# CTK Insights

• ## Pages

23 Feb

### A First Look at "The Population Explosion" Book

The book in fact has a longer title: The Population Explosion and Other Mathematical Puzzles. The title warrants an observation.

I once wrote of the difference in attitude of mathematicians and puzzlists to solving problems. While, for a puzzlist, solving a problem is a goal in itself, for a mathematician it may serve as a starting point for further investigation, for turning the problem each other way, for trying to generalize, and learn something from. As Murray S. Klamkin once wrote

...small solved and unsolved problems lead to larger solved and unsolved problems which in turn lead to important mathematical results.

At first glance, the author of the book, Dick Hess, is fully justified to refer to the book as a collection of mathematical puzzles. And this is not only because the problems in the book - the puzzles - need some kind of mathematics to be solved successfully, but also because the author exhibits a remarkably "mathematician's attitude" in his approach to generating the puzzles. Many a problem in the book come as modifications of each other or of various better known puzzles. This is pretty uncommon for a puzzle book. Some readers might want to have a greater puzzle variety, but in my view, having problems turned around, seen from different angles, even munched under slightly modified conditions, makes the book a valuable resource not just for puzzle fans, but for teachers of mathematics who may want to introduce their students to problem solving strategies and instill in them the right kind of attitude to problem solving in general.

Here, for example, the opening of Chapter 2, Geometric Puzzles that come as a modification of a well-known geometric conundrum:

Mining on Rigel IV An amazing thing about the planet Rigel IV is that it is a perfectly smooth sphere of radius 4,000 miles. Like the earth it rotates about a north pole so a latitude and longitude system of coordinates referenced to the poles serves to locate positions on Rigel IV just as it does on earth. Three prospectors make the following reports to headquarters.

(a) Prospector A: "From my base camp I faced north and went 1 mile in that direction without turning. Then I went east for 1 mile. I rested for lunch before facing north again and going 1 mile in that direction without turning. Finally, I went west for 1 mile and arrived precisely at my base camp." What are the possible locations for base camp A?

(b) Prospector B: "From my base camp I went 1 mile north; then I went 1 mile east. I next went 1 mile south and, finally, I went 1 mile west and arrived precisely at my base camp:" What are the possible locations for base camp B?

(c) Prospector C: "From my base camp I went 1 mile north; then I went 1 mile east. I next went 1 mile south and, finally, I went 1 mile west and arrived at a point the most distance from my base camp under these conditions." What are the possible locations for base camp C and how far from base camp C does the prospector end up?

Chapter 9 is devoted in its entirety to an enormous number of variations on a single puzzle:

Jeeps in the Desert The problems in this chapter deal with fleets of jeeps in the desert initially located at point A where there is a fuel depot with unlimited fuel supply. All jeeps end up at either point A or at a delivery point B, as far from point A as possible. The jeeps are all identical, can go a unit distance on a tank of fuel and consume fuel at a constant rate per mile. Jeeps may not tow each other or carry more than a tankful of fuel.

One-Way Trip with a Single Jeep Your fleet consists of one jeep, which is trying to get to point B as far away as possible from the depot and finish at point B. It is permitted to cache fuel unattended in the desert for later use.

(a) How far can you get if you may use only 2 tanks of fuel?
(b) How far can you get if you may use only 1.9 tanks of fuel?
(c) How much fuel is required to go 1.33 units of distance?
(d) How far can you get if you may use only 3 tanks of fuel?
(e) How far can you get if you may use only 2.5 tanks of fuel?
(f) How much fuel is required to go 1.5 units of distance?
(g) How much fuel is required to go 2 units of distance?

These are followed with subsections Round trip with a single jeep, One One-Way Trip with Two Jeeps, and so on. The chapter is stretched over five full pages of questions. There are 10 chapters in all: Playful puzzles, Geometric Puzzles, Digital Puzzles, Logical Puzzles, Probability Puzzles, Analytical Puzzles, Physical Puzzles, Trapezoid Puzzles, Jeeps in the Desert, and MathDice Puzzles.

Among the "Analytical puzzles" one caught my eye. Rate Race:

Three cats and a rat are confined to the edges of a tetrahedron. The cats are blind but catch the rat if any cat meets the rat. One cat can travel 1% faster than the rat's top speed and the other two cats can travel 1% faster than half the rat's speed. Devise a strategy for cats to catch the rat.

The essential point in the cats being blind is of course their inability to detect the location of the mouse. Thus, when solving the problem, the reader should assume that the cats also have dysfunctional olfactory and hearing faculties.

The book is titled after Problem 6 The Population Explosion, in the first Chapter:

In March 2015 the estimated population of the earth reached 7.3 billion people. The average pewrson is estimated to occupy a volume of 0.063 m3, s the volume ov the total population is 0.4599 km3.

(a) Model the earth as a sphere with a radius of 6,371 km and spread the volume of people over the surface of the earth in a shell of constant thickness. How thick is the shell?

(b) The population currently grows geometrically at 1.14% a year. How long will it take at this rate for the population to fill a shell one meter thick covering the earth? What will the population be then?

(c) At the 1.14% geometric rate how long will it take and what will the population be to occupy a sphere whose radius is expanding at the speed of light (= 9.4605284 x 1012 km/yr)? Ignore relativistic effects.

The framework of the question is rather unexpected and funny, although I'd prefer comparing the total volume of the population to that of the Grand Canyon than to smearing the people all over the surface of the Earth.

The book is small but offers a rich collection of interesting problems - puzzles, if you will - not requiring math knowledge beyond high, and many middle, school level.