In the long division below, each asterisk represents a whole number — any digit from 0 to 9. Reconstruct all the calculations, given that there is no remainder.
[Solution]

## Lucky Derby

The bugle sounds, and 20 horses make their way to the starting gate for the first annual Lucky Derby. These horses, all trained at the mysterious Riddler Stables, are special. Each second, every Riddler-trained horse takes one step. Each step is exactly one meter long. But what these horses exhibit in precision, they lack in sense of direction. Most of the time, their steps are forward (toward the finish line) but the rest of the time they are backward (away from the finish line). As an avid fan of the Lucky Derby, you’ve done exhaustive research on these 20 competitors. You know that Horse One goes forward 52 percent of the time, Horse Two 54 percent of the time, Horse Three 56 percent, and so on, up to the favorite filly, Horse Twenty, who steps forward 90 percent of the time. The horses’ steps are taken independently of one another, and the finish line is 200 meters from the starting gate.
Handicap this race and place your bets! In other words, what are the odds (a percentage is fine) that each horse wins?
[Solution]

## Paint Balls

You play a game with four balls: One ball is red, one is blue, one is green and one is yellow. They are placed in a box. You draw a ball out of the box at random and note its color. Without replacing the first ball, you draw a second ball and then paint it to match the color of the first. Replace both balls, and repeat the process. The game ends when all four balls have become the same color. What is the expected number of turns to finish the game?
[Solution]

## Highest Card

From a shuffled deck of 100 cards that are numbered 1 to 100, you are dealt 10 cards face down. You turn the cards over one by one. After each card, you must decide whether to end the game. If you end the game on the highest card in the hand you were dealt, you win; otherwise, you lose.
What is the strategy that optimizes your chances of winning? How does the strategy change as the sizes of the deck and the hand are changed?
[Solution]

## Supreme Vacancies

Imagine that U.S. Supreme Court nominees are only confirmed if the same party holds the presidency and the Senate. What is the expected number of vacancies on the bench in the long run?
[Solution]