Calculator And Computer Technology User Service
with Hartley Hyde
[email protected]
An Old Tale
If we consider the number of grains on each
square and the accumulated total we can
construct the following Excel 5 spreadsheet.
I asked my class to consider the following
story, based on an extract from Kasner and
Newman's Mathematics and the Imagination,
which describes adding wheat grains to a
chess board. I chose to use rice because my
students were more likely to find rice in their
home pantry.
The Grand Vizier Sissa Ben Dahir
was granted a boon for hav ing
invented chess for the Indian King,
Shirhâm. Since this game is played
on a board with 64 squares, Sissa
addressed the king: “Majesty, give
me a grain of rice to place on the
f irst square, and two grains of rice
to place on the second square, and
four grains of rice to place of the
third square, and eight grains of rice
to place on the fourth, and so, Oh,
King, let me cover each of the 64
squares of the board.”
The students were asked to find how much
grain Sissa was talking about and to describe
this is more meaningful terms. Karen and Lisa
counted and weighed 100 grains of rice and
then they weighed a metric cup (250 ml) of
rice. From this they found that a metric cup
holds about 11 300 grains. Because the grains
are ellipsoidal, the number varies with how
much the cup is shaken as it is filled.
Assuming that a cup of rice would feed one
person for a day, they hoped to find how many
days the rice would feed all the people of India.
They found that there would be enough rice to
sustain the present world population for over
seven hundred years!
Karen and Lisa weigh a cup of rice
Spreadsheet 1
To build this spreadsheet, use the first row
for titles and enter 1 in the three cells of Row 2.
The cells of Row 3 each contain a formula:
A3 = 1 + A2, B3 = 2 * B2 and C3 = C2 + B3.
We then select these three cells and copy them
down to Row 65 as shown above.
Clearly there are 2n-1 grains on the nth
square and 2 n–1 grains have accumulated.
Thus when the task is completed there should
be 264–1 grains on the board. Karen and Lisa
found that the average weight of a grain of rice
is 0.00198 gram. This means that Sissa was
describing 3.653 × 1010 tonne of rice. The
students quickly realised that the chess board
would long since have disappeared and some
rice would have drifted off the board.
Suppose it drifted evenly across the land
masses of the earth. The rice would cover the
surface to a depth of 2.7 metre. Knowing that
rice swells as it absorbs water we did not
trouble to calculate the effects of tipping the
rice into the oceans. This much rice is
certainly more than Shirhâm had to offer.
Several students were disappointed that the
larger numerals of the first spreadsheet were
expressed in scientific notation using the E
format: they wanted to know the exact value of
264–1. One student did the calculation by hand
and probably spent less time than some of the
other students whose efforts are described on
this page.
Using the first spreadsheet, select the cells
of Columns B and C, choose the Format Menu
and selected a number format with zero
decimal places. At first glance it appears that
we have obtained an exact value. However, on
closer inspection (Spreadsheet 2) we find that
the values beyond the fiftieth square have
been rounded to fifteen significant figures. I
guess the good folk at Microsoft think that this
should be enough to calculate the boss’s
income for the foreseeable future.
Spreadsheet 2
And so began the search for a spreadsheet
which allowed the calculation of more
significant figures. We acquired Excel 98 but
despite a number of interesting improvements,
there has been no change of precision. An
Apple dealer recommended WingZ, but
although it claimed 64 digits, fifteen figures
was the best we could conjure from it.
Eventually we tried Microsoft Works 2 on a
Mac Plus and the result is shown on the third
spreadsheet. Works 2 has calculated precisely
up to cell B64 and has made an error when
adding B64 to C63. It then admits defeat when
asked to find C65 which has twenty significant
figures. At any rate, this was close enough to
find that 264–1 = 18 446 744 073 709 551 615
by doing one pencil and paper addition.
The real limitation is determined at a binary
level. The version of Works 2 written for the
eight-bit 8088 machines overflows at 232 and
we have seen that the version written for the
sixteen-bit Mac Plus overflows at 264. Clearly
the code for both versions used four memory
locations to store the value of each cell. The
software then performs multiple-precision
arithmetic on all four locations whenever the
value of a cell changes. Why, I wonder, do later
versions of Works follow the Excel policy of
rounding to only fifteen significant figures?
Spreadsheet 3
This type of thinking led us to explore some
simple multiple precision arithmetic using two
cells to store each value of the spreadsheet.
In Spreadsheet 4, Columns A and B are the
same as those of Spreadsheet 2. We can then
used Columns C and D to perform some very
simple 12-digit multiple-precision doubling.
Down to Row 41, Column D is the same as
Column B, but in Row 42, the thirteenth digit
must overflow into Column C.
To do this we change the following cells:
C42 = 2*C41+((2*D41)>10^12) and
D42 = 2*D41–10^12*((2*D41)>10^12)
The key to understanding these entries is
the boolean expression (2*D41)>10^12. If this
expression were allocated its own cell it would
appear as either ‘True’ or ‘False’ but, when it is
incorporated in a more extensive expression, it
takes the value 1 when ‘True’ and 0 when
‘False’. Thus, if doubling the value in D41
exceeds the 12-digit limit, the value in C42 is
increased by 1 and the value in D42 is
decreased by 10 12.
These cells are then copied down to Row 66
to give us an accurate value of 264. All of the
cells have been given a (# ##0) number format
so that the digits are grouped in threes. This
helps us to see where the leading zeros are
missing in Column D.
And so we were able to find 264 precisely, but
it was Karen and Lisa who helped us
understand how large this number is.
Reference:
Kasner, E. and Newman, J., 1949,
Mathematics and the Imagination,
London: G. Bell and Sons, Ltd.