Reasoning

A cat knocks 3 things off a table every hour. The owner puts 2 things back every hour. There are 12 things on the table. When does the table become empty?

You have 8 identical-looking balls. One is slightly heavier than the rest. Using a balance scale, what is the minimum number of weighings needed to guarantee you find the heavy ball? Explain your strategy.

A farmer needs to cross a river with a cat, a mouse, and a bag of catnip. The boat can carry the farmer and one item at a time. If left alone together: the cat will eat the mouse, and the cat will destroy the catnip (cats gonna cat). The mouse ignores the catnip. How does the farmer get everything across? Show all steps.

In a round-robin tournament with 6 players, each player plays every other player exactly once. A win is worth 2 points, a draw is 1 point each, and a loss is 0 points. After the tournament: Player A has 10 points, Player B has 8 points, and Player C has 6 points. Players D, E, and F each have the same score. What score do D, E, and F each have? Show your reasoning.

There are 100 closed lockers in a hallway. 100 cats walk by. The first cat toggles every locker. The second cat toggles every 2nd locker. The third cat toggles every 3rd locker. And so on — the nth cat toggles every nth locker. After all 100 cats have passed: (a) which lockers are open? (b) Why? Prove your answer mathematically.