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# The Fifteen Cent Flip

The Puzzle Corner activity this week comes from the great puzzlist Martin Gardner. In his book Perplexing Puzzles and Tantalizing Teasers, this game appears with the title The Dime-and-Nickel Switcheroo. Six squares are shown forming a two by three rectangle. In every square but one a coin is to be placed three pennies, one dime, and one nickel. The object of the puzzle is to switch the dime and the nickel using the fewest possible moves. A coin can only move by sliding into an adjacent empty spot to the right, left, up, or down (no diagonals).

As Gardner notes, this puzzle is easy to do if you slide the coins for long enough; the challenge is to solve the puzzle in the fewest possible moves. This kind of puzzle is ideal for the elementary classroom and beginning puzzle solvers because it allows for success at every level. If a student can improve upon his or her original solution, success has been achieved (assuming it wasn’t solved in the fewest number of moves on the first try).

Gardner gives some broad guidelines concerning the best possible solution saying, “It can be done in fewer than 20 moves, but it takes more than 12.” I would encourage you not to give your students this information until they have already discovered at least one solution and improved upon it. That way their sense of achievement will not be diminished if their original solution did not come close to the best solution.

I would also encourage you to try this problem yourself before giving it to your students. See how long it takes you to discover the best possible solution. It is always good to be able to empathize with students as they go through the problem-solving experience. Not only will you appreciate their task much more, you will be better able to guide their discovery without giving away the answer.

Although this puzzle calls for coins, other small objects can be substituted as long as there are three distinct types. Colored tiles, beans, buttons, or other small manipulatives will all work equally well. Each student should have a set of manipulatives to use while working on the puzzle you can even have the necessary coins brought from home the day before so that the puzzle can be taken home and shared with families when students are finished with it.

Place the appropriate coins on top of their pictures in the grid below. The object of the game is to switch the positions of the nickel and the dime. A coin can move by sliding into an empty space that is to the left, right, above or below. Coins cannot move diagonally. Try to switch the coins in the fewest number of moves possible. Once you get a solution, see if you can get another that uses fewer moves. Continue to improve upon each solution until you cannot solve the puzzle in any fewer moves.

My original solution took me ____________ moves.

My best solution took me ___________ moves.

I improved by ___________ moves.

Solutions

Click the arrow below to view the solutions.

The challenge in the puzzle was to switch the position of a nickel and a dime in a rectangular array using the fewest number of moves possible. It was stated that the minimum number of moves needed to switch the coins is fewer than 20, but more than 12. The puzzle can actually be solved in 17 moves by following the sequence given below. In each step the direction the coin moves is indicated. Due to the arrangement of the array, there is only one coin that can move in the direction indicated for a given step, so which coin is moving has not been specified.

1. Left
2. Down
3. Right
4. Right
5. Up
6. Left
7. Down
8. Left
9. Up
10. Right
11. Right
12. Down
13. Left
14. Up
15. Left
16. Down
17. Right