An illusionist lays a full deck of 52 cards in a row and announces that, after 51 removals, only the Three of Clubs will remain. At each step a spectator names a valid position number. The illusionist chooses whether to count that position from the left or from the right and removes the corresponding card. For which initial positions of the Three of Clubs can the illusionist guarantee success?
Starting with 1 on a blackboard, one may replace the current integer $x$ by either $3x+1$ or $\lfloor x/2\rfloor$, the greatest integer not exceeding $x/2$. Prove that every positive integer can be obtained.
In a polygon, every two adjacent sides are perpendicular. Two vertices are called unfriendly if their angle-bisector lines are perpendicular. Prove that every vertex has an even number of unfriendly vertices.
There is a counter in each of 100 consecutive squares. Swapping adjacent counters costs 1 dollar; swapping counters with exactly four counters between them is free. What is the minimum cost to reverse their order? Give the number of dollars.
