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Although we don't have the tools to prove these are the minimal number of moves to solve the puzzle, we have gathered some evidence. The class was unable to solve the puzzle for 1, 2, 3, and 4 discs with fewer moves, and we notice a pattern in the list of numbers. Tell students that the numbers in the table are, in fact, the minimum number of moves required to solve the puzzle for each number of discs, and they may have opportunities to prove it in future courses.
Write this list for all to see: 1, 3, 7, 15, 31, 63, 127
Tell students that in mathematics we often call a list of numbers a sequence. A speci c number in the list is called a term of the sequence. Ask students how they would describe the rule for the next term in the pattern. Write down any notation they come up with to describe the recursive rule, such as There is no need to introduce formal notation or discuss a closed-form rule at this time, but if students suggest these (time permitting), welcome their explanations.
1.3 Checker Jumping Puzzle
Optional: 35 minutes
This optional activity is provided in case you need an additional activity to reinforce the lesson goal. It can be safely skipped if you don’t need it. The instructions are identical to the previous activity, but the mathematical context results in a di erent sequence. The recursive rule is something like “ rst add 5, and then keep adding the next odd number.” The sequence is 3, 8, 15, 24, . . .
SILVERMATH applet requested
Student Task Statement
Some checkers are lined up, with red on one side, blue on the other, with one empty space between them. A move in this checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backwards is not allowed, and 2 checkers cannot occupy the same space.
You complete the puzzle by switching the colors completely: ending up with red on the left, blue on the right, with 1 empty space between them.
1. Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?
2. Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?
3. Make guesses about the number of moves for 2 and 4 checkers on each side, then test your guesses.
Unit 1
Lesson 1: A Towering Sequence 11