Problems

You have in your hands a royal flush! That is, Ace, King, Queen, Jack and \(10\) of spades. How many shuffles of your hand swap the Ace and Jack?

Munira wants to put \(6\) books on her shelf, \(4\) of which are red and \(2\) of which are blue. The four red ones are a small paperback, a small hardback, a large paperback and a large hardback. The two blue ones are both paperback, one small and one large. She doesn’t want the two blue ones next to each other. In how many ways can she do this?

Imagine a cube that’s turquoise on the front, pink on top, yellow on the right, white on left, dark blue on back and orange on the bottom. If Arne rotates this \(180^{\circ}\) about the line through the middles of the turquoise and dark blue sides, then does it again, he gets back to the original cube. If Arne rotates this \(90^{\circ}\) about that same line, then does that three more times, then he also gets back to the original cube.
Is there a rotation he could do, and then do twice more, to get back to the original cube?

Arne has a cube which is pink on top and orange on bottom, yellow on right and white on left, turquoise on front and dark blue at the back. He rotates this once so that it looks different. Could he perform the same rotation four more times and get back to the original colouring?

Sam the magician shuffles his hand of six cards: joker, ace (\(A\)), ten, jack (\(J\)), queen (\(Q\)) and king (\(K\)). After his shuffle, the relative order of joker, \(A\) and \(10\) is now \(A\), \(10\), joker. Also, the relative order of \(J\), \(Q\) and \(K\) is now \(Q\), \(K\) and \(J\).
For example, he could have \(A\), \(Q\), \(10\), joker, \(K\), \(J\) - but not \(A\), \(Q\), \(10\), joker, \(J\), \(K\).

How many choices does Sam has for his shuffle?