Solution of Main Problems:
1.
2 n
2

=
3
1 p

=
1
2 p
1
+
6 p
3
=
6 p
+
1
6 p
4
=
6 p
2
=
3 p
=
2
. n
2.
And,
2
21
=
2
3


1
7


=
1
2 ( 7 )
1
+
6 ( 7 )
1
=
14
+
1
42
.
2 n
1
=

 n
+
2
1 

1
+

 n
( n
2
+ 1 ) 

= n
2
+ 1
+
2 n
( n
+ 1 )
= n
(
2 n n
+ 1 )
+ n
( n
2
+ 1 )
=
2 ( n
+ 1 n
( n
+ 1 )
)
=
2 n
.
3.
And,
2
21
=
1


21
2
+ 1 

+


1
21 ( 21 + 1 )
2


=
1
11
+
1
231
.
2 n
=



2 pq



=


1 p
( p
+
2 q
) 

1
+

 q
( p
+ q
)
2


=
2 q pq
( p
+ q
)
+
2 p pq
( p
+ q
)
And,
2
21
=



2
( 3 )( 7 )



=
1



3 ( 3
2
+ 7 ) 


+
1



7 ( 3
2
+ 7 ) 


1
=
15
1
+
35
.
=
2 ( p
+ q
) pq
( p
+ q
)
=
2 pq
=
2 n
.
4.
2
=
1 n
1
+
2 n
1
+
3 n
1
+
6 n
6
=
6 n
3
+
6 n
2
+
6 n
1
+
6 n
=
12
6 n
=
2 n
. And, n
2
=
1
21
+
1 1
+
3 ( 21 )
1
+
6 ( 21 )
=
1
21
+
1
42
+
1 1
63
+
126
.
21 2 ( 21 )
Extension 1: To expand 2/45 into distinct unit fractions using this process, we have:
•
45 = (3)(15)
•
3 + 15 = 18 or or
45 = (5)(9)
5 + 9 = 14
•
18/2 = 9
•
(9)(3) = 27 and (9)(15) = 135 or 14/2 = 7 or (7)(5) = 35 and (7)(9) = 63
•
Thus,
2
=
1
+
1
→
1
+
1 or
2
=
1
+
1
→
1
+
1
45 45 45 27 135 45 45 45 35 63
In general, to expand 2/n into distinct unit fractions where n is odd and n = pq, then:
•
n = (p)(q)
•
p + q
•
(p+q)/2 = w or p+q = 2w
•
(w)(p) and (w)(q)

•
2
= n
1 pq
+
1 pq
→
1 wp
+
1 wq
Finally, this an identity, since
1
+
1
= q
+ p
= p
+ q
=
2 w
=
2
=
2
. wp wq wpq wpq wpq wpq pq n
Van Der Waerden (1983, pp. 165-166) is the source of this problem and its analysis.
Also, it should seem familiar because it is basically the same as Problem 3 above.
Extension 2: By the previous identities, first express the fraction 2/n as a sum of unit fractions. Order the unit fractions from least to greatest denominator. Then, take the last fraction 1/m and apply the identity to it, producing a new sum for unit fractions for
2/n with one more term involved. Also, by the construction process, we know that the two new fractions are new and distinct because their denominators are greater than m, the previously greatest denominator.
Extension 3: Given the fraction 5/11, 1/3 is the largest unit fraction less than 5/11. By subtraction, 5/11 - 1/3 = 4/33. Now, the next largest unit fraction is 1/9. By subtraction, 4/33 – 1/9 = 3/297 = 1/99. Thus, 5/11 = 1/3 + 1/9 + 1/99.
Given the fraction m/n, the denominator d of the largest unit fraction less than m/n is found by d = [[n/m]] + 1, where [[?]] represents the greatest integer function. If students have a solid knowledge of proof techniques in number theory, ask them to prove that this “intuitive” algorithm produces the largest unit fraction.
Open-Ended Exploration: Encourage students to investigate these questions on their own before consulting the literature (e.g. Mathematics Magazine or American
Mathematical Monthly or http://www.ics.uci.edu/~eppstein/numth/egypt/ ). The task puts students in the role of being mathematicians because these problems are reflective of current research questions. For example, Paul Erdös, after observing that the fractions 2/n and 3/n can always be expanded into sums of three or less unit fractions, claimed that the same is true for 4/n. No proof of the claim currently exists
(that I know of). And what about 5/n? 6/n? etc.
When discussing these problems, show students translated copies of the Rhind
Mathematical Papyrus. Chace (1978 reprint) and Robbins & Shute (1987) perhaps are the two standard resources. Students will gain an appreciation for the translation effort when they see pictures of the actual scroll, as well as the original hieratic text, the hieroglyphic transcription, the transliteration, and the literal translation.
Discuss the special use of Eye of Horus fractions, which are all powers of two. These
“sacred fractions” are equivalent to fractions of the hekat
, which was the Egyptions’ fundamental unit for measuring grain. Two references are Gillings (1982) and http://www.absoluteastronomy.com/topics/Ancient_Egyptian_units_of_measurement .
The algebraic identities being explored were probably not known as such to the
Egyptian scribes. Rather, the identities are our modern conjectures of the underlying

