The Legend of the Chessboard
 Long long ago in an ancient land, there lived a very wise man who happened to be the Vizier at the court of a great Sultan.  Months and years passed by and the great Sultan died, and his young prince replaced him. Being young, the prince lacked experience. He started spending more than what his father used to. The wise Vizier decided to teach the brash prince a lesson!  The prince set a contest and as a reward, decided to give the winner whatever he wishes, boasting of his wealth, being under the illusion that his wealth is virtually endless. The Vizier won, and asked the prince for the prize: a single grain of wheat and a chessboard!  "What?! Just a grain of wheat! Are you insulting my wealth?" yelled the prince. "No! Your majesty!" The Vizier explained. "You have to promise to double that grain of wheat until the chessboard is full, so on the first day you give me one grain of wheat on the first square of the chessboard, on the second day you double it on the second square (giving me two grains), on the third, you double that on the third square (giving me four grains), and so on, until the sixty fourth square on the chessboard." "I would thought you being so smart", the young prince said. "You would ask for something more substantial. Anyway, if this is your wish I will grant you that." And so, on the second day, the Vizier got 2 grains, on the third, he got 4 grains, and the young prince couldn't help himself making fun of the Vizier.  By the sixth day, the Vizier got 32 grains of wheat. By the eighth day and the end of the first row, he got a mere 128 grains. By the sixteenth day and the end of the second row, he got 32,768 grains.  Where was this leading to? Was it worth it for the Vizier? By the end of the game (it was a mind game, wasn't it?) can you guess how many grains the Vizier would get in all? The prince could not provide enough grains to give the Vizier a chessboard's worth of grains. Why? Study the progress of the number of grains on the individual squares and the corresponding running total in the table below. There are not enough grains in the whole world to give the Vizier the sum of 18,446,744,073,709,551,615 grains!  Indeed, that many grains would cover the entire earth several inches deep.

 Square Running Total Grains on Square 1 1 1 2 2 3 3 4 7 4 8 15 5 16 31 6 32 63 7 64 127 8 128 255 9 256 511 10 512 1,023 11 1,024 2,047 12 2,048 4,095 13 4,096 8,191 14 8,192 16,383 15 16,384 32,767 16 32,768 65,535 17 65,536 131,071 18 131,072 262,143 19 262,144 524,287 20 524,288 1,048,575 21 1,048,576 2,097,151 22 2,087,152 4,194,303 23 2,097,152 8,388,607 24 8,388,608 16,777,215 25 16,777,216 33,554,431 26 33,554,432 67,108,863 27 67,108,864 134,217,727 28 134,217,728 268,435,455 29 268,435,456 536,870,911 30 536,870,912 1,073,741,823 31 1,073,741,824 2,147,483,647 32 2,147,483,648 4,294,967,295 33 4,294,967,296 8,589,934,591 34 8,589,934,592 17,179,869,183 35 17,179,869,184 34,359,738,367 36 34,359,738,368 68,719,476,735 37 68,719,476,736 137,438,953,471 38 137,438,953,472 274,877,906,943 39 274,877,906,944 549,755,813,887 40 549,755,813,888 1,099,511,627,775 41 1,099,511,627,776 2,199,023,255,551 42 2,199,023,255,552 4,398,046,511,103 43 4,398,046,511,104 8,796,093,022,207 44 8,796,093,022,208 17,592,186,044,415 45 17,592,186,044,416 35,184,372,088,831 46 35,184,372,088,832 70,368,744,177,663 47 70,368,744,177,664 140,737,488,355,327 48 140,737,488,355,328 281,474,976,710,655 49 281,474,976,710,656 562,949,953,421,311 50 562,949,953,421,312 1,125,899,906,842,623 51 1,125,899,906,842,624 2,251,799,813,685,247 52 2,251,799,813,685,248 4,503,599,627,370,495 53 4,503,599,627,370,496 9,007,199,254,740,991 54 9,007,199,254,740,992 18,014,398,509,481,983 55 18,014,398,509,481,984 36,028,797,018,963,967 56 36,028,797,018,963,968 72,057,594,037,927,935 57 72,057,594,037,927,936 144,115,188,075,855,871 58 144,115,188,075,855,872 288,230,376,151,711,743 59 288,230,376,151,711,744 576,460,752,303,423,487 60 576,460,752,303,423,488 1,152,921,504,606,846,975 61 1,152,921,504,606,846,976 2,305,843,009,213,693,951 62 2,305,843,009,213,693,952 4,611,686,018,427,387,903 63 4,611,686,018,427,387,904 9,223,372,036,854,775,807 64 9,223,372,036,854,775,808 18,446,744,073,709,551,615