Using all the numbers 0 1 2 3 4 5 6 7 8 9 10 and all the operators + + + + and - - - - - - exactly once, write an RPN expression with maximal value. (Example: using 1 2 3 and + -, the largest expression value is 3 1 - 2 + = 4)
Similar to problem #1, what is the minimal possible value?
Given a solution to #1, can you swap two numbers in your expression without changing the value? (e.g. 2 1 - 3 + = 4)
Follow up to #3 – if the answer is “yes”, count how many different expressions have the same value, where the order of numbers and operators is the same, but the numbers’ values can be rearranged. (e.g. 1 2 3 4 + + + and 4 2 3 1 + + + are the same)
Binary Search
If you play the clock game (binary search) to guess an integer in the range [1,1023], what is the maximum number of guesses required? What are three numbers that require that many guesses?
Playing the clock game, over a min-max range of [1,1023], write a formula for $g(n)$ where $g(n)$ is the number of prices that require exactly $n$ guesses. Follow-up does the range effect the definition of $g()$?
(Follow up to #2) – for the range [1,1023], what is the expected (average) number of guesses needed to win, assuming each price is equally likely?