1,073,741,823 Grains of Rice

In the children’s book “One Grain of Rice: A Mathematical Folktale” a girl uses her knowledge of exponential growth to trick a greedy king into turning over his stockpile of rice. Hidden in the story are mathematical concepts related to doubling: powers of two, geometric sequences, geometric series, and exponents. I will analyze the story from this perspective, and then discuss my experience reading it to first and third grade students.

Synopsis

In the story, a girl named Rani is given an amount of rice each day for thirty days: one grain of rice on day one, two grains of rice on day two, four grains of rice on day three, eight grains of rice on day four, etc. In other words, on each day after the first, she receives double the amount of rice she received on the previous day. Since the doubling starts from one grain of rice, the amounts received correspond to the first 30 nonnegative powers of two.

The story progresses through select days, describing the number of grains received, the capacity and number of containers used to hold them, and the total received to each point. The amounts of each measure of rice are not spelled out for each day, so I filled in the details.

Containers

To manage the description of the number of grains of rice, and to put the growth in perspective, the author (Demi) uses these containers: bowls, bags, baskets, and storehouses:

• 1 bowl = 4,096 grains
• 1 bag = 8 bowls = 32,768 grains
• 1 basket = 32 bags = 256 bowls = 1,048,576 grains
• 1 storehouse = 128 baskets = 4,096 bags = 32,768 bowls = 134,217,728 grains

(I inferred some of these relationships from the story.)

The units of the containers themselves progress through powers of two — not only through nonnegative powers of two, but through negative powers of two as well. Nonnegative powers of two represent whole number units, and negative powers of two represent fractional units. For example, 1,024 grains of rice is 1/4 of a bowl and 2,048 grains is 1/2 a bowl.

A sequence of powers of two is a geometric sequence, also known as a geometric progression; in this case, each number in the sequence is double the previous.

I made a table showing the progression of powers of two by day, by measure; negative powers of two amounts are shown in gray (I generated this table, and the ones that follow, with PARI/GP scripts I wrote):

Amount of Rice Received By Day
Day Grains Bowls Bags Baskets Storehouses
1 1 1/4096 1/32768 1/1048576 1/134217728
2 2 1/2048 1/16384 1/524288 1/67108864
3 4 1/1024 1/8192 1/262144 1/33554432
4 8 1/512 1/4096 1/131072 1/16777216
5 16 1/256 1/2048 1/65536 1/8388608
6 32 1/128 1/1024 1/32768 1/4194304
7 64 1/64 1/512 1/16384 1/2097152
8 128 1/32 1/256 1/8192 1/1048576
9 256 1/16 1/128 1/4096 1/524288
10 512 1/8 1/64 1/2048 1/262144
11 1024 1/4 1/32 1/1024 1/131072
12 2048 1/2 1/16 1/512 1/65536
13 4096 1 1/8 1/256 1/32768
14 8192 2 1/4 1/128 1/16384
15 16384 4 1/2 1/64 1/8192
16 32768 8 1 1/32 1/4096
17 65536 16 2 1/16 1/2048
18 131072 32 4 1/8 1/1024
19 262144 64 8 1/4 1/512
20 524288 128 16 1/2 1/256
21 1048576 256 32 1 1/128
22 2097152 512 64 2 1/64
23 4194304 1024 128 4 1/32
24 8388608 2048 256 8 1/16
25 16777216 4096 512 16 1/8
26 33554432 8192 1024 32 1/4
27 67108864 16384 2048 64 1/2
28 134217728 32768 4096 128 1
29 268435456 65536 8192 256 2
30 536870912 131072 16384 512 4

(The thirteen days mentioned explicitly in the story are shown in bold).

Here’s the same table, but using exponents to make the powers of two progressions more obvious:

Amount of Rice Received By Day (Using Exponents)
Day Grains Bowls Bags Baskets Storehouses
1 20 2-12 2-15 2-20 2-27
2 21 2-11 2-14 2-19 2-26
3 22 2-10 2-13 2-18 2-25
4 23 2-9 2-12 2-17 2-24
5 24 2-8 2-11 2-16 2-23
6 25 2-7 2-10 2-15 2-22
7 26 2-6 2-9 2-14 2-21
8 27 2-5 2-8 2-13 2-20
9 28 2-4 2-7 2-12 2-19
10 29 2-3 2-6 2-11 2-18
11 210 2-2 2-5 2-10 2-17
12 211 2-1 2-4 2-9 2-16
13 212 20 2-3 2-8 2-15
14 213 21 2-2 2-7 2-14
15 214 22 2-1 2-6 2-13
16 215 23 20 2-5 2-12
17 216 24 21 2-4 2-11
18 217 25 22 2-3 2-10
19 218 26 23 2-2 2-9
20 219 27 24 2-1 2-8
21 220 28 25 20 2-7
22 221 29 26 21 2-6
23 222 210 27 22 2-5
24 223 211 28 23 2-4
25 224 212 29 24 2-3
26 225 213 210 25 2-2
27 226 214 211 26 2-1
28 227 215 212 27 20
29 228 216 213 28 21
30 229 217 214 29 22