mathematics found in the 2/n tables. At most, they were techniques that worked, but the Egyptians did not have a way to prove them.
As a good source of writing projects, students can explore any of the following ideas relative to both unit fractions and the “greedy” algorithm:
•
These problems explore the world of unit fractions as “required” by the
Egyptians. Investigate their mathematics and their culture to determine why the Egyptians used this limited format? Why the special allowance for using the fraction 2/3? And, was their policy inconsistent, given the scribes in the
Kahun Papyrus
used vulgar fractions (i.e. fractions with numbers other than1 in the numerator) for calculations. A good resource is Darrah Chavey’s website remarks http://www.ics.uci.edu/~eppstein/numth/egypt/why.html
•
The Egyptians’ dependence on unit fractions has often been criticized because of its stifling impact. For example, the respected mathematician
André Weil characterized this dependence as "a wrong turn" (Hoffman 1998, pp. 153-154). Mathematical author Martin Gardner (1978, p. 23) adds that
“some historians have maintained that the long preoccupation with unit fractions was a cultural bias that delayed progress in mathematics as much as the Roman system of writing numerals did.” Investigate the merits of these criticisms.
•
In 1972 in number theory, Slovak mathematician Štefan Znám posed an interesting “visual” problem about number congruence systems: What sets of m integers are such that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Investigate how Znám’s
Problem is connected to the Egyptian’s unit fraction arithmetic. A good resource is Brenton & Vasiliu (2002), which also shows the problems
“surprising applications.”
•
Investigate the role of Italian mathematician Leonardo of Pisa (aka Fibonacci) in the development of his “greedy” algorithm. For example, Fibonacci supposedly published eight algorithms for expanding a fraction as a sum of distinct unit fractions, of which the seventh is the given “greedy” algorithm.
Hurd (1991) would be a helpful resource.
•
The “greedy” algorithm in Extension Problem #3 is an example of the broader category of “greedy algorithms” that have become important techniques in mathematical problem solving. Investigate and document a history of “greedy” algorithms, which basically help problem solvers make an “optimum” choice at a given stage in a problem solution, expecting that it will lead towards an
“optimum” solution overall. For example, consult books on algorithms (e.g.
Cormen et al, 2003) or related web sites (e.g. http://www.cse.ohiostate.edu/~gurari/course/cis680/cis680Ch17.html
).
Brenton, L. and Vasiliu, T. (2002). “Znam’s Problem,” Mathematics Magazine . Vol
75#1, pp. 3-11.

Chase, A. (1978). The Rhind Mathematical Papyrus . NCTM Reprint of 1927-1929
Edition published by the MAA.
Cormen, T., Leiserson, C., Rivest, R., and Stein, C. (2003). “Greedy algorithms” in
Introduction to Algorithms . McGraw-Hill.
Friberg, J. (2005).
Unexpected Links Between Egyptian and Babylonian
Mathematics . World Scientific.
Gardner, M. (1978). “Puzzles and number-theory problems arising from the curious fractions of ancient Egypt.” Scientific American , (October), pp. 23-30.
Gillings, R. (1982).
Mathematics in the Time of the Pharaohs
. Dover Publications.
Hoffman, P. (1998). T he Man Who Loved Only Numbers: The Story of Paul Erd ő s and the Search for Mathematical Truth
. New York: Hyperion, pp. 153-157.
Hurd, S. (1991). “Egyptian fractions: Ahmes to Fibonacci to Today.” Mathematics
Teacher
. (October), pp. 561-568.
Imhausen, A. (2007). “Egyptian Mathematics” in V. Katz’s (ed.) The Mathematics of
Egypt, Mesopotamia, China, India, and Islam . Princeton University Press, pp. 7-57.
Kavett, H. and Kavett, P. (1975). “The Eye of Horus is upon you.”
Mathematics
Teacher . (May), pp. 390-394.
Newman, J. (1952). “The Rhind Papyrus,”
Scientific American
, (August), pp. 24-27.
Robins, G. and Shute, C. (1987).
The Rhind Mathematical Papyrus: An Ancient
Egyptian Text
. Dover Publications.
Van der Waerden, B. (1983). Geometry and Algebra in Ancient Civilizations .
Springer-Verlag.