Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of a young moon (see the picture below)
Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a ring.
Nick has written in some order all the numbers \(1,2,...33\) at the vertices of a regular \(33\)-gon. His little sister Hannah assigned to each side of the \(33\)-gon the number equal to the sum of the numbers at the ends of that side. It turns out that Hannah obtained \(33\) consecutive numbers in certain order. Can you find an arrangement of numbers as written by Nick which lead to this situation?
Cut a \(7\times 7\) square into \(9\) rectangles, out of which you can construct any rectangle whose sidelengths are less than \(7\). Show how to construct the rectangles.
There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?
Find all the prime numbers \(p\) such that the number \(2p^2+1\) is also prime.
Is it possible to arrange the numbers \(1,\, 2,\, ...,\, 50\) at the vertices and middles of the sides of a regular \(25\)-gon so that the sum of the three numbers at the ends and in the middle of each side is the same for all sides?
Consider a set of numbers \(\{1,2,3,4,...n\}\) for natural \(n\). Find the number of permutations of this set, namely the number of possible sequences \((a_1,a_2,...a_n)\) where \(a_i\) are different numbers from \(1\) to \(n\).
We want to wrap \(12\) Christmas presents in different coloured paper. We have \(6\) different patterns of paper and we want to use each one exactly twice. In how many ways can we do this?
Mr Roberts wants to place his little stone sculptures of vegetables on the different shelves around the house. He has \(17\) sculptures in total and three shelves that can fit \(7\), \(8\) and \(2\) sculptures respectively. In how many ways can he do this?
The order of sculptures on the shelf does not matter.