A Classic Chess Fable

(Reposted in honor of Martian Chess (HT BMF))

Continuing the Chess train of thought from earlier this week. I was reminded of one of my favorite legends surrounding Chess. Before the game had a King and Queen, in Persia, where the Chess was invented, it had a Shah and Vizier. The Shah is a Persian King (“Shah Mat” in Persian means “The King is Ambushed / Dumbfounded/ Dead“, and is where we get the term “Checkmate.”)

As the story goes, the Vizier, Shah’s closest advisor, had invented the game of Chess, and the Shah was so delighted with it that he offered the Vizier anything he wanted as payment for creating such a wonderful past time.

The Vizier answered that he would like to have one grain of wheat for the first square on the Chess board, two for the second, four for the third, eight for the fourth, and so on, doubling the amount until all 64 squares were accounted for.

“That’s all?” the Shah was surprised at the seemingly meager request, but quickly had his treasurer tasked with fulfilling it.

The Treasurer did the math, as we can see here:

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

There wasn’t enough grain on the entire planet to fulfill the Vizier’s request! No one knows what happened to the Vizier when the Shah discovered the trick, but the fable does illustrate the power of exponential growth.