Problems

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?

Can you tile the plane with regular octagons?

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?

Draw how to tile the whole plane with figures, composed from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), and \(5\times 5\) where squares of all sizes are used the same amount of times in the design of the figure.

Anna and Beth played rock paper scissors ten times. Rock beat scissors, scissors beat paper and paper beat rock. Anna used rock three times, scissors six times and paper once. Beth used rock twice, scissors four times and paper four times. None of the ten games was a tie. Who won more games?

Consider the \(4!\) possible permutations of the numbers \(1,2,3,4\). Which of those permutations keep the expression \(x_1x_2+x_3x_4\) the same?

In the picture below, there are the \(12\) pentominoes. Is it possible to tile a \(6\times10\) rectangle with them, using each pentominoe exactly once?

Show how to tile a \(5\times12\) rectangle with the twelve pentominoes.

Show how to tile a \(4\times15\) rectangle with the twelve pentominoes.