One gambler had a pair of dice. Rolling them was something that kept him concentrated. As a result of frequent usage all the numbers were wiped off from both of the dice. In January the gambler went through a rough patch and decided to take a break from gambling. He understood he could not rely only on his luck which has recently failed him. Therefore, our gambler started doing mathematical puzzles to master his mind. The first puzzle is to paint digits on each side of both dice (one digit per one side) in such a way that any natural number between 1 and 31 inclusive can be obtained by putting one dice next to the other. We do not allow the digit “6” to be used as the digit “9” and vice versa. Is there any solution to this problem?
(a) The second puzzle for our gambler is a bit similar to the first:
“To paint digits on each side of both dice (one digit per one side) in such a way that any combination from 01 and 31 can be obtained by putting one dice next to the other.”
The digit “6” cannot be used as the digit “9” and vice versa. Is there any solution?
(b) What is the answer to (a) if we allow rotations (i.e. we allow the usage of “6” instead of “9” and vice versa)?
(a) After building the garden the successful businesswoman had another idea in mind. She is keen to re-build the terrace in front of her country house. Now the goal is to plant nine sakura trees in such a way that one can count eight rows of trees each consisting of three trees (obviously, a tree can be counted in several rows). How the landscape gardener can satisfy this requirement?
(b) The neighbour of the businesswoman learned about her plans from the talk with the same landscape gardener and decided to outdo her with a similar but more complicated request. He is planning to plant nine sakura trees so that there can be found ten rows of three trees each. Is there a configuration of nine trees satisfying this condition?
A group of three smugglers is offered to smuggle a chest full of treasures across the dangerous river. The boat they possess is old and frail. It can carry three smugglers without the chest, or it can carry the chest and only two smugglers. The price for this job is extremely high, and the gang is more than interested in completing the job. Think of a strategy the smugglers should follow to successfully transit the chest and themselves to the other shore.
My mum once told me the following story: she was walking home late at night after sitting in the pub with her friends. She was then surrounded by a group of unfriendly looking people. They demanded: “money or your life?!” She was forced to give them her purse. She valued her life more, since she was pregnant with me at that time. According to her story she gave them two purses and two coins. Moreover, she claimed that one purse contained twice as many coins as the other purse. Immediately, I thought that the mum must have made a mistake or could not recall the details because of the shock and the amount of time that passed after that moment. But then I figured out how this could be possible. Can you?
Becky and Rishika play the following game: There are 21 biscuits on the table. Each girl is allowed to take 1, 2 or 3 biscuits at once. The girl who cannot take any more biscuits loses. Rishika starts – show that she can always win.
Alice and Bob play a game, Alice will go first. They have a strip divided into \(2018\) identical squares. In one move, they put a \(2 \times 1\) domino block on the strip, covering two full squares. One that is not able to make their move, loses. Who has the winning strategy?
Ana and Daniel are playing a game that involves a chocolate bar. The top left square of the bar is poisoned. In each move, a player has to pick a square and take all the pieces contained in the rectangle whose top left corner is the selected square and the bottom right corner is the bottom right corner of the whole bar. The person who takes the poisoned square loses. Who will win, if Daniel starts?
Two pirates are playing a game. They have \(42\) gold coins on a table. Each of them is allowed to take either \(1\) or \(5\) coins from the table. The pirate who takes the last coin wins. Who will win – the first pirate or the second pirate?
Rekha and Misha also play with coins. They have an unlimited supply of 10p coins and a perfectly round table. In each move, one of them places a coin somewhere on that table, but not on top of any other coins already there. A person that cannot place any more coins loses. Who will win, if Rekha goes first?