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
Grains on Square
Running Total
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



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12-September-2008
Copyright Jill Britton