Mathematical Doodlings

Geoffrey Marnell

• Publisher: Burdock Books
• Year: 2017
• ISBN:
  9780995440500
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

This is a journal about a trip to the edges of the mathematical universe. It is testament to the fact that even the amateur lover of numbers—one both daunted and humbled by the sledgehammer tools of the professional mathematician—can discover new terrains, new patterns and new interconnections. All that’s needed is a pen and paper, an inquisitive mind and a modicum of patience—and perhaps a simple spreadsheet to see if a keyhole discovery can be scaled up into something worthy of a conjecture or even a theorem. This book is full of curiosities and conjectures, all the product of idle number-play. In it you’ll find new ways to multiply, simple mathematical tricks to baffle your dinner party guests, various ways of building integers from prime numbers, and much more. And nothing in it needs more than classroom arithmetic. May it inspire the timid numerophile to seek the joy of discovery undaunted by the esoteric complexities of professional mathematics.

Side by side with the playfulness in this book is a serious concern. One’s dexterity in seeing patterns in numbers is often used to judge and to classify. Aptitude and intelligence tests invariably test this, and it is often fed into an assessment of one’s suitability for a scholarship or a job. And yet the immense richness of the mathematical universe, the staggering interconnections between numbers, challenges even the most skilful test designer to create numeracy questions that have just one correct answer. As this book shows, one very common type of question on such tests—what number comes next in a given sequence—never has just one answer, contrary to the assumption of test designers. In fact, each such question has an infinite number of answers. The ramifications of this are explored in detail, especially from the perspective of natural justice. If someone is to be denied power, privilege, responsibility or reward on the basis of a test, the test had better be a good one. Alas, many are not.

Geoffrey Marnell has a masters degree and doctorate from the University of Melbourne, gained by research in philosophy at the universities of Melbourne and Oxford. He has published widely, on such topics as language, psychology and mathematics. This is his sixth book.

Part 1

Curiosities, conjectures and challenges

2, 4, 6, 8, 10, –3243: Cognitive disarray or rational answer?

Anyone who has sat a typical IQ or scholastic test will have encountered questions asking for the next number in a numerical sequence or for the numbers missing from a given portion of a numerical sequence. A simple example is this: {2, 4, 6, 8, 10, ?}, to which 12 springs to mind as the required answer. Many will have experienced the niggling doubt that there may well be more than one answer to such questions, a doubt likely to be well rooted if they had previously encountered IQ tests (or done number-sequence puzzles in puzzle books) and observed that a numerical sequence need not always be understood as a strict arithmetic series. It may, instead, be merely an ordered set of like numbers (for example, prime numbers) rather than a series of numbers that progress by some identifiable mathematical formula. And so, