by Dave Youngs
This month's puzzle is an adaptation of one I found in Martin Gardner's book Perplexing Puzzles and Tantalizing Teasers from Dover Publications. Gardner's puzzle, which is almost identical to the second part of Star Crossings, looks deceptively easy, but proves difficult. In order not to frustrate students, I have included a much easier problem for a warm-up before students tackle the harder puzzle. I have also challenged students to make a record of their answers when they find them-there are multiple answers for both puzzles-and to see if they can find any patterns in those answers.
To do Star Crossings students will need a copy of the sheet and four small objects like pennies, buttons, or beans. In both puzzles, students are to place the first object on any of the numbered vertices of the star and slide it along the line to a vertex opposite it and leave it there. They then place the second object on any vertex without an object and slide it into an unoccupied vertex and leave it there. They continue this process until they have placed all four objects. They then make a record of their moves by listing the number pairs making up their solution as shown below for the six-point star. In this example the first object was placed on one and slid to five, the second was placed on six and slid to two, the third placed on one and slid to three, and last placed on four and slid to six. For this solution the record would be written: 1-5, 6-2, 1-3, 4-6.
While the six-point star puzzle is quite easy, the five-point star is a different case altogether. It is possible, although it may not seem so after a few unsuccessful tries. With persistence however, students should be able to come up with at least one of the many solutions. When they do, have them record this solution using the technique already described. With several solutions recorded, students can look for patterns. One extension to this problem is to ask students if it is possible to place five objects on the six-point star in the same manner as four were placed. After numerous tries, they will discover that it is not possible. At that point try to get them to figure out why it is not possible. (Hint: Have them work backwards, starting with five objects on five of the points and seeing if they can take them off by sliding instead of putting them on that way.)
I hope your students enjoy this month's puzzle.