Student
Explorations
in Mathematics
Teacher Notes Version
November 2009
This icon indicates a link to the Teacher Notes page:
A Powerful
✍
Little Number
Formerly Student Math Notes
According to an Indian myth, Grand Vizier Sissa Ben Dahir invented the
game of chess and gave the game to King Shirham of India. The king
offered a reward of gold for the game, but the grand vizier stated that
he would prefer to instead have some grains of wheat—one grain for
the first square of the chessboard, two grains for the second square,
four grains for the third, and so on—doubling the amount each time. The
king granted this seemingly modest request. This myth became known
as the Wheat and Chessboard problem.
The king was surprised when he determined how many grains of wheat
are required to cover all sixty-four squares of the chessboard in this manner!
How Many Grains of Wheat?
1. Predict how many grains of wheat you think will be needed to cover all sixty-four squares of the chessboard.
2. Fill in table 1 at the bottom of the page with the number of
grains of wheat found on each of the first eight squares of
the chessboard.
6. What do you think about using Pat’s method to find the
number of grains of wheat to be placed on square 64?
7. Describe a shortcut you could use to find the number of
grains for square 64.
✍
1 (This
3. What pattern(s) do you notice in the table?
icon indicates a link to the Teacher Notes page)
4. Pat wrote 2 • 2 • 2 • 2 • 2 • 2 to find the number of grains of
wheat for square 7. Does this method work? Why, or why
not?
This form of repeated multiplication can be represented
using exponents. Exponents (sometimes referred to as
powers) are a shorthand method of writing repeated
multiplication. The expression 26 (called an exponential
expression) means that two is used as a factor six times
(2 • 2 • 2 • 2 • 2 • 2) and is read as “two times two times two
times two times two times two.” Instead of saying all that,
we could say, “two to the power of six” or “two to the sixth
power” or “two to the sixth.” In the expression 26, 2 is the
base, and 6 is the exponent.
exponent
2 • 2 • 2 • 2 • 2 • 2 ⇔ 26base
5. Use Pat’s method to find the number of grains of wheat for
square 9 and for square 10.
Table 1
Place on the Chessboard
1
2
3
4
5
6
7
8
Number of Grains
Show how you
determined your answer
Copyright © 2009 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use only.
A Powerful Little Number—continued
8. F
or which square of the chessboard can the number
of grains of wheat be represented by the exponential
2
expression 2 • 2 • 2 • 2 • 2 • 2 or 26?
✍
13. Beginning with square 3, complete table 2 by
rewriting the value for the number of grains of wheat
as exponential expressions. (We will do the first two
squares soon).
Special Patterns and Powers of 2
9. W
hat exponential expression can be used to represent
the number of grains of wheat on squares 9 and 10?
Why does this work? What relationship do you notice
between the exponent and the place (the square) on
the chessboard?
10. W
hat exponential expression could you write to represent the number of grains of wheat on square 64? Why
does it work?
14. Now use any patterns you see in the table to fill in the
exponential expression for square 2 and square 1.
Let’s talk about 21… If the exponent of an exponential
expression is 1, then the expression equals the base number. In the expression 21, since 2 is the base and 1 is the
exponent, the expression equals 2. Another way of thinking about 21 is that we are using the base 2 as a factor
1 time. So, for square 2, you wrote 21 as the exponential
expression to represent the two grains of wheat.
15. (a) What is the value of 51?
(b) What is the value of 2341?
(c) What might ☺1 represent?
11. W
hat exponential expression could you write to represent the number of grains of wheat on the following?
(d) What is y1?
(a) square 30
(e) What is (x+1)1?
(b) square 47
(c) square n (where n is any number less than or equal
to 64)
Are you doing it
again ... algebra?
Cool! You are
doing algebra!!
In a general sense, a1 = a, where a represents any
number.
Now let’s talk about 2 0… Any number to the 0 power will
always equal 1. But why is any number to the 0 power
equal to 1? Here, think about how dividing an exponential
expression by our base (factor) is the same as decreasing
the power by 1.
12. W
hat is it about the problem that makes 2 the base?
