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Bronya from Tattingstone and Alistair from Histon and Impington Infant School both drew a table to show the number of grains on each square (as suggested in the clue) to answer this problem. Here is Alistair's table:

Square Grains
1 1
2 2
3 4
4 8
5
16
6 32
7 64
8 128

Bronya carried on her table up to square 15. She explained that she saw the number of grains of rice were doubling each time and that these numbers were also powers of two. She wrote these powers down the side of the table like this:

Square Grains Powers of Two
1 1
2 2
3 4 $2^2$
4 8 $2^3$
5
16
$2^4$
6 32 $2^5$
7 64 $2^6$
8 128 $2^7$

Bronya then went on to say:

I noticed that the power of 2 was one less than the number of the square. So the number of grains of rice on the nth square would be $2^{n-1}$

Alistair then drew another table to show the number of grains altogether:

Square Total Grains
1 1
2 3
3 7
4 15
5 31
6 63
7 127

He then continued the table using approximate answers:

Square Total Grains
$7$ $12 \times10$
$8$ $25 \times10$
$9$ $51 \times10$
$10$ $10 \times 10^2$
$11$ $20 \times10^2$
$12$ $40 \times 10^2$
$13$ $80 \times 10^2$
$14$ $16 \times 10^3$
$15$ $32 \times 10^3$
$16$ $64 \times 10^3$
$17$ $12 \times 10^4$

Then Alistair said he noticed that square number 17 was 1000 times square number 7. So he took square 14 (which is equal $16 \times 10^3$) and multiplied it by 1000 to get an estimate for square 24, and so on... He continued the table in this way and arrived at an answer of $16 \times10^{18}$ by the 64th square. In other words approximately 16 000 000 000 000 000 000 grains of rice! There would actually be 18,446,744,073,709,551,616 on the last square.

Both Alistair and Bronya noticed that the total number of grains up to and including a particular square was one less than the number of grains on the next square. They both wrote this down as a general rule using $n$:
$$(2^{n-1} \times2) -1$$ or more simply
$$2^n-1$$

As Bronya indicates, this would mean that the exact number of grains of rice on the entire board would be:

$$(2^{63} \times 2)-1$$ or
$$2^{64}-1$$

Fantastic work! Thank you to Bronya and Alistair for such logically reasoned solutions.