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.
A goat and a cow would take \(45\) days to eat a full cart of hay. It would take a cow and a sheep \(60\) days, but a sheep and a goat would need \(90\) days. How many days would it take for all three animals to eat all the hay?
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?
Prove the \(GM-HM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1} + ... \frac{1}{a_n}}.\]
Find all pairs of whole numbers \((x,y)\) so that this equation is true: \(xy+1 = y+x\).
Albert was calculating consecutive squares of natural numbers and looking at differences between them. He noticed the difference between \(1\) and \(4=2^2\) is \(3\), the difference between \(4\) and \(9=3^2\) is \(5\), the difference between \(9\) and \(16=4^2\) is \(7\), between \(16\) and \(5^2=25\) is \(9\), between \(25\) and \(6^2=36\) is \(11\). Find out what the rule is and prove it.
Find all pairs of whole numbers \((x,y)\) so that this equation is true: \(xy = y+1\).
After Albert discovered the previous rule, he began looking at differences of squares of consecutive odd numbers. He found the difference between \(1^2\) and \(3^2\) is \(8\), the difference between \(3^2\) and \(5^2\) is \(16\), the difference between \(5^2\) and \(7^2\) is \(24\), and that the difference between \(7^2\) and \(9^2\) is \(32\). What is the rule now? Can you prove it?