Finally, here’s the same information, condensed into simple formulas:

• Number of grains on day d = 2d-1
• Number of bowls on day d = 2d-13
• Number of bags on day d = 2d-16
• Number of baskets on day d = 2d-21
• Number of storehouses on day d = 2d-28

These relationships are simple: each measure doubles each day, so the exponent of each power of two increases by one each day. The offset in the exponent of the container formulas shows where each transitions from negative powers of two to nonnegative powers of two.

I made a table showing the cumulative number of grains of rice by day, by measure. The amounts are not powers of two, although they follow a pattern, which I’ll explain below. The table is broken into two parts (to fit on this page), with fractional amounts less than one shown in gray:

Amount of Rice Accumulated Through Each Day (Part 1: Grains, Bowls, and Bags)
Day Grains Bowls Bags
1 1 1/4096 1/32768
2 3 3/4096 3/32768
3 7 7/4096 7/32768
4 15 15/4096 15/32768
5 31 31/4096 31/32768
6 63 63/4096 63/32768
7 127 127/4096 127/32768
8 255 255/4096 255/32768
9 511 511/4096 511/32768
10 1023 1023/4096 1023/32768
11 2047 2047/4096 2047/32768
12 4095 4095/4096 4095/32768
13 8191 1 4095/4096 8191/32768
14 16383 3 4095/4096 16383/32768
15 32767 7 4095/4096 32767/32768
16 65535 15 4095/4096 1 32767/32768
17 131071 31 4095/4096 3 32767/32768
18 262143 63 4095/4096 7 32767/32768
19 524287 127 4095/4096 15 32767/32768
20 1048575 255 4095/4096 31 32767/32768
21 2097151 511 4095/4096 63 32767/32768
22 4194303 1023 4095/4096 127 32767/32768
23 8388607 2047 4095/4096 255 32767/32768
24 16777215 4095 4095/4096 511 32767/32768
25 33554431 8191 4095/4096 1023 32767/32768
26 67108863 16383 4095/4096 2047 32767/32768
27 134217727 32767 4095/4096 4095 32767/32768
28 268435455 65535 4095/4096 8191 32767/32768
29 536870911 131071 4095/4096 16383 32767/32768
30 1073741823 262143 4095/4096 32767 32767/32768
Amount of Rice Accumulated Through Each Day (Part 2: Grains, Baskets, and Storehouses)
1 1 1/1048576 1/134217728
2 3 3/1048576 3/134217728
3 7 7/1048576 7/134217728
4 15 15/1048576 15/134217728
5 31 31/1048576 31/134217728
6 63 63/1048576 63/134217728
7 127 127/1048576 127/134217728
8 255 255/1048576 255/134217728
9 511 511/1048576 511/134217728
10 1023 1023/1048576 1023/134217728
11 2047 2047/1048576 2047/134217728
12 4095 4095/1048576 4095/134217728
13 8191 8191/1048576 8191/134217728
14 16383 16383/1048576 16383/134217728
15 32767 32767/1048576 32767/134217728
16 65535 65535/1048576 65535/134217728
17 131071 131071/1048576 131071/134217728
18 262143 262143/1048576 262143/134217728
19 524287 524287/1048576 524287/134217728
20 1048575 1048575/1048576 1048575/134217728
21 2097151 1 1048575/1048576 2097151/134217728
22 4194303 3 1048575/1048576 4194303/134217728
23 8388607 7 1048575/1048576 8388607/134217728
24 16777215 15 1048575/1048576 16777215/134217728
25 33554431 31 1048575/1048576 33554431/134217728
26 67108863 63 1048575/1048576 67108863/134217728
27 134217727 127 1048575/1048576 134217727/134217728
28 268435455 255 1048575/1048576 1 134217727/134217728
29 536870911 511 1048575/1048576 3 134217727/134217728
30 1073741823 1023 1048575/1048576 7 134217727/134217728

The pattern is easier to see if you rewrite the values using exponents, and write mixed numbers as improper fractions:

