will get at least 6 more chocolate squares than the second player.
Initially, a natural number A is written on a board. You are allowed to add to it one of its divisors, distinct from itself and one. With the resulting number you are permitted to perform a similar operation, and so on.
Prove that from the number A = 4 one can, with the help of such operations, come to any given in advance composite number.
What figure should I put in place of the “?” in the number 888 … 88? 99 … 999 $($ eights and nines are written 50 times each $)$ so that it is divisible by 7?
Two players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?
Two play a game on a chessboard 8 × 8. The player who makes the first move puts a knight on the board. Then they take turns moving it $($ according to the usual rules $)$, whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?
True or false? Prince Charming went to find Cinderella. He reached the crossroads and started to daydream. Suddenly he sees the Big Bad Wolf. And everyone knows that this Big Bad Wolf on one day answers every question truthfully, and a day later he lies, he proceeds in such a manner on alternate days. Prince Charming can ask the Big Bad Wolf exactly one question, after which it is necessary for him to choose which of the two roads to go on. What question can Prince Charming ask the Big Bad Wolf to find out for sure which of the roads leads to the Magic kingdom?
15 points are placed inside a $4 \times 4$ square. Prove that it is possible to cut a unit square out of the $4 \times 4$ square that does not contain any points.
Cowboy Joe was sentenced to death in an electric chair. He knows that out of two electric chairs standing in a special cell, one is defective. In addition, Joe knows that if he sits on this faulty chair, the penalty will not be repeated and he will be pardoned. He also knows that the guard guarding the chairs on every other day tells the truth to every question and on the alternate days he answers incorrectly to every question. The sentenced person is allowed to ask the guard exactly one question, after which it is necessary to choose which electric chair to sit on. What question can Joe ask the guard to find out for sure which chair is faulty?
Decipher the following rebus $\\$
All the digits indicated by the letter “E” are even $($ not necessarily equal $)$; all the numbers indicated by the letter O are odd $($ also not necessarily equal $)$.