Table 2
Place on the Chessboard
1
2
3
4
5
6
7
8
Number of Grains
Exponential
Expression
2
Student Explorations in Mathematics, November 2009
A Powerful Little Number—continued
Let’s look at a sequence of numbers beginning with 25 to
3
see why this is true:
number of grains of wheat that would be
✍
25 =
(a) on all sixty-four squares.
(b) on a total of n squares.
2 • 2 • 2 • 2 • 2 = 32
24 = 2 • 2 • 2 • 2 = 32 = 16
2
16
23 = 2 • 2 • 2 =
=8
2
8
22 = 2 • 2 =
=4
2
4
21 = 2 =
=2
2
2
20 =
=1
2
Some students incorrectly think that 2 0 = 0. But if we follow the pattern established in the table and the pattern
above, we can see that 2 0 = 1.
Think of the deal the grand vizier made with the king and
the total amount of grain he requested. According to the
Kansas Wheat Commission, roughly one million grains of
wheat are in a bushel; one bushel of wheat yields enough
flour for seventy-three one-pound loaves of white bread.
The total U. S. wheat production is 2.4 billion bushels per
year. The total number of grains of wheat from the problem
is 18,446,744,073,709,551,615. Using this information, it
would take U.S. farmers about 7686 years to grow enough
wheat.
Deal or No Deal?
20. Is 7686 years a reasonable request from the king?
Explain your thoughts.
16. L
ook back at your table in question 2. What is the total
number of grains of wheat the grand vizier received
after eight squares are filled?
(Hint: square 1 + square 2 + square 3 + … )
21. Suppose the grand vizier had invented tic-tac-toe
instead of chess. Should he have made his request to
the king to take his reward in wheat in a similar manner? Why, or why not?
17. How
many grains of wheat will be placed on the ninth
square? How does this relate to the number of grains
4
of wheat on just the eighth square?
✍
22. Suppose the number of grains of wheat were to triple
each time rather than double. Fill in table 3 with the
number of grains in row 2 and the exponential notation
in row 3. In row 4, record the total number of grains of
wheat to that point.
18. W
hat is the relationship between the total number of
grains of wheat on the first eight squares and the number of grains on only the ninth square? Express your
answer using exponents.
23. Write an exponential expression for the number of
grains of wheat on square 64 when you triple each
time. How many grains of wheat would be on square n
if you triple each time?
19. In question 10, you wrote an exponential expression
for the number of grains of wheat on the sixty-fourth
square. Write an exponential expression for the total
Table 3
Square Number
1
2
3
4
5
6
7
8
Number of Grains
Exponential Notation
Total Grains
Student Explorations in Mathematics, November 2009
3
A Powerful Little Number—continued
The Tower of Hanoi
The Tower of Hanoi is another classic mathematical problem. The Tower of Hanoi consists of three towers arranged
so that they are collinear. A player positions the towers in
front of himself or herself and places disks of increasing
diameter on the leftmost tower, the disk with the largest
diameter on the bottom. The objective of the game is to
move the disks to the rightmost tower in as few moves as
possible. The middle tower is used as a temporary holding area.
24. First, try the Tower of Hanoi with just a dime. What is
the fewest number of moves needed? In the chart, tally
5
the number of moves.
✍
Dime
• Only one disk at a time may be moved.
Next try the Tower of Hanoi using a dime and a penny. Be
sure that a larger (in size) coin is never on top of a smaller
coin (meaning that the penny cannot go on top of the
dime). Use the chart to tally the number of moves of each
coin. When you think you have moved the coins using
the least number of moves, record your total in the chart
below.
• A larger disk can never be placed on top of a
smaller disk.
25. What is the least number of moves needed for each
6
coin?
Rules of the game
✍
Penny
• You do not need to use the holding area for every
move.
One version of the Tower of Hanoi uses four coins—a
dime, a penny, a nickel, and a quarter—stacked in that
order on an index card, plus two other empty index cards
beside the tower as shown.
dime
penny
nickel
quarter
_____________
_____________
Play the game using a dime, a penny, a nickel, and a
quarter or by visiting the NCTM Illuminations Web site at
http://illuminations.nctm.org/ActivityDetail.aspx?id=40.
