Alice wants to write down the numbers from 1 to 16 in such a way that the sum of two neighbouring numbers will be a square number. The Hatter tells Alice that he can write down the numbers with this property in a line, but he believes that it is absolutely impossible to write the numbers with this property in a circle. Show that he is right.
In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?
On her birthday, my grandma was asked how old she was. She said: "Start with the year I was born. Add the current year to it. Then, from the sum subtract the year I celebrated by \(20\)th birthday. From that, take away the year I was \(30\). The result will be \(16\)." How old is my grandma?