# The Population Explosion and Other Mathematical Puzzles

This is a sequel to the books *Mental Mathematics* (Dover, 2011) and *Golf on the Moon* (Dover, 2014) and like its predecessors the present book contains a set of mathematical puzzles and recreations.

The 116 problems (many of them have sub-questions a,b,c,d,...) are grouped in chapters referring to a common property such as being geometric, logical, digital, probabilistic, analytical,... The formulation of the problem is mostly short and basic. They are just enumerations but easy to understand for everyone. With only few exceptions, the solutions though are far from easy and will be quite challenging, even for advanced puzzlers. Most of them can be solved with paper and pencil, some may require a computer to do the harder calculations. There is no explicit use required of advanced mathematics beyond the secondary school curriculum. All the problems are listed in just 51 pages. The second half of the book has all the answers, but they are basic and minimalistic. It might say for example something like the angle is 30 degrees and hence the answer is such and such, but finding out why the angle has to be 30 degrees might be a bit of a challenge on its own.

I have seen quite a few of these books collecting this kind of puzzles. Sometimes the same kind of problems are reformulated in different disguises, but in this case almost all of the problems are fresh or at least have an extra complication to it. Some of them are "straightforward" in the sense that it requires careful application of the "obvious" rules, while others do require the wit to apply a "trick" or require "outside the box thinking" to find a solution. For example the title refers to a problem of the first kind where the Earth is assumed to be a perfect sphere with a given radius. Given the total volume of its human population, how thick would the layer be if this volume is uniformly spread out over its surface, which is a problem of the straightforward type. The "recreational effect" lies in the surprising result. An example of the second type that needs a computer to solve plays on a planet Rigel IV which is again a perfect sphere with radius of 4000 miles. In point A you face north and walk for 1 mile, then walk east for 1 mile and finally walk north again for 1 mile to arrive back in A. What are the possible locations of A?

There is a surprizing chapter on geometric problems introducing integer-sided isoscale trapezoids (IIT's). Another one about jeeps in the desert has more familiar puzzles of a fleet of identical jeeps that consume one tank of fuel per unit of distance. No extra fuel can be taken and jeeps can not tow one another. Several tasks have to be performed in an optimal way like travelling the longest distance or the longest round trip with a prescribed number of jeeps. The last chapter formulates several MathDice-like puzzles. MathDice is a dice game invented in 2004 by Sam Ritchie where two 12-sided dice are thrown to give two numbers that should be multiplied. Then 3 classical 6-sided dice are thrown giving 3 numbers between 1 and 6. The goal is to approximate as well as possible the product of the 12-sided dice by using the 3 numbers from the second throw once and only once, and combine them in concatenation or by prescribed arithmetical operations possibly using parenthesis. A (trivial) example: use 0, 1 and 2 in ascending order to form 12 [answer: 0+12 = 12].

This is a marvelous collection that I can warmly recommend to occasional as well as diehard puzzlers.

**Submitted by Adhemar Bultheel |

**29 / May / 2016