11 scouts are working on 5 different badges. Prove that there will be two scouts \(A\) and \(B\), such that every badge that \(A\) is working towards is also being worked towards by \(B\).
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
A piece fell out of a book, the first page of which is the number 439, and the number of the last page is written with those same numbers in some other order. How many pages are in the fallen out piece?
A rectangle of size \(199\times991\) is drawn on squared paper. How many squares intersect the diagonal of the rectangle?
Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?
Each cell of a \(2 \times 2\) square can be painted either black or white. How many different patterns can be obtained?
\(N\) young men and \(N\) young ladies gathered on the dance floor. How many ways can they split into pairs in order to participate in the next dance?
Prove that the product of any three consecutive natural numbers is divisible by 6.
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).