# Number of grains of sand... **URL:** \ **Category:** Factual Questions\ **Created:** [May 2, 2007, 2:08pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457 "2007-05-02T14:08:14Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![cobberone](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@cobberone](https://boards.straightdope.com/u/cobberone)\ **Post date:** [May 2, 2007, 2:08pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/1 "2007-05-02T14:08:14Z")
G’day folks, greetings from Perth, Western Australia. This is my first post, so please be gentle with me. While playing a game of Chess with my nephew the other nite, a thought occured to me: If I was to place 1 grain of sand on the first square of a Chess-Board, 2 on the second, 4 on the third, 256 on the fourth, 65,536 on the fifth, how many grains of sand would there be on the last (64th) square? I’ve tried to work this out on using calculators, but they are not able to calculate such a seemingly large number. Thanking you in advance, cobberone. ---
**Author:** ![ReuvenB](https://avatars.discourse-cdn.com/v4/letter/r/9fc348/32.png) [@ReuvenB](https://boards.straightdope.com/u/ReuvenB)\ **Post date:** [May 2, 2007, 2:15pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/2 "2007-05-02T14:15:18Z")
If you’ve made a mistake in your calculating and just want to find 2^64, that is, have double the number of grains of sand as the previous square (represented by the pattern 1, 2, 4, 8, 16, 32, etc) , then your answer is 18446744073709551616 grains of sand. If you’re looking for a different pattern than that, you’re going to need to define it further, because I really have no idea what you’re looking for. ---
**Author:** ![Shagnasty](https://avatars.discourse-cdn.com/v4/letter/s/9dc877/32.png) [@Shagnasty](https://boards.straightdope.com/u/Shagnasty)\ **Post date:** [May 2, 2007, 2:25pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/3 "2007-05-02T14:25:28Z")
Answer at the end of this list: 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 16777216 33554432 67108864 134217728 268435456 536870912 1073741824 2147483648 4294967296 8589934592 17179869184 34359738368 68719476736 137438953472 274877906944 549755813888 1099511627776 2199023255552 4398046511104 8796093022208 17592186044416 35184372088832 70368744177664 140737488355328 281474976710656 562949953421312 1125899906842620 2251799813685250 4503599627370500 9007199254740990 18014398509482000 36028797018964000 72057594037927900 144115188075856000 288230376151712000 576460752303423000 1152921504606850000 2305843009213690000 4611686018427390000 9223372036854780000 18446744073709600000 ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [May 2, 2007, 2:26pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/4 "2007-05-02T14:26:24Z")
**cobberone** The answer would be 2x10[sup]63[/sup] or 9.22 x 10[sup]18[/sup] grains os sand. The total grains of sand on the board would be twice this number minus 1. Total = 1.844 x 10[sup]19[/sup] (and don’t for get to subtract 1 grain.) 🙂 And welcome to the boards. ---
**Author:** ![cobberone](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@cobberone](https://boards.straightdope.com/u/cobberone)\ **Post date:** [May 2, 2007, 2:28pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/5 "2007-05-02T14:28:31Z")
Hi XWalrus2 Thanks for the reply. Sorry for not being as clear as I could have. What I meant to ask is if I started with 1 grain of sand, ‘powered’ (is that the right term?) it to 2, then 4 (2x2), 16 (4x4), 256 (16x16), 65,536 (256x256), 4,294,967,296 (65,536x65,536), and so-on. Hope that’s clearer. cobberone. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [May 2, 2007, 2:31pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/6 "2007-05-02T14:31:28Z")
I believe the OP asked secifically for the number of grains on the 64th square which would be 9.22 x 10[sup]18[/sup] ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [May 2, 2007, 2:32pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/7 "2007-05-02T14:32:41Z")
