ABSTRACT
This chapter introduces the Intermediate Value Theorem, which says that if a number begins at A and changes continuously until it reaches B, then it must pass through every value in between. The usefulness of this took is explored in 9 puzzles and a theorem. The puzzles include inscribing West Virginia in a square, dividing a necklace, making a fair decision, and bisecting a polygon. One puzzle even exhibits a case where the Intermediate Value Theorem mysteriously works for a number that jumps around instead of moving continuously. Another puzzle uses a three-dimensional analog of the intermediate Value Theorem to prove the surprising fact that in any set of 100 baskets containing apples, bananas and cherries, there are 51 baskets that together contain at least half of each fruit. The chapter concludes with a classical theorem: every odd-degree polynomial with real coefficients has a real root.
