The Rice Problem
A long time ago (so it is written), a wise man in India saved the life of a great
rajah. Out of gratitude, the rajah offered to give the wise man anything his heart
desired. “Only this,” the wise man said. “On a chessboard, place one grain of rice
on the first square, two grains on the second square, four grains on the third, eight
grains on the fourth, and so on to the end.” Greatly pleased by this humble
request, the rajah agreed.
Here is the problem we are solving today: If all of the promised rice was delivered to Michigan, how
deeply would it cover the state?
Warmup
Number the squares of the chessboard starting with zero: 0, 1, 2, 3 … What is the number of the last
square? Let S be the number of grains of rice on the square n, and let T be the total rice on the chessboard from squares 0 to n. Complete the table below up to n = 5.
n
0
1
2
3
4
5
n
S
1
2
T
1
3
S(n)
T(n)
Come up with a formula for S in terms of n. That is, given a value for n, how to you compute S? Now
give a formula for (a) T in terms of S, and (b) T in terms of n.
Problem A: Find the total number of grains of rice.
1. Method 1 – use your calculator. Use n = the number of the last square on the chessboard, and
enter the formula we derived in the warmup for T. Write your answer in scientific notation.
2. Method 2 – assume your calculator is broken. You will use the fact that
210 = 1024 ≈ 1000 = 103 (≈ means approximately equal to). You will also use the rules for exponents. Complete the missing entries in the table.
=
220
221
222
223
224
230
231
264
=
220
(210)2
1x
2 x 220
22 x 220
1x
2 x (210)2
4 x (210)2
1 x 230
1 x (210)3
≈
1 x (103)2
2 x (103)2
4 x (103)2
=
1 x 106
2 x 106
4 x 106
How well does this agree with the first method?
11-TheRiceProblem.doc
© Jan Gombert, 2006
Problem B: Find number of grains of rice that fit in one cubic inch.
Determine the dimensions of one grain of rice.
3. Find the length. Carefully line up 10 grains of rice along one edge of your ruler. I suggest that
you put the ruler on a piece of paper and carefully tap the rice into place with an eraser. Measure the length to the nearest 1/16”. Using your calculator, divide the total length by 10 to get
the length of 1 grain. Round off your answer to the nearest .001 inch.
4. Find the width = diameter. Now lay the 10 grains side-by-side, and repeat what you did in a.
5. When you have the dimensions, write the values for your group on the board and compare to
what the other groups got.
Find the number of grains that will fit in one cubic inch.
6. If the length of a grain of rice is ¼”, say, how many grains will fit in one inch? In general, if you
have the length of a grain of rice (which you do), how do you find the number that will fit in
one inch? Compute this number, rounding the answer to 3 decimal places.
7. Find the number of grains that will fit in one inch if you lay them sideways (the smaller dimension).
8. Think of the grains of rice as tiny bricks. You are trying to stack as many of these bricks inside a
small room that is one inch by one inch by one inch.
9. When you have your answer, write it on the board.
Problem C: Find the area of Michigan, in square inches.
10. Using the map, estimate the
area of Michigan in square
miles.
Hint: Draw a rectangle that
you think roughly
corresponds to the area of
Michigan (including the
Upper Peninsula). Then use
the map scale to determine
the area of the rectangle.
11. Convert the area from square miles to square inches.
Facts you will need to know: one mile = 5,280 feet, one foot = 12 inches.
Problem D: If we cover Michigan with the rice, how deep will it be?
12. From problems A and B, determine the total number of cubic inches of rice there will be. Divide
by the area of Michigan to get the depth of rice, in inches.
11-TheRiceProblem.doc
© Jan Gombert, 2006