Amount of Rice Accumulated Through Each Day, Using Exponents
Day Grains Bowls Bags Baskets Storehouses
1 21 – 1 (21 – 1)/212 (21 – 1)/215 (21 – 1)/220 (21 – 1)/227
2 22 – 1 (22 – 1)/212 (22 – 1)/215 (22 – 1)/220 (22 – 1)/227
3 23 – 1 (23 – 1)/212 (23 – 1)/215 (23 – 1)/220 (23 – 1)/227
4 24 – 1 (24 – 1)/212 (24 – 1)/215 (24 – 1)/220 (24 – 1)/227
5 25 – 1 (25 – 1)/212 (25 – 1)/215 (25 – 1)/220 (25 – 1)/227
6 26 – 1 (26 – 1)/212 (26 – 1)/215 (26 – 1)/220 (26 – 1)/227
7 27 – 1 (27 – 1)/212 (27 – 1)/215 (27 – 1)/220 (27 – 1)/227
8 28 – 1 (28 – 1)/212 (28 – 1)/215 (28 – 1)/220 (28 – 1)/227
9 29 – 1 (29 – 1)/212 (29 – 1)/215 (29 – 1)/220 (29 – 1)/227
10 210 – 1 (210 – 1)/212 (210 – 1)/215 (210 – 1)/220 (210 – 1)/227
11 211 – 1 (211 – 1)/212 (211 – 1)/215 (211 – 1)/220 (211 – 1)/227
12 212 – 1 (212 – 1)/212 (212 – 1)/215 (212 – 1)/220 (212 – 1)/227
13 213 – 1 (213 – 1)/212 (213 – 1)/215 (213 – 1)/220 (213 – 1)/227
14 214 – 1 (214 – 1)/212 (214 – 1)/215 (214 – 1)/220 (214 – 1)/227
15 215 – 1 (215 – 1)/212 (215 – 1)/215 (215 – 1)/220 (215 – 1)/227
16 216 – 1 (216 – 1)/212 (216 – 1)/215 (216 – 1)/220 (216 – 1)/227
17 217 – 1 (217 – 1)/212 (217 – 1)/215 (217 – 1)/220 (217 – 1)/227
18 218 – 1 (218 – 1)/212 (218 – 1)/215 (218 – 1)/220 (218 – 1)/227
19 219 – 1 (219 – 1)/212 (219 – 1)/215 (219 – 1)/220 (219 – 1)/227
20 220 – 1 (220 – 1)/212 (220 – 1)/215 (220 – 1)/220 (220 – 1)/227
21 221 – 1 (221 – 1)/212 (221 – 1)/215 (221 – 1)/220 (221 – 1)/227
22 222 – 1 (222 – 1)/212 (222 – 1)/215 (222 – 1)/220 (222 – 1)/227
23 223 – 1 (223 – 1)/212 (223 – 1)/215 (223 – 1)/220 (223 – 1)/227
24 224 – 1 (224 – 1)/212 (224 – 1)/215 (224 – 1)/220 (224 – 1)/227
25 225 – 1 (225 – 1)/212 (225 – 1)/215 (225 – 1)/220 (225 – 1)/227
26 226 – 1 (226 – 1)/212 (226 – 1)/215 (226 – 1)/220 (226 – 1)/227
27 227 – 1 (227 – 1)/212 (227 – 1)/215 (227 – 1)/220 (227 – 1)/227
28 228 – 1 (228 – 1)/212 (228 – 1)/215 (228 – 1)/220 (228 – 1)/227
29 229 – 1 (229 – 1)/212 (229 – 1)/215 (229 – 1)/220 (229 – 1)/227
30 230 – 1 (230 – 1)/212 (230 – 1)/215 (230 – 1)/220 (230 – 1)/227

Each value is simply the number of grains of rice accumulated, divided by the size of the corresponding measure in grains:

• Number of grains through day d: 2d – 1
• Number of bowls through day d: (2d – 1)/212
• Number of bags through day d: (2d – 1)/215
• Number of baskets through day d: (2d – 1)/220
• Number of storehouses through day d: (2d – 1)/227

The sum of grains is a geometric series, being the sum of numbers from a geometric sequence — in this case, the first d nonnegative powers of two. Conveniently, we don’t need to add them to know their total; this well-known formula gives their sum directly:

(Numbers of the form 2n – 1 are called Mersenne numbers.)

For example, on day 15, 215 – 1 = 32767 grains of rice have accumulated, an amount that would fill (215 – 1)/212 = 32767/4096 = 7 4095/4096 bowls.

The Formulas for Representation as Mixed Numbers

The simple formulas above, expressed as improper fractions, hide the pattern that lies in the mixed number representation of the values of the containers.

