Mathematical Nuggets from Austria

The following sections are included:

• Preface
• About the Authors
• Contents
• Introduction

Sometimes, the most fun you can have with numbers comes from just playing around with their digits. There are all kinds of weird and enjoyable things that turn up when you start thinking about combinations of digits that numbers with certain properties can have. This chapter contains some problems of this type. You may be confronted with numbers whose digits have interesting sums or products, or whose order might change in interesting ways under certain circumstances. The most important thing is to keep an open mind; you’ll be surprised at what you will be able to figure out. And remember, if you can’t get the solution right away, you can find a tip to help you on your way at the end of the problem list.

There are problems for which it can be really useful to introduce a variable and solve an equation. That doesn’t necessarily mean that this is the only way to proceed, but a little bit of simple algebra can often make life easier. Often the advantage just lies in the fact that a complicated text is really not so complicated if you can write it using variables as short-cuts…

Thinking in three spatial dimensions does not come easily to everyone. A fun way to approach this type of question is through problems concerning things built out of rectangular bricks. Sometimes the bricks are even cubic blocks, with all sides of the same length. Here are a few problems dealing with objects that are either built up out of rectangular bricks or cut up into such bricks.

Sports are fun and solving problems is fun, so solving problems about sports must be at least twice as much fun, right? Or is it fun squared?

Some readers might take issue with some of the activities classified as “sports” here (chess? folk dancing?), but try to keep an open mind. We start off with a number of problems concerning races: foot races, bicycle races, and because the problems were developed in Austria, ski races. Then we will advance to some games played in a circle, and then finally to some problems concerning more classical team sports.

There are several interesting types of problems that deal with circles in some way. Such problems can have some aspect of counting, or possibly deal primarily with such fundamental geometric properties as distance, angle or area. Circles are such basic objects that all such problems end up with a common flavor, however.

Some problems don’t really require any mathematical knowledge at all to solve. (That is, unless you include counting as mathematical knowledge.) All it takes to solve one of these gems is some insight into the situation and some argumentative savvy. One type of such problems deals with people who are known to tell the truth or known to lie. Another type asks us to find the number of objects with some specific obscure trait, while others deal with equally unlikely situations. What they all have in common is the fact that pure logical reasoning will get you to a solution, without requiring any serious calculation.

A lot of people are wary of fractions. Sometimes the problem is that they simply find it difficult to understand what they mean. A number on top and a number on the bottom . . . what is that exactly? Sometimes it has to do with the calculations involved . . . how do you get a handle on them? The following problems all deal with fractions in some way. The first few are pure calculations, but all of them have a twist of some kind, and the answers are always surprisingly simple once you understand the structure. You will be asked to “simplify” or to “calculate”. Both of these terms simply mean that you are meant to find a way of expressing the resulting number in as simple a way as possible, which usually involves very small integers. The next group of problems deals with relative sizes of fractions with different numerators and denominators (recall that those are the numbers on top and on the bottom, respectively). Finally, you will find a mixed bag of problems, all of which deal with fractions in some unique way. Some of these problems are posed in a multiple choice format. Just doing the calculations will get you to the answer, of course, but you may try guessing the right answer first. Once you calculate the correct answer, you may be quite surprised!

Number relationships are often expressed with percentages. Since this particular mathematical tool is so common, you might think that most everyone would be able to interpret number expressions expressed with it quite easily. This is not always the case, however. In fact, many people find correct manipulation of percentages to be quite a tricky business. Here are some problems involving percentages, starting with very elementary one-step problems, and moving slowly to slightly more involved questions.

We do live in an essentially three-dimensional (3D)world, but as was already noted in Chapter 3, many people find 3D problems to be among the most difficult to solve. You may want to try your hand at the problems there before you attack these slightly more general ones. Most of the problems in this section can be done with pure spatial reasoning, though, maybe combined with a little bit of simple calculation. A lot of them have pictures, and if you find the situation difficult to comprehend at first, you should definitely take a closer look at the picture.

We all know that money makes the world go round. It is therefore quite natural for a mathematics competition to contain problems dealing with money.

You will find that the statements in these problems involve different units of currency. One reason for this is the fact that Austria replaced its previously used currency 1S (Austrian Schilling)=100 g (Groschen) by the euro in 2002. In addition, you will find some fictional currencies showing up here and there.

The passing of time is a surprisingly multi-faceted thing. There are interesting questions to be asked about brief snippets of time or long stretches, as well as just the passing of an indeterminate amount of time. To reflect this, this section is sub-divided into several bits, each of which contains problems of a certain type.

All the problems in this book are mathematical puzzles, but there are some that look more like the type of puzzle you might see in a typical puzzle book or a newspaper. The problems presented in this chapter are of this type. You probably wouldn’t be too surprised to see one of these alongside the crossword and the Sudoku in your daily paper.

A lot of people find puzzles dealing with speeds and rates of change to be much harder than other types of problems. One reason for this may be the fact that such problems inevitably involve multiple intertwined units of measurement, like miles per hour combined with miles and hours. This certainly forces a certain level of concentration in dealing with such questions, but that doesn’t mean that they can’t be a lot of fun to take on! Try your hand at these.

Most geometric questions can be reduced somehow to considerations of triangles. We might be asking about angles or lengths of line segments or areas of certain shapes; most anything can be chopped up into triangles in some way. Considering this, it isn’t surprising to note that there are a lot of different kinds of fairly elementary questions we can ask about just triangles themselves. Here are some interesting ones. We have chosen these because they each have some special characteristic to separate them from the pack. You might find these a bit more challenging than the problems in other chapters, but that should make it all the more satisfying, when you get the solutions!…

In this chapter, we will take a look at some problems dealing with geometric figures with more than three sides. As usual in geometry, we will be dealing with the fundamental geometric properties of distance, angle and area, but there will also be some problems that have more to do with counting or matters of relative position.

There are a lot of interesting things we can do with numbers that just result from ordinary division. Here are some enjoyable problems of this type. It will prove to be really useful in solving many of these problems if you are familiar with the rules for deciding under which circumstances a number is divisible by small numbers like 2, 3, 4, 5, 8 or 9.

Here are some fun problems concerning ways that shapes can — or can’t — be cut up in regular ways. You may even want to try cutting out some of these shapes from paper and playing with them that way.

A lot of interesting things can happen when we put things in a row according to some underlying mathematical principle. When we put numbers in a row in this way, we get a sequence. Here are some enjoyable problems concerning sequences of positive integers.

A large number of interesting questions arise just by asking how many things can have a certain property or how many ways there are to do something. In this section, we will take a look at a number of problems of this type. Some are quite abstract, dealing with number digits or points and lines in the plane. Others are quite practical. In how many ways can we sit on a bench, if we consider a number of restrictions? How many rooms are there in a certain building if we know something specific about the rooms?