Harry Potter and the Boggling Bottles
1. The two possible combinations that satisfy the clues are PWPFPWB and PWFPPWB, so the F bottle must be in either position 3 or 4.
2. The biggest bottle must be in either position 2 or 6, so that Hermione could eliminate possible solutions that have poison in both these positions.
3. The only safe bottle in all possible solutions is bottle 7.
4. In this case there are two possible orders for the bottles: BPPWFPW and BPWFPPW. In either case Hermione could safely choose the first bottle as B. But not knowing whether bottle 4 or 5 was F, the best she could do would be to give bottle 4 to Harry to drink. If it proved to be wine, which would not protect him from the fire in front, he could then drink bottle 5, knowing it was F. But that would work only if there were a way to distinguish between W and F without being burned. A Pentomino Odyssey
The insight for problem 4 is that there are only six pentominoes that fit in a strip 2 squares wide. If we place all six of these pieces on the board, then we can leave as many empty 2-square-wide strips as we please. Mathematics Takes a Bow
1. Two paths in three steps: BCD and BCE
2. Fourteen paths in four steps: eight ways that bounce back from point 3 (ADDD, ADDE, ADED, ADEE, AEDD, AEDE, AEED, AEEE), four ways that bounce back from point 2 (AAAD, AAAE, ACCD, ACCE), and two ways that double back from point 4 (BBAD, BBAE).
3. Eighteen paths in five steps.
4. There are 9,829,858,162 paths in 25 steps. To solve the problem with a spreadsheet program, make a table listing the number of ways to reach point 1, 2, 3, and 4 in a given number of steps. The entries in each row can be computed from the entries in the previous row using the formulas shown below. To reach a point in n moves, you must first reach one of the neighbors of the point with one move fewer, so the number of ways to reach a point in n moves is the sum of the number of ways to reach the neighboring points in n-1 moves.
|Steps||# Paths to Point 1||# Paths to Point 2||# Paths to Point 3||# Paths to Point 4|
Want to go back to the puzzle
Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com