Problems

Age
Difficulty
Found: 1923

Cut a packet of 4 cards. Is any of the cards in the same place as it was before?

We have a packet 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 interchange 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 1×2 rectangles of a 6×6 rectangle?

Irreducibly tile a floor with 1×2 tiles in a room that is a 6×8 rectangle.

Prove that every pair of consecutive Fibonacci numbers are coprime. That is, they share no common factors other than 1.

Calculate the following: F12F0F2, F22F1F3, F32F2F4, F42F3F5 and F52F4F6. What do you notice?

Work out F32F0F6, F42F1F7, F52F2F8 and F62F3F9. 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 i=0nFi2=F02+F12+F22+...+Fn12+Fn2 in terms of just Fn and Fn+1?