one grain of rice on the first square of a chess board....?

Question

in relation to http://answers.yahoo.com/question/;_ylt=A9FJujW.s_ZEDj0B5gIjzKIX?qid=1006042630384

After getting to the end of the chessboard, how many people would it feed? (assuming one person eats one bowl of rice)

Answer 1

There's something wrong with jay Z's answer:

He calculated the number of grains of rice in every square of the chessboard, but in the last squares you will notice that the ending digits are zeroes, but there is no power of 2 which is divisible by 10. This means that those numbers are rounded off, but there is a way of getting the exact number of grains in the chessboard:

Since there is:
1 grain of rice in the 1st square,
2 grains of rice in the 1st square,
4 grains of rice in the 1st square, and
8 grains of rice in the 1st square,

There are 2^(n - 1) grains of rice in the nth square: The number of grains we are looking for is (since there are 64 squares in the chessboard):
2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + ... + 2^62 + 2^63 + 2^64

If we multiply 1 to it in the form (2^0 - 2^1)/(2^0 - 2^1), then it is equal to
= (2^0 - 2^1)/(2^0 - 2^1) · (2^0 + 2^1 + 2^2 + 2^3 + 2^4 + ... + 2^63 + 2^64)

= (2^0 - 2^1)(2^0 + 2^1 + 2^2 + 2^3 + 2^4 + ... + 2^63 + 2^64)/(2^0 - 2^1)

Trying to distribute (2^0 - 2^1),
= (2^0 - 2^1 + 2^1 - 2^2 + 2^2 - 2^3 + 2^3 - 2^4 + 2^4 - ... - 2^64 + 2^64 - 2^65)/(2^0 - 2^1)
= (2^0 - 2^65)/(2^0 - 2^1)

Try to substitute 2^0 = 1 and 2^1 = 2:
= (1 - 2^65)/(1 - 2)
= (1 - 2^65)/(-1)
= 2^65 - 1

Therefore, there are exactly 2^65 - 1 or 36893488147419103231 grains of rice in the chessboard.

If we assume there 15000 grains of rice in one bowl of rice, then we would be able to feed (2^65 - 1)/15000 or approximately 2459565876494607 people. That's almost 100000 times the human population on the earth!

^_^

Answer 2

Kevin, I am a bit surprised. You ususally do a clear, clean job. I am just joking.

You are right that there are 2^(n-1) grains on the nth square.

The sum is wrong though:

2^(n-1) + 2^(n-2) + 2^(n-3)+.......2^3 + 2^2 +2 + 1+1 = 2^n

Since there are 2^63 grains on the 64th square the number of grains on the board is simply 2^64 - 1 = 18446744073709551615 But the exact number is irrelevant since we want to see how many people we can feed. I'd figure more realistically that you'd have about 100 grains in a spoon and maybe 50 spoons in a bowl so thats about 5000 per bowl. Let's be generous and make it a power of 2. Say 2^13 = 8192. So the number of people is aprroximately 2^64 / 2^13 = 2^51. Obviously we can't feed more people than there are on earth, so the earth's population is the limit with lots left over for 2nds..and the next day etc....lol

Answer 3

It seems that this would be more than enough to feed my whole people :) The population of my country is less than the population of London.

Answer 4

the answer is 9223372036854780000.00 grains of rice



check it out

Squareno of grains
1-1
2-2
3-4
4-8
5-16
6-32
7-64
8-128
9-256
10-512
11-1024
12-2048
13-4096
14-8192
15-16384
16-32768
17-65536
18-131072
19-262144
20-524288
21-1048576
22-2097152
23-4194304
24-8388608
25-16777216
26-33554432
27-67108864
28-134217728
29-268435456
30-536870912
31-1073741824
32-2147483648
33-4294967296
34-8589934592
35-17179869184
36-34359738368
37-68719476736
38-137438953472.00
39-274877906944.00
40-549755813888.00
41-1099511627776.00
42-2199023255552.00
43-4398046511104.00
44-8796093022208.00
45-17592186044416.00
46-35184372088832.00
47-70368744177664.00
48-140737488355328.00
49-281474976710656.00
50-562949953421312.00
51-1125899906842620.00
52-2251799813685250.00
53-4503599627370500.00
54-9007199254740990.00
55-18014398509482000.00
56-36028797018964000.00
57-72057594037927900.00
58-144115188075856000.00
59-288230376151712000.00
60-576460752303423000.00
61-1152921504606850000.00
62-2305843009213690000.00
63-4611686018427390000.00
64-9223372036854780000.00


my 10 points please

;-)