Andy,

1. I think that finding 8²¹ is probably a bad start; the best solution would never find this number explicitly. Look at the sums of the first few squares:
   
   1 square 1 grain
   2 squares 3 grains total
   3 squares 7 grains total
   
   [get them to do a few more]
   
   and look for the pattern. Which power of 2 do you want to find the sum through the 64th square?
   
   
2. The number is very approximately 10²⁰ [remember 2¹⁰ is about 10³]. Do you really need all the digits or will scientific notation do? If so, any scientific calculator [or Google] will get you the answer to more decimal places than you need. If you need higher precision use MAPLE or some other specialized software package.
   
   
3. For hand calculation the probability of an error somewhere is extremely high. If you are set on this, I would have each stage of the calculation done in parallel by all (or several) students, taking the majority answer as correct and having everybody use it in the next stage. Repeated multiplication by a one-digit number (8 except 2 at the last stage) is probably optimal. Just be aware that doing this by hand is a stunt, not serious mathematics. It could be fun if approached in the right spirit - maybe with a prize [an inexpensive chess set? rice cakes?] for the last one to make a mistake? But don't have people drop out, you'll need them to check the others.

Good Hunting!
RD

The ``basic calculator'' (bc) on my unix machine does it in milliseconds:

ctardif@mastray2% bc
8^21
9223372036854775808

In my mind I can do 8²¹ = 2⁶³, and I know that 2¹⁰ is 1024, a bit more than 1000, so that

2⁶³ = 2^{3 + 6 × 10} > 8 × 1000⁶ = 8 000 000 000 000 000 000.

Since the actual answer is 9 223 372 036 854 775 808, my estimate of ``eight billions of billions'' is not that bad.

The quickest ``human computation'' is by repeated squaring:
8¹ = 8
8² = 64
8⁴ = 64 × 64 = 4096
8⁸ = 4096 × 4096 = 16777216
8¹⁶ = 16777216 × 16777216 = 281474976710656

So now 8²¹ = 8^{16 + 4 + 1} = 8¹⁶ × 8⁴ × 8¹ = 281474976710656 × 4096 × 8.

Long, but it can be done.
If I were to do it by hand, I'd start with
8 × 281474976710656 = 2251799813685248,
and then do 2251799813685248 × (5000 - 4).

Claude