Mass of Rice on a Chessboard

**Problem created by Tao Steven Zheng (郑涛)**

## **Problem**

There is a famous Persian legend[1] of a man who presented a new game he had invented (apparently chess) to the Shah. The Shah was so delighted that he would grant whatever reward the man wished. The clever man asked for one grain of rice on the first square, two on the second, four on the third, doubling until the last square on the board. What would be the mass of all the rice in kilograms? Give your answer as an *order of magnitude*.

This problem appeared in the 2015 UBC Physics Olympics Fermi problems event.

**Hint:**

64
log
⁡

2

≈
19.26592

{\displaystyle 64 \log{2} \approx 19.26592 }

[1] The earliest telling of this story was first recorded in 1256 AD by the Persian scholar Ibn Khallikan (1211 - 1282 AD). Another version had Sissa, the legendary inventor of Chaturanga (Indian chess), requesting the king to give him wheat in a similar fashion.

## **Solution**

**Rough Estimate**

Using geometric series, the number of grains of rice is approximately

2

64

{\displaystyle {2}^{64} }

.

1
2
4
.
.
.

2

63

=

2

64

−
1

{\displaystyle 1 + 2 + 4 + ... + {2}^{63} = {2}^{64} - 1 }

To estimate the number

2

64

{\displaystyle {2}^{64} }

 as a power of 10, use the hint

64
log
⁡

2

≈
19.26592

{\displaystyle 64 \log{2} \approx 19.26592 }

. Subsequently,

2

64

≈

10

19.26592

{\displaystyle {2}^{64} \approx {10}^{19.26592} }

or more roughly

2

64

≈

10

19.25

≈

10

19

×

10

0.25

{\displaystyle \begin{align} {2}^{64} &\approx {10}^{19.25} \\
&\approx {10}^{19} \times {10}^{0.25}
\end{align} }

Since

10

≈
3.16

{\displaystyle \sqrt{10} \approx 3.16 }

, approximate

10

0.25

=

3

{\displaystyle {10}^{0.25} = \sqrt{3} }

.

Since

3

≈
1.73

{\displaystyle \sqrt{3} \approx 1.73 }

, we let

10

0.26592

≈
2

{\displaystyle {10}^{0.26592} \approx 2 }

 (the true value is closer to 1.8447).

Thus, the number of grains of rice is roughly

2
×

10

19

{\displaystyle 2 \times {10}^{19} }

.

Now we need to approximate the mass of 1 grain of rice. Estimate that there is roughly 50 grains of rice per cubic centimeter, and that 1 cubic centimeter of rice weighs about 1 gram. Therefore, each gram of rice weighs 0.02 grams.

Thus, the mass of all the rice is

(
2
×

10

19

)
(
0.02

g
# )
4
×

10

17

g
.

{\displaystyle (2 \times {10}^{19})(0.02 \; g) = 4 \times {10}^{17} \; g. }

Converting to kilograms (1 kg = 1000 g) gives

(
4
×

10

17

g
)

(

1

k
g

1000

g

)

=
4
×

10

14

k
g
.

{\displaystyle (4 \times {10}^{17} \; g) \left( \frac{1 \; kg}{1000 \; g} \right) = 4 \times {10}^{14} \; kg. }

**More Accurate Estimate**

Number of rice:

1
2
4
.
.
.

2

63

=

2

64

−
1
≈
1.8447
×

10

19

{\displaystyle 1 + 2 + 4 + ... + {2}^{63} = {2}^{64} - 1 \approx 1.8447 \times {10}^{19} }

Mass of 1 grain of rice: 0.025 grams

Mass of all rice on the chessboard:

(
1.8447
×

10

19

)
(
0.025

g
# )
4.61
×

10

17

g

{\displaystyle (1.8447 \times {10}^{19})(0.025 \; g) = 4.61 \times {10}^{17} \; g }

(
4.61
×

10

17

g
)

(

1

k
g

1000

g

)

=
4.61
×

10

14

k
g

{\displaystyle (4.61 \times {10}^{17} \; g) \left( \frac{1 \; kg}{1000 \; g} \right) = 4.61 \times {10}^{14} \; kg }

The order of magnitude is 14.