Total Moves
26. Now try the Tower of Hanoi puzzle with a dime, a
7
penny, and a nickel.
✍
Nickel
_____________
Dime
Penny
Dime
Total Moves
27. Finally try the Tower of Hanoi puzzle using all four
coins. Conduct three trials using the chart below. What
is the fewest number of moves you used to solve the
puzzle?
Quarter
Nickel
Penny
Dime
Total Moves
1
Directions for using the Tower of Hanoi applet
• The object of the game is to move all of the disks
to the peg on the right. Click-and-drag to move a
disc. Only one disc may be moved at a time. A disc
can be placed either on an empty peg or on top of
a larger disc.
2
3
• Click the Timer checkbox to turn the timer on or off.
• Vary the difficulty by changing the number of discs.
The + and – buttons increase and decrease the
number of discs. You will likely have no problem
with three discs, but can you solve the puzzle with
twelve discs?
28. What do you observe if you compare the total number
of moves in the Tower of Hanoi to the table you created
in question 2? What do you observe about the total
number of moves and the total number of grains of
wheat that we found in question 19?
• If you need help, select Solution to watch the computer solve the puzzle. The Speed slide determines
how fast the discs move.
• Restart returns all the disks to the left peg.
4
Student Explorations in Mathematics, November 2009
A Powerful Little Number—continued
29. Predict
the fewest number of moves if you use a fifth
coin, a half dollar. Try this. Was your prediction correct?
30. T
he Tower of Hanoi puzzle can also be done with any
number of disks. What is the fewest number of moves
needed to move a tower of sixty-four different diameter
disks? The previous chessboard patterns you discovered should help you.
Can You …
• figure out the exponential expressions for the wheat
and chessboard problem if the grains of wheat are
quadrupled each time? Quintrupled?
• compute the volume of wheat using the total number
of grains on all sixty-four squares of the chessboard?
• compute the dimensions of a conical pile using the
volume you found for the total wheat on the sixty-four
squares of the chessboard?
• find the angle of repose (the slope) of a natural pile of
wheat?
• determine how many regions are created from the first
fold when you fold a sheet of paper in two? Fold the
sheet again. How many regions are created by this
second fold? Fold the paper one more time, determine
the number of regions, and record the results for the
first three folds. What pattern do you see? How many
folds would be needed to equal the height of the
Sears (Willis) Tower in Chicago?
Did You Know …
• that Arabic mathematician Ibn Kallikan first published
the Wheat and Chessboard problem in 1256? Many
other versions of the problem exist. For a literacy
connection, a recommended version of the story is
A Grain of Rice (Pittman, Helena. Bantam Doubleday
Dell Books, 1986).
• that at www.freerice.com, for each vocabulary word
defined correctly, twenty grains of rice are donated to
the United Nations World Food Program?
• that the number of grains of wheat on all sixty-four
squares is the sum of a geometric sequence? The first
term in the sequence is 1, and the common ratio is 2.
• that the number of moves required to solve the Tower
of Hanoi puzzle is related to Pascal’s Triangle?
of the world? Within the dome, priests moved golden
disks between diamond needle points that were a
cubit high and as thick as a bee’s body. God placed
sixty-four golden disks on one needle at the time of
creation. It was said that when the priests completed
their task, the universe would end. (It would take at
least 264 – 1 (18,446,744,073,709,551,615) moves to
complete the task! Assuming one move per second
and no wrong moves, the task would take almost 580
billion years to complete.)
• that for computer storage capacity, 210 (1024) is actually the number of bytes (the basic storage cell) in a
kilobyte (Kb); 220 (1,048,576) is the number of bytes in
a megabyte (Mb); 230 (1,073,741,824) is the number
of bytes in a gigabyte (Gb); a terabyte is 240 power,
or approximately a thousand billion bytes (that is, a
thousand gigabytes); a petabyte is 250 power of bytes,
or approximately a thousand terabytes; an exabyte
(EB) is 260 power bytes? The prefix exa means one
billion billion, or one quintillion. So, 260 is actually
1,152,921,504,606,846,976 bytes, or somewhat over a
quintillion (or 1018 power) bytes.