[QUOTE=cobberone] If I was to place 1 grain of sand on the first square of a Chess-Board, 2 on the second, 4 on the third, 256 on the fourth, 65,536 on the fifth, how many grains of sand would there be on the last (64th) square? [/QUOTE] We have: Square 1 - 2[sup]0[/sup] Square 2 - 2[sup]1[/sup] Square 3 - 2[sup]2[/sup] Square 4 - 2[sup]8[/sup] Square 5 - 2[sup]16[/sup] We thus need to figure out terms 6 through 64 in the sequence that begins 0, 1, 2, 8, 16. We probably need a bit more info. ---
**Author:** ![What\_Exit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/what_exit/32/10652_2.png) [@What\_Exit](https://boards.straightdope.com/u/What_Exit)\ **Post date:** [May 2, 2007, 2:35pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/8 "2007-05-02T14:35:35Z")
[QUOTE=cobberone] If I was to place 1 grain of sand on the first square of a Chess-Board, 2 on the second, 4 on the third, 256 on the fourth, 65,536 on the fifth, how many grains of sand would there be on the last (64th) square? [/QUOTE] Your pattern is a little odd, but you could use Excel to do it once your explain the pattern. Why is your pattern 1 2 4 256 65536? You are not doubling or squaring each square. What is the pattern you want to achieve? You took one and doubled it. Then you either took 2 to the second or 2x2. Then you took 4[sup]4[/sup]. Then you simply squared 256 to get 65536. Your pattern does not make a lot of sense and the previous answers appear wrong for the information provided by you. Welcome to the SDMB. Enjoy the stay. Jim (I see you update the question, give me a few minutes) ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [May 2, 2007, 2:39pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/9 "2007-05-02T14:39:07Z")
The OP doesn’t appear to be using a progression of powers of two, but rather, the next number is the square of the previous one (except in the first instance, or we’d never get past 1) ---
**Author:** ![CJJ](https://avatars.discourse-cdn.com/v4/letter/c/ecc23a/32.png) [@CJJ](https://boards.straightdope.com/u/CJJ)\ **Post date:** [May 2, 2007, 2:40pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/10 "2007-05-02T14:40:47Z")
[QUOTE=cobberone] What I meant to ask is if I started with 1 grain of sand, ‘powered’ (is that the right term?) it to 2, then 4 (2x2), 16 (4x4), 256 (16x16), 65,536 (256x256), 4,294,967,296 (65,536x65,536), and so-on. Hope that’s clearer. [/QUOTE] This is squaring the total of square n to get the total on square n+1, and since there are 2 grains on square n=2, the formula for any square n is 2^(2^(n-1)). Thus the 64th square has 2^(2^63) = 2^9223372036854775808. The number is too large for my calculator to compute, but I can estimate it; since 2^10 = 1,024 ~= 1,000, the number is equal to 1024^9007199254740992, so it’s in the neighborhood of 1 followed by 10^16 = 10 quadrillion zeros. ---
**Author:** ![What\_Exit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/what_exit/32/10652_2.png) [@What\_Exit](https://boards.straightdope.com/u/What_Exit)\ **Post date:** [May 2, 2007, 2:43pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/11 "2007-05-02T14:43:12Z")
[QUOTE=cobberone] Hi XWalrus2 Thanks for the reply. Sorry for not being as clear as I could have. What I meant to ask is if I started with 1 grain of sand, ‘powered’ (is that the right term?) it to 2, then 4 (2x2), 16 (4x4), 256 (16x16), 65,536 (256x256), 4,294,967,296 (65,536x65,536), and so-on. Hope that’s clearer. cobberone. [/QUOTE] This progression quickly exceeds any tool I have. Good luck with it. It exceed 10[sup] 1,387,108,685,230,110,000 [/sup] ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [May 2, 2007, 2:43pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/12 "2007-05-02T14:43:19Z")
From the OP’s second post, he missed one out. The sequence should be 1, 2, 4, 16, 256, 65536… i.e. 2^0, 2^1, 2^2, 2^4, 2^8, 2^16… The exponents themselves double at each step, so for square _n_, the exponent is 2^(_n_-2) (after the first square, where the exponent is zero). So on square 64, the answer is 2^(2^62), which is 2^(4.61168602 x 10^18). That number is pretty huge. ---
**Author:** ![What\_Exit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/what_exit/32/10652_2.png) [@What\_Exit](https://boards.straightdope.com/u/What_Exit)\ **Post date:** [May 2, 2007, 2:43pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/13 "2007-05-02T14:43:26Z")
Hamsters don’t like my post. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [May 2, 2007, 2:47pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/14 "2007-05-02T14:47:16Z")
