of half, has the advantage of receiving more then 50
percent of the slice according to his own measure. One
can avoid this perceived advantage with a different
sharing scheme. In the following “knife holding”
scheme, each brother is guaranteed to feel that he has
the advantage:
Each brother holds a knife vertically across the
cake at the location he believes cuts the cake
precisely in half. (If the brothers have different
estimation of what constitutes “half,” then the
knives will be at different locations.) Cutting
the cake anywhere between these two positions
guarantees each brother a piece, in his estima-
tion, greater than a 50 percent.
This scheme generalizes to sharing among any
number of players. For instance, it is possible to share a
cake among three players in such a manner that each
player honestly believes he is receiving more than one-
third of the cake.
Each player holds a knife vertically across the
cake at the location he believes cuts off
exactly one-third of the cake from the left.
The cake is then cut between the two leftmost
positions, giving that piece to the player hold-
ing the leftmost knife. This player has
received, in his estimation, more than one-
third of the cake, and each of the remaining
two players believes that more than two-
thirds of cake remains. The second two broth-
ers then perform the cake division scheme
described above for two players.
Any cake-sharing scheme among nplayers that guaran-
tees each player, in his estimation, at least 1/nof the
cake is called a fair-division scheme.
The scheme described above among three players,
however, is not “envy free.” Although each player
believes that he received more than his fair share of the
cake, it is not assured that he also believes that he
received the largest piece ever cut. (Every fair division
scheme between two players is envy free.) Complicated
envy-free fair-division schemes do exist for sharing
cake among any number of players. There are also fair-
division methods for dividing collections of indivisible
objects (such as the furniture in an estate) among two
or more people using cash payments to even up the
final division.
Farey sequence (Farey series) For a given positive
whole number n, the sequence of all proper fractions,
written in reduced form, with denominators no larger
than nand arranged in order of magnitude is called the
nth Farey sequence. For example, the fifth Farey
sequence is:
,,,,,,,,,,
These sequences have a number of arithmetic properties.
For instance, if and are consecutive terms of a
Farey sequence, then the numbers ad and bc, arising
from taking their cross product, are always consecutive
integers. (For example, the consecutive pair and
yield consecutive integers 9 and 10.) Also, if , ,
are three successive terms of a Farey sequence, then the
middle term is the mediant fraction . (For
instance, the term between and is = = .)
This observation allows one to quickly build up from
one Farey sequence to the next: simply compute the
mediants between all terms present and retain those
whose denominators are not too large.
The Farey sequence was first studied by C. Haros
in 1802, but interest in the topic did not stir until
British geologist John Farey (1766–1826) published his
1816 piece, “On a Curious Property of Vulgar Frac-
tions” in Philosophical Magazine. (Farey was unaware
of Haros’s work.) In 1938 American mathematician
Lester R. Ford presented a remarkable geometric inter-
pretation of the Farey sequence:
Above each reduced fraction on the number
line, draw a circle of radius touching the
number line at that point. Despite expectation,
these circles never overlap, although they do
often touch. Moreover, two circles at positions
and touch precisely when ad and bc are
consecutive integers, and furthermore the
largest circle that fits in the space between
them above the number line is the circle at the
mediant .
a+ c
–—
—
b+ d
c
–
d
a
–
b
1
–
b2
a
–
b
2
–
3
6
–
9
3 + 3
–—
—
5 + 4
3
–
4
3
–
5
a+ e
–––
b+ f
c
–
d
e
–
f
c
–
d
a
–
b
2
–
3
3
–
5
c
–
d
a
–
b
1
–
1
4
–
5
3
–
4
2
–
3
3
–
5
1
–
2
2
–
5
1
–
3
1
–
4
1
–
5
0
–
1
188 Farey sequence