Mathematical Content
Exponents, bases, exponential operations, constructing
tables
References
Enchev, Rodin. “Towers of Hanoi.” Boston University.
http://math.bu.edu/DYSYS/applets/hanoi.html.
“Exponents: A Lesson on Exponents.” About.com, a part
of the New York Times Company. 2009. www.math
.about.com/library/weekly/aa072002a.htm.
Mahoney, John F. “NUMB3RS Activity: Pile It On.” Washington, DC: Texas Instruments. 2007. http://classrooms
.tacoma.k12.wa.us/tps/kmath/documents/act2_pileiton_
januslist_final.pdf.
“Math Review: Useful Math for Everyone, Section 3: What
Is an Exponent?” University of Minnesota School of
Public Health. 2004. www.mclph.umn.edu/mathrefresh/
exponents.html.
“Tower of Hanoi.” National Council of Teachers of Mathematics. 2008. http://illuminations.nctm.org/ActivityDetail.
aspx?id=40.
“Tower of Hanoi.” The Regents of the University of California, Lawrence Hall of Science Math around the World.
2009. http://www.lhs.berkeley.edu/java/tower/.
• that the legend that accompanied the Tower of
Hanoi puzzle stated that during the reign of the Emperor Fo Hi, a temple with a dome marked the center
Student Explorations in Mathematics, November 2009
5
A Powerful Little Number—continued
Answers
(b) In square 47 the number of grains of wheat can be
represented by 246.
How Many Grains of Wheat?
(c) In square n, the number of grains of wheat can be
represented by 2n – 1.
1. Answers will vary.
2. See table 1 below.
12. In each of the expressions, 2 is used as a factor.
3. S
ample answer: Starting with one grain of wheat, the
number of grains of wheat doubled as the chessboard
square number increased.
13. and 14. See table 2.
Special Patterns and Powers of 2
4. A
nswers could include the following: Beginning with
15. (a) 5
the second square, the number of grains of wheat were
(b) 234
multiples of 2. The number of grains of wheat at each
square was equal to the value of 2 used as a factor
(c) ☺
one less than the square number so that the number of
(d) y
grains for square 3 is equal to the product of 2 used as
a factor two times (3 – 1 = 2). So, for square 7, we use
(e) x + 1
2 as a factor six times (because 7 – 1 = 6).
5. S
quare 9 would have 256 grains of wheat (2 • 2 • 2 • 2 •
2 • 2 • 2 • 2), and square 10 would have 512 (2 • 2 • 2 •
2 • 2 • 2 • 2 • 2 • 2).
Deal or No Deal?
16. After eight squares are filled, the total number of wheat
grains is 255.
6. Students may say that there would be sixty-three factors 17. 256 = 2 • 128 will be placed on the ninth square; it is
of 2, and this would be too many to write.
double the number of grains of wheat on the eighth
square.
7. Students could discuss using exponents to avoid writing very long expressions.
18. The total number of grains of wheat on the first eight
8. The seventh square represents this exponential expression.
squares is one less than the number of grains on the
ninth square; 255 = 256 – 1 = 28 – 1.
19. (a) The total number of grains of wheat on
9. The number of grains of wheat in square 9 could be
all sixty-four squares is 264 – 1 (which is
represented by the exponential expression 28. The
18,446,744,073,709,551,615 grains of wheat).
number of grains of wheat in square 10 could be represented by the exponential expression 29. The place on (b) The total number of grains of wheat on n squares
the chessboard is one greater than the exponent.
is 2n – 1.
10. T
he number of grains of wheat in square 64 could
20. Answers will vary. The request is not really reasonable.
be represented by the exponential expression 263
Certainly a king would not live that long, and neither
because 64 –1 = 63.
have many civilizations, historically.
11. (a) In square 30, the number of grains of wheat can be
represented by 229.
