6.7: Induction in geometry, combinatorics and number theory
We turn next to a mixed collection of problems designed to highlight a range of applications.
Problem 256 Let , . Prove by induction that has at least k distinct prime factors.
Problem 257
(a) Prove by induction that n points on a straight line divide the line into parts.
(b)(i) By experimenting with small values of n , guess a formula for the maximum number of regions which can be created in the plane by an array of n straight lines.
(ii) Prove by induction that n straight lines in the plane divide the plane into at most regions.
(c)(i) By experimenting with small values of n , guess a formula for the maximum number of regions which can be created in 3-dimensions by an array of n planes.
(ii) Prove by induction that n planes in 3-dimensions divide space into at most regions.
Problem 258 Given a square, prove that, for each , the initial square can be cut into n squares (of possibly different sizes).
Problem 259 A tree is a connected graph, or network, consisting of vertices and edges, but with no cycles (or circuits). Prove that a tree with n vertices has exactly edges.
The next problem concerns spherical polyhedra . A spherical polyhedron is a polyhedral surface in 3-dimensions, which can be inflated to form a sphere (where we assume that the faces and edges can stretch as required). For example, a cube is a spherical polyhedron; but the surface of a picture frame is not. A spherical polyhedron has
- faces (flat 2-dimensional polygons, which can be stretched to take the form of a disc),
- edges (1-dimensional line segments, where exactly two faces meet), and
- vertices (0-dimensional points, where several faces and edges meet, in such a way that they form a single cycle around the vertex).
Each face must clearly have at least 3 edges; and there must be at least three edges and three faces meeting at each vertex.
If a spherical polyhedron has V vertices, E edges, and F faces, then the numbers V , E , F satisfy Euler's formula
For example, a cube has vertices, edges, and faces, and .
Problem 260
(a) (i) Describe a spherical polyhedron with exactly 6 edges.
(ii) Describe a spherical polyhedron with exactly 8 edges.
(b) Prove that no spherical polyhedron can have exactly 7 edges.
(c) Prove that for every , there exists a spherical polyhedron with n edges.
Problem 261 A map is a (finite) collection of regions in the plane, each with a boundary, or border, that is `polygonal' in the sense that it consists of a single sequence of distinct vertices and – possibly curved – edges, that separates the plane into two parts, one of which is the polygonal region itself. A map can be properly coloured if each region can be assigned a colour so that each pair of neighbouring regions (sharing an edge) always receive different colours. Prove that the regions of such a map can be properly coloured with just two colours if and only if an even number of edges meet at each vertex.
Problem 262 (Gray codes) There are sequences of length n consisting of 0s and 1s. Prove that, for each , these sequences can be arranged in a cyclic list such that any two neighbouring sequences (including the last and the first) differ in exactly one coordinate position.
Problem 263 (Calkin-Wilf tree) The binary tree in the plane has a distinguished vertex as `root', and is constructed inductively. The root is joined to two new vertices; and each new vertex is then joined to two further new vertices – with the construction process continuing for ever ( Figure 11 ). Label the vertices of the binary tree with positive fractions as follows:
- the root is given the label
- whenever we know the label of a ‘parent’ vertex, we label its `left descendant' as , and its `right descendant' .
(a) Prove that every positive rational occurs once and only once as a label, and that it occurs in its lowest terms.
(b) Prove that the labels are left-right symmetric in the sense that labels in corresponding left and right positions are reciprocals of each other.
Problem 264 A collection of n intervals on the x -axis is such that every pair of intervals have a point in common. Prove that all n intervals must then have at least one point in common.