face A flat surface on the outside of a solid figure,
typically a
POLYHEDRON
, is called a face of the figure.
For example, a cube has six identical faces, and a cylin-
der has two faces. (The lateral surface of a cylinder is
not flat.) In the mid-1700s, Swiss mathematician L
EON
-
HARD
E
ULER
established that if all the outside surfaces
of a convex solid are flat, then the number of faces f
the figure possesses is given by the formula:
f= 2 – v+ e
Here vis the number of vertices and eis the number of
edges the figure has.
The angle between two edges of a polyhedron meet-
ing at a common vertex is sometimes called a face angle.
In
GRAPH THEORY
, any region of plane bounded by
edges of a planar graph is sometimes called a face of
the graph. E
ULER
’
S FORMULA
v– e+ f= 2 also holds
for connected planar graphs if one is willing to regard
the large unbounded region outside the graph as a face.
See also
DIHEDRAL
.
factor The term factor is used in two mathematical
settings:
NUMBER THEORY
and
ALGEBRA
. In number
theory, if a, b, and nare whole numbers and if atimes
bequals n, then aand bare called factors of n. For
example, 3 and 4 are both factors of 12 (since 12 = 3 ×
4), as are the numbers 1, 2, 6, and 12 (2 ×6 = 12 and 1
×12 = 12). Any number that divides the given number
evenly is a factor. For this reason, factors are sometimes
called divisors.
The factors of a given number have a geometric
interpretation. For example, one can arrange 12 peb-
bles into six different rectangular arrays: a 1 by 12
rectangle, a 2 by 6 rectangle, a 3 by 4 rectangle, a 4 by
3 rectangle, a 6 by 2 rectangle, and finally a 12 by 1
rectangle. The dimensions of these rectangles are pre-
cisely the factors of 12. This interpretation shows that
the factors of a number come in pairs—unless, one of
the rectangles formed is a perfect square (in which case,
one factor is “paired with itself”). This shows:
Square numbers have an odd number of fac-
tors. All other numbers have an even number
of factors.
For example, 36, which equals 6 ×6, has an odd num-
ber of factors: 1 and 36, 2 and 18, 3 and 12, 4 and 9,
and 6. This simple observation solves the famous
prison warden puzzle:
A prison warden and 100 inmates, residing in
cells numbered 1 through 100, agree to per-
form the following experiment over 100 days.
In the process of the experiment some cell
doors will be left unlocked and the prisoners
agree not to escape.
On the first day, the prison warden will turn
the key of each cell door and leave all the
doors unlocked.
On the second day, the warden will turn the
key of every second door. This will lock doors
numbered 2, 4, …, 100 and leave the odd-
numbered doors open.
185
F