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AMBITION

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