Copyright 2016 Mario V. Farina
Shakespir Edition, License Notes
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The game of Chess was invented by a little-known mathematician named the Thaddeus, Thad, for short in the land named Chessanovia, a land led by King Chester, the Great. Thad named the game in honor of the king and of the land in which they lived.
The king loved the game so much he offered to grant the inventor one wish. Thad’s response was, “Great King, grant to me ownership to a plot of land equal to the size of one square inch for the first square on the chessboard, another plot with the size of two square inches for the second square, another plot with the size of four square inches for the third plot, etc, doubling the size of each subsequent plot for the fifth through the sixty-fourth square.”
The king was not sure he had heard correctly. He asked, “let me see if I understand your request. As I understand it you want sixty-four plots of land. The first six plots will have these sizes: one square inch, two square inches, four square inches, eight square inches, sixteen square inches, thirty-two square inches. Am I correct?”
“Absolutely correct, oh Great King,” responded Thad.
The king was amazed at the modesty of the wish and said, “Thaddeus, thou art the full for requesting so little. I will gladly grant your wish!”
The first few plots were easy. A one square inch plot of land was signed over to Thad, then a two square inch plot, then a four square inch plot, and so forth. But soon, it was found that the plots were becoming alarmingly large. For example, when the size of the plots exceeded the size of the throne room, not all the squares had been covered. When the plots required exceeded all of the available land in the kingdom, Thad mysteriously disappeared.
Since Thad could not be found, the king was not able to give him a reward for his invention. Actually, the king never gave anyone a reward.
Many years later, a mathematician named Dr. Llewellyn calculated exactly how much land it would have required for Thad’s wish to have been granted. He stated that it would have required land in square inches represented by the number two raised to the sixty-fourth power minus one power. In actual numbers the number of square inches would have equaled 18,446,744,073,709,551,615.
This number is too large to give us a good understanding of what Thad had requested, so Dr. Llewellyn converted the number to square feet. Dividing the large number by 144 (there are 144 square inches in one square foot) gives a result of about 128,102,389,400, 761,000 square feet.
Dr. Llewellyn converted that number to square miles. It requires 27,878,400 square feet to equal one square mile. Dividing the preceding large number by 27,878,400, Dr. Llewellyn showed that Thad’s request had been to receive a plot of land having a size of 4,595,040,942 square miles.
This figure is larger than the entire land and water surface of the entire world. The entire land and water surface of the world is a little less than 200,000,000 square miles. It seems clear that Thad’s reward would have required the surface area of many worlds. If we divide 4,595,040,942 square miles by 200,000,000, we’ll see that it would have required about twenty-three worlds to satisfy Thad’s reward.
Thad had mysteriously disappeared. It was whispered that he had been banished to the twenty-third world.
Numbers that double have a way of getting very large very fast. Following are some numbers that show what happens when a number begins with 1 and doubles 63 times:
Time Doubled, Result