(a) A traveller decided to stay in the motel. He has no money but he has a golden chain consisting of 7 links (the chain is not closed). The host agreed on one golden link to be the payment for one day of staying. The traveller wants to stay for the next 7 days. What is the smallest number of links he has to disunite to be able to make the payment every day? (Take into account that the host can give the change “in links” if he already got some from the traveller.)
(b) Assume we have a chain consisting of 23 golden links and now the traveller has to spend 23 days in the motel. Is it enough to disunite 2 links to be able to make the daily payments? As before the host can give the change with the links he gets from the traveller.
(c) Consider 24 links and 24 days now. Can we manage to make daily payments after we disunite some 2 links?
Comment: In all questions above after we disunite the chain at some link in general we obtain three parts: the link itself, the left part of the chain and the right of the chain. Note that there might be no left or no right part.)
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?
The seller with weights. With four weights the seller can weigh any integer number of kilograms, from 1 to 40 inclusive. The total mass of the weights is 40 kg. What are the weights available to the seller?
Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?
We are looking for the correct statement. In a notebook one hundred statements are written:
1) There is exactly one false statement in this notebook.
2) There are exactly two false statements in this notebook.
...
100) There are exactly one hundred false statements in this notebook.
Which of these statements is true, if it is known that only one is true?
Solve the rebus \(AC \times CC \times K = 2002\) (different letters correspond to different integers and vice versa).