Each value is a sum of consecutive powers of two. Some values are sums of negative powers of two, and some are sums of both negative and nonnegative powers of two. Values for measures that start out accumulating in fractional units (each measure except grains) are sums of negative powers of two. At some point, when whole number units start to add in, sums of nonnegative powers of two become part of the total. This is when the value becomes an improper fraction, or mixed number.

These formulas take into account the point at which the values become improper fractions, but expressing them as mixed numbers:

• Number of bowls through day d:

If day ≤ 12: (2d – 1)/212

If day > 12: (2d-12 – 1) + (212 – 1)/212

• Number of bags through day d:

If day ≤ 15: (2d – 1)/215

If day > 15: (2d-15 – 1) + (215 – 1)/215

• Number of baskets through day d:

If day ≤ 20: (2d – 1)/220

If day > 20: (2d-20 – 1) + (220 – 1)/220

• Number of storehouses through day d:

If day ≤ 27: (2d – 1)/227

If day > 27: (2d-27 – 1) + (227 – 1)/227

Fractional values accumulate as a Mersenne number over a power of two; once whole number values begin to add in, the accumulated fractional part remains unchanged while the whole number part accumulates as a Mersenne number.

It’s easy to show that the mixed number formulas are equivalent to the improper fraction formulas. For example, take the mixed number formula for bowls after day 12:

You can rewrite this, using simple algebra and the laws of exponents, as follows:

This is the same formula as for before day 12, and thus the same formula for all days.

In the Classroom

This is a good story upon which to base a math lesson, and you can vary the mathematical sophistication according to grade level. I read it to first and third graders, so I had to keep it simple. My goal was to show how numbers grow through doubling, and to introduce place value of large numbers.

When I read the story to a first grade class, I got lots of “oohs” and “aahs” whenever I read a big number from the text and whenever they saw an illustration of the many animals needed to transport the rice. At the end of the story, I wrote 1,073,741,823 on the board to show them how it looks in numerals. Although I knew they’d only been taught up to the hundreds place, I went through each place and named it. I finished by saying “when you get to third grade I can explain the math a little more.”

For the third grade class, I read the story and spent about fifteen minutes afterwards discussing the numbers. First I wrote 1,073,741,823 on the board and went through the places. Even though they had been taught only up to the thousands place, most of them knew to the billions place (and beyond). Next, I asked them to say the number in words, and many had difficulty — particularly the “seven-hundred forty one thousand” part (I don’t know why it was harder than “seventy-three million” — I would have thought the zero in the hundred millions place would be confusing).

Next, I wrote the number 1 and had them tell me the next numbers in the sequence. A couple of kids answered correctly up to 8,192 (214). When I asked one boy if he had them memorized, he said “No, I can add them in my head.”

I continued the sequence using a pocket calculator. I had the students take turns computing the next number with it (they found it very hard to recite the numbers back to me, because there were no commas). By design, one unlucky kid had 227 (134,217,728) for his turn, which caused the calculator to overflow. I told the class that the numbers get so big that you need a computer to figure them out.

At the end, I referred to these numbers as “powers of two,” and told them they would be useful someday if they wanted to work with computers (can you blame me for plugging my profession?).

One boy asked a good question: “what if you got 20 grains every minute for 30 days — would that be more?” I said “Great question: I think you’d still get more by doubling though.” The calculator showed him this to be true — his method accumulated only 864,000 grains.

Lesson Plan Resources

If you search the Internet for “One Grain of Rice” you’ll find other ideas on how to incorporate this story into a math lesson. For example, this worksheet uses graphs and logarithms.

Errors in the Story

I found two errors and one inconsistency in the story:

Before the Rice Is Delivered

Rani describes her plan for receiving the rice as follows:

“Today, you will give me a single grain of rice. Then, each day for thirty days you will give me double the rice you gave me the day before. Thus, tomorrow you will give me two grains of rice, the next day four grains of rice, and so on for thirty days.”

The doubling occurs for twenty-nine days, not thirty days.

On Day 16

On day 16 it says

“On the sixteenth day, Rani was presented with a bag containing thirty-two thousand, seven hundred and sixty-eight grains of rice. All together she had enough rice for two full bags”

Through day 16 she had 65,535 grains of rice, one grain less than two full bags; she actually had 1 32767/32768 = 1.999969482421875 bags.

On Day 9

On day 9 it says

“On the ninth day, Rani was presented with two hundred and fifty-six grains of rice. She had received in all five hundred and eleven grains of rice, only enough for a small handful.”

Technically, there’s nothing wrong with this, but the measure “small handful” — 511 grains — is not a power of two. It doesn’t fit neatly into the scheme of other units (and hence is why I didn’t include it in the tables).