It looks like starting with square 2, there are 2^2^(n - 2) grains on the nth square. Based on that, there should be 2^2^62 grains on the 64th square, which has about 1.4 \* 10[sup]19[/sup] digits. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [May 2, 2007, 2:50pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/15 "2007-05-02T14:50:14Z")
[QUOTE=CJJ\*] This is squaring the total of square n to get the total on square n+1, and since there are 2 grains on square n=2, the formula for any square n is 2^(2^(n-1)). Thus the 64th square has 2^(2^63) = 2^9223372036854775808. . [/QUOTE] Not quite. If you start with 1 grain on square number 1, then the formula is 2[sup]2[sup]n-2[/sup][/sup]: ```auto Square No of grains 1 1 = 2^0 2 2 = 2^1 = 2^(2^0) 3 4 = 2^2 = 2^(2^1) 4 16 = 2^4 = 2^(2^2) 5 256 = 2^8 = 2^(2^3) 6 65536 = 2^16 = 2^(2^4) ``` So the answer, as I said, is 2[sup]2[sup]62[/sup][/sup], or 2[sup]4.61x10[sup]18[/sup][/sup]. Edit: as **ultrafilter** says. 😛 ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [May 2, 2007, 3:00pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/16 "2007-05-02T15:00:04Z")
[QUOTE=cobberone] What I meant to ask is if I started with 1 grain of sand, ‘powered’ (is that the right term?) it to 2, then 4 (2x2), 16 (4x4), 256 (16x16), 65,536 (256x256), 4,294,967,296 (65,536x65,536), and so-on. Hope that’s clearer. [/QUOTE] It’s definitely clearer. You’re saying that the number of grains on square n is the square on the number on n - 1. Unfortunately, when you start this at 1, it never grows - square 2 has 1[sup]2[/sup], which is again 1. The simple way to escape this difficulty is to express the sequence as Colophon has done. The number of grains on the final square would then be far greater than the number of subatomic particles in the known universe. ---
**Author:** ![CJJ](https://avatars.discourse-cdn.com/v4/letter/c/ecc23a/32.png) [@CJJ](https://boards.straightdope.com/u/CJJ)\ **Post date:** [May 2, 2007, 3:05pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/17 "2007-05-02T15:05:17Z")
[QUOTE=Colophon] Not quite. If you start with 1 grain on square number 1, then the formula is 2[sup]2[sup]n-2[/sup][/sup]: [/QUOTE] Ack; should have been more careful; it’s always those stupid mistakes that catch me…thx **Colophon** 🙂 ---
**Author:** ![Nature\_s\_Call](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/nature_s_call/32/19006_2.png) [@Nature\_s\_Call](https://boards.straightdope.com/u/Nature_s_Call)\ **Post date:** [May 2, 2007, 3:13pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/18 "2007-05-02T15:13:32Z")
Here’s the real answer: zero! The chessboard would have collapsed under the weight of all that sand long before square 64. ---
**Author:** ![Sapo](https://avatars.discourse-cdn.com/v4/letter/s/ec9cab/32.png) [@Sapo](https://boards.straightdope.com/u/Sapo)\ **Post date:** [May 2, 2007, 10:18pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/19 "2007-05-02T22:18:47Z")
Can we have both answers (1 2 4 8 and 1 2 4 16) in tonnes, or cubic meters or some other unit that is easier to imagine than gazillions of grains? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 2, 2007, 10:36pm UTC](https://boards.straightdope.com/t/number-of-grains-of-sand/402457/20 "2007-05-02T22:36:14Z")
> [@](#): > > Not quite. If you start with 1 grain on square number 1, then the formula is 2[sup]2[sup]n-2[/sup][/sup]: That still doesn’t match for square 1: That formula would give sqrt(2) grains on the first square. > [@](#): > > Can we have both answers (1 2 4 8 and 1 2 4 16) in tonnes, or cubic meters or some other unit that is easier to imagine than gazillions of grains? You can have it in tons if you’d like, but it won’t make things any easier. These numbers are easily huge enough that, the way we’re writing them, even if you used neutrinos instead of grains of sand and used the mass of the Universe for our mass units, we still wouldn’t be able to write the answer any differently. [Next page](https://boards.straightdope.com/t/number-of-grains-of-sand/402457.md?page=2)