Puzzle with pebbles
This unit is adapted from Mathematical Journeys by P. D. Schumer, published by John Wiley & Sons, Inc., 2004. The mathematical theory that deals with this type of problem is called "the theory of Petri nets".
Students can do this puzzle with pebbles, pennies, beans, or any other small objects.
The pebbles are arranged into three piles (a pile can be empty). A move consists of selecting one pile, taking out one pebble from each of the remaining two, and adding these two pebbles to the selected pile.
The goal is to put all pebbles into one pile.
An example.
Here we show the number of pebbles after each move in piles A, B, and C
| move | A: | B: | C: | |
| 10 | 2 | 1 | ||
| 1. | 9 | 1 | 3 | C is selected. |
| 2. | 8 | 0 | 5 | C is selected. |
| 3. | 7 | 2 | 4 | B is selected. |
| 4. | 6 | 1 | 6 | C is selected. |
| 5. | 5 | 3 | 5 | B is selected. |
| 6. | 4 | 5 | 4 | B is selected. |
| 7. | 3 | 7 | 3 | B is selected. |
| 8. | 2 | 9 | 2 | B is selected. |
| 9. | 1 | 11 | 1 | B is selected. |
| 10. | 0 | 13 | 0 | B is selected. |
Ten moves were made to reach the 11th position from the first one.
Remark.
The goal can be achieved only for some initial configurations.
Tasks.
Find when the goal can be achieved, and when it cannot.
Describe a strategy for achieving the goal (when it is possible).
Answers
>Let x, y, and z be the number of pebbles in the three piles. If the remainders from division by 3 of x, y, and z are all different, then the goal cannot be achieved.
Reason: Observe that if the remainders from division by 3 of the numbers of elements of two piles are different, then they remain different after any move. But when the goal is achieved, remainders for both empty piles are 0.
Observe the example above. See for which two piles the remainders are the same (A and C above), and work toward making these numbers equal. Afterwards, the rest is easy.
Remark.
The puzzle can be presented in any grade, but analyzing the strategy requires knowledge of arithmetic that is rarely achieved before middle school.