a) What is the answer in case we are asked to split the figure below into \(1\times4\) rectangles instead of \(1\times5\) rectangles?
(b) In the context of Example 1 what is the answer in case we are asked to split the figure into \(1\times7\) rectangles instead of \(1\times5\) rectangles?
(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?
A hedge fund is intending to buy 50 computers and connect each of them with eight other computers with a cable. Please do not ask why they need to do that, that is a top secret never to be made public! A friend of mine said that it’s related to some cryptocurrency research, but you should immediately forget all I just told you; it would be unwise to spread rumours! Let’s go back to the mathematical part of this story and stop the unrelated talk. The question is, how many cables do they need?
At a party there are people dressed in either blue or green. Every person dressed in blue had a chance to dance with exactly \(7\) people in green, only once with each one. On the other hand, every person in green danced exactly with \(9\) people in blue, also only once with each. Were there more people dressed in blue or in green at the party?
Lady X has 3 different black skirts, and 5 different jackets – 3 blue, and 2 green. She also has 10 different hats – 6 blue and 4 green. Lady X’s outfit consists of a skirt, a jacket, and a hat of the matching colour.
In how many ways can the Lady choose her outfit?
Let us call a number super-odd if it is made of odd digits only. (For example, numbers \(5\), \(33\), \(13573\) are all super-odd.) How many \(3\)-digit super-odd numbers with all digits different are there?
Among 7 girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all 7 girls in a row such that the two girls wearing red shirts are not sitting adjacent to each other?