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 cylinder has two faces. (The lateral surface of a cylinder is
not flat.) In the mid-1700s, Swiss mathematician LEONHARD EULER 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 v is the number of vertices and e is the number of
edges the figure has.
The angle between two edges of a polyhedron meeting 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. EULER’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 n are whole numbers and if a times
b equals n, then a and b are 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 pebbles 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 precisely 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 factors. All other numbers have an even number
of factors.
For example, 36, which equals 6 × 6, has an odd number 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 perform 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 oddnumbered doors open.
185
F
typically a POLYHEDRON, is called a face of the figure.
For example, a cube has six identical faces, and a cylinder has two faces. (The lateral surface of a cylinder is
not flat.) In the mid-1700s, Swiss mathematician LEONHARD EULER 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 v is the number of vertices and e is the number of
edges the figure has.
The angle between two edges of a polyhedron meeting 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. EULER’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 n are whole numbers and if a times
b equals n, then a and b are 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 pebbles 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 precisely 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 factors. All other numbers have an even number
of factors.
For example, 36, which equals 6 × 6, has an odd number 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 perform 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 oddnumbered doors open.
185
F