Problems

Prove the divisibility rule for \(25\): a number is divisible by \(25\) if and only if the number made by the last two digits of the original number is divisible by \(25\);
Can you come up with a divisibility rule for \(125\)?

Which of the following numbers are divisible by \(11\) and which are not? \[121,\, 143,\, 286, 235, \, 473,\, 798, \, 693,\, 576, \,748\] Can you write down and prove a divisibility rule which helps to determine if a three digit number is divisible by \(11\)?

On a chessboard (an \(8 \times 8\) grid), we place eight identical rooks. A rook can move any number of squares in a straight line horizontally (along a row) or vertically (along a column). In chess, a piece can take another piece if it can move to the other piece’s square in a single move.

In how many ways can we arrange the eight rooks so that no rook can take any other?

How many five-digit numbers are there which are written the same from left to right and from right to left? For example the numbers \(54345\) and \(12321\) satisfy the condition, but the numbers \(23423\) and \(56789\) do not.

A coin is tossed six times. How many different sequences of heads and tails can you get?

Each cell of a \(3 \times 3\) square can be painted either black, white, or grey. How many different ways are there to colour in this table?

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 \(n\).

Eleven people were waiting in line in the rain, each holding an umbrella. They stood closely together, so that the umbrellas of the neighbouring people were touching (see the figure)

The rain stopped and everyone closed their umbrellas. They then shuffled closer, keeping a distance of \(50\) cm between neighbours. What is the ratio of the old queue length to the new queue length? People can be considered points, and umbrellas are circles with a radius of \(50\) cm.

Cut a square into five triangles in such a way that the area of one of these triangles is equal to the sum of the area of other four triangles.

A circular triangle is a triangle in which the sides are arcs of circles. Below is a circular triangle in which the sides are arcs of circles centered at the vertices opposite to the sides.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of the circular triangle.