Table 1
Place on the Chessboard
1
2
3
4
5
6
7
8
Number of Grains
1
2
4
8
16
32
64
128
Show how you
determined your answer
Students may say that they kept doubling the previous result or kept multiplying by two.
Table 2
Place on the Chessboard
1
2
3
4
5
6
7
8
Number of Grains
1
2
4
8
16
32
64
128
21-1 = 20
22-1 = 21
23-1 = 22
24-1 = 23
25-1 = 24
26-1 = 25
27-1 = 26
28-1 = 27
Exponential
Expression
6
Student Explorations in Mathematics, November 2009
A Powerful Little Number—continued
21. T
ic-tac-toe has nine squares, so the grand vizier
would have received 511 grains of wheat. He would
have been better off getting gold instead.
(Note: 29 – 1 = 512 – 1 = 511)
27. Fifteen moves is the minimum number.
22. See table 3.
28. The number of moves is the same as the total number
of grains of wheat.
Quarter
1
23. When you triple each time, square 64 would have 363
grains of wheat. There would be 3n – 1 grains of wheat
on square n.
Nickel
2
Penny
4
Dime
8
Total Moves
15
29. You can do this with a minimum of 31 moves:
25 – 1 = 31.
30. The fewest number of moves needed to move a tower
of sixty-four different diameter disks is 264 – 1 moves.
The Tower of Hanoi
24. O
ne move with one coin (21 – 1 = 1) is the fewest
number.
25. Three moves with two coins is the minimum (22 – 1 = 3):
One move for the penny, two for the dime.
26. Seven moves with three coins is the minimum
(23 – 1 = 7): One move for the nickel, two for the penny,
and four for the dime.
Table 3
Square Number
1
2
3
4
5
6
7
8
Number of Grains
1
3
9
27
81
243
729
2187
Exponential Notation
0
3
31
32
33
34
35
36
37
Total Grains
1
4
13
40
121
364
1093
3280
Student Explorations in Mathematics is published electronically as a supplement to the bimonthly Summing Up by the National Council of Teachers of
Mathematics, 1906 Association Drive, Reston, VA 20191-1502. The five issues a year appear in September, November, January, March, and May.
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Production Specialist:
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Luanne Flom, NCTM
Rebecca Totten, NCTM
Student
Explorations
in Mathematics
Teacher Notes
A Powerful
November 2009
Little Number
1. In questions 3 – 8, we want students to see the pattern in the table. We began with one grain of wheat on the first
square and doubled the number of grains of wheat on each square after the first one. We multiply by 2 one time fewer
than the square number.
2. In questions 9–12, we want students to connect the exponent to what is happening in the problem. The exponent is
one number less than the square number. We want students to generalize that the number of grains of wheat on any
square n is 2n – 1.
3. In question 15c, we use a smiley face to show students how whenever you use the number 1 as an exponent, you end
up with the factor itself. We are not trying to state that ☺ is a type of mathematical expression, only that the exponent 1
returns the factor as the simplified form.
4. In the section entitled “Now let’s talk about 20 ”, a common misconception is that 20 = 0. Students may incorrectly
interpret 20 as 2 * 0. Depending on the grade level of your students, you may want to connect to laws of exponents to
25
25
show that 5 = 1 (since any value divided by itself equals 1) and 5 = 25 – 5 = 20 = 1.
2
2
You might also want to use a variable to show the general pattern of 25 to 20 given in the activity.
5. For questions 17–20, we want students to see the pattern that the total number of grains of wheat up to square 8 is
one number less than the number of grains of wheat we will place on square 9. In general, the total number of grains
of wheat up to any square n is 2n + 1 – 1. (In other words, double the amount of wheat and subtract one).
6. If you have access to a computer lab, have students play the Tower of Hanoi game on the NCTM Illuminations Web
site. If you have access to at least one computer in your classroom, you might demonstrate how the game is played
and have students first play the game with coins or other manipulatives.
7. When you get to question 26, you may want to discuss students’ answers and processes up to this point to make sure
students are on the right track and recording the fewest number of moves for each coin.