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There's a room in your school that has three tables in it with plenty of space for chairs to go round. Table $1$ has one block of chocolate on it, table $2$ has two blocks of chocolate on it and, guess what, table $3$ has three blocks of chocolate on it.
Now ... outside the room is a class of children. Thirty of them all lined up ready to go in and eat the chocolate. These children are allowed to come in one at a time and can enter when the person in front of them has sat down. When a child enters the room they ask themself this question:
"If the chocolate on the table I sit at is to be shared out equally when I sit down, which would be the best table to sit at?"
It is fairly easy for the first few children to decide where to sit, but the question gets harder to answer, e.g.
It maybe that when child $9$ comes into the room they see:
Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?