Cut a deck of \(4\) cards. Are any of the cards in the same place as they were before?
We have a deck of \(13\) cards from Ace to King. Let Ace be the first card, \(2\) the second card and so on with King being the thirteenth card. How can you swap \(4\) and \(7\) (and leave all other cards where they are) by only switching adjacent pairs of cards?
How many permutations of 13 cards leaves the third card where it started?
Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(6\times 6\) rectangle?
Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(6\times8\) rectangle.
Prove that every pair of consecutive Fibonacci numbers are coprime. That is, they share no common factors other than 1.
Calculate the following: \(F_1^2-F_0F_2\), \(F_2^2-F_1F_3\), \(F_3^2-F_2F_4\), \(F_4^2-F_3F_5\) and \(F_5^2-F_4F_6\). What do you notice?
Work out \(F_3^2-F_0F_6\), \(F_4^2-F_1F_7\), \(F_5^2-F_2F_8\) and \(F_6^2-F_3F_9\). What pattern do you spot?
Can every whole number be written as the sum of two Fibonacci numbers? If yes, then prove it. If not, then give an example of a number that can’t be. The two Fibonacci numbers don’t have to be different.
What’s \(\sum_{i=0}^nF_i^2=F_0^2+F_1^2+F_2^2+...+F_{n-1}^2+F_n^2\) in terms of just \(F_n\) and \(F_{n+1}\)?