## Answer to the chess game grain counting quiz:

Michael Bullard correctly replied that the total count of grains for the
chessboard was 18,446,744,073,709,551,615 . This is one less than 2 raised
to the 64th power [= (2^64)-1].

Note the first square has 2^0=1 grain, second square 2^1=2
grains, third 2^2=4 grains... and the last, or 64th square has 2^63 =
9,223,372,036,854,775,808 grains. Also note he sum of grains on all the
squares previous to any square is one grain less than that square itself
has.

The number written in Bill Gates' book was 18,446,744,073,709,551,600.
The reason for the discrepancy is that Mr. Gates' copy of Microsoft Excel®
rounded off the number. In fact, the regular 64 bit numbers that a
computer's microprocessor works with can't hold that large a number.
Spreadsheets and other programs that compute larger numbers use special
techniques such as binary coded decimal numbers.

Michael found by using the MOD function (remainder after division) he
could recover the missing digits within his spreadsheet, so spreadsheets
clearly deal with numbers greater than 64 binary digits long, internally.