Is it possible to colour the cells of a \(3\times 3\) board red and yellow such that there are the same number of red cells and yellow cells?
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.
There are six letters in the alphabet of the Gloops. 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 Gloops?
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 \(n\).
A rectangular parallelepiped of the size \(m\times n\times k\) is divided into unit cubes. How many rectangular parallelepipeds are formed in total (including the original one)?