This week’s puzzle is a good one to use early in the school year. It is fairly easy and shouldn’t frustrate students too much in their early exposure to the puzzle-solving process. To do this puzzle, students need only the student sheet depicting the nine-penny arrays and a pencil. The puzzle challenges students to draw four “pens” around the nine pennies in the array in such a way that there is an odd number of pennies in each pen. Students should begin to realize, after several trial and error attempts, that there is no way to divide nine objects into four separate sets of odd numbers since any combination of four odd numbers produces an even sum. Once this realization has been made, the puzzle seems unsolvable. This is where an important puzzle-/problem-solving skill comes into play—thinking divergently. Since there is no way to draw four separate pens (with no overlap) that each contain an odd number of pennies, one or more of the pens must overlap. Once this insight is made, it is easy to see that there are several valid solutions to the puzzle.
As in any Puzzle Corner activity, students should be encouraged to work independently and asked not to share their solutions until the appropriate sharing time at the end of the week. While you should encourage students not to give away their solutions, you may want to allow them to offer hints to their fellow students. These hints can help those students who have not yet developed their abilities to think outside the box. Your role as the teacher is to facilitate this problem-solving process—not give students the answer.
Click the arrow below to view the solution.
The Penny Penning Challenge asked students to draw four “pens” around the nine pennies in a three by three array in such a way that there were an odd number of pennies in each pen. There is no way to divide nine objects into four separate sets of odd numbers since any combination of four odd numbers produces an even sum. Therefore, to solve this puzzle students had to think divergently and realize that in any solution, one or more of the pens must overlap. When this insight has been made, there are several valid solutions to the puzzle. One of these solutions is shown below.