Chess and grains
The legend of chess
For many centuries chess has been a game well known and popular. Its very creation is surrounded by different variants of the following legend:
Once upon a time an Indian ruler led his country and citizens into poverty and misery. A wise man intended to call the ruler’s attention to his failures, yet feared his wrath. So he devised the game chess: While in chess the king is the most important piece beyond doubt, it is utterly helpless without the other pieces. Even pawns play a crucial role.
Getting to know chess, the Indian ruler understood the message and became more gentle and gracious. Highly impressed he offered the wise man a reward of his choice. When the wise man asked for one grain for the first square, two for the second, four for the third, and so on, the ruler thought this a modest and humble wish and granted it gladly.
Task
Create a mathematical model that allows you to determine the number of grains on each of the 64 chess squares. Use this model to answer the questions below.
Instructions
| 1. | A1 | Enter 1 in cell A1 and 2 in A2. |
| 2. |  | Highlight both cells A1 and A2 and drag the little square in the lower right corner of the highlighted area down to cell A12. |
| 3. | B1 | Enter 1 in cell B1 and the formula =2*B1 in cell B2. |
| 4. | | Relative copy the formula from cell B2 down to cell B12. Hint: Only select cell B2 and drag the lower right corner of the highlighted area to relative copy its formula to the other cells. |
| 5. |  | Highlight the cells A1 to B12 and use the tool List of Points.
Note: The points will be displayed in the  Graphics View.
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Try it yourself...
Task
Try to find a formula that allows you to calculate the number of grains on an arbitrary square of the chess board.
Instructions
| 6. | C2 | Enter the Command =Factors(B2)
in cell C2 and press the Enter key.
Note: The Factors command lists the prime factors of a number together with their corresponding exponents.
Example: ab is displayed as (a b). |
| 7. | | Relative copy the content of cell C2 down to cell C12. |
| 8. | Compare the values in column A (number of square on the chess board) to the corresponding values in column C. Can you come up with a formula to directly calculate the number of grains on each square? | |
| 9. | D1 | Enter your formula as a function g(x)=... in cell D1 and press the Enter key to immediately display its graph in the Graphics View.
Hint: The x in your formula is supposed to be the number of a chess square. |
| 10. | Check whether the points you created in step 5 match the graph of your function. If not, try to adjust your formula accordingly. |