Solution to: Weighing Marbles
Number the ten bags from 1 up to and including 10. Take the following number of marbles from each bag:
| Bag | Number of marbles |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 5 |
| 5 | 8 |
| 6 | 13 |
| 7 | 21 |
| 8 | 34 |
| 9 | 55 |
| 10 | 89 |
In the numbers of marbles, you can recognize the famous Fibonacci sequence, in which every number after the first two is the sum of the two preceding ones: (0,) 1, 1, 2, 3, 5, 8, 13, ...
Place all 231 marbles that you took from the bags together on the balance. The number of grams that the total weight of these marbles is less than 2310 grams indicates the two bags with the marbles of 9 grams, according to the table shown below:
| Bag | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 g | - | - | - | - | - | - | - | - |
| 3 | 4 g | 5 g | - | - | - | - | - | - | - |
| 4 | 6 g | 7 g | 8 g | - | - | - | - | - | - |
| 5 | 9 g | 10 g | 11 g | 13 g | - | - | - | - | - |
| 6 | 14 g | 15 g | 16 g | 18 g | 21 g | - | - | - | - |
| 7 | 22 g | 23 g | 24 g | 26 g | 29 g | 34 g | - | - | - |
| 8 | 35 g | 36 g | 37 g | 39 g | 42 g | 47 g | 55 g | - | - |
| 9 | 56 g | 57 g | 58 g | 60 g | 63 g | 68 g | 76 g | 89 g | - |
| 10 | 90 g | 91 g | 92 g | 94 g | 97 g | 102 g | 110 g | 123 g | 144 g |
For example, if the weight of the marbles is 2268 grams, which is 42 grams less than 2310 grams, then bags 5 and 8 contain the marbles of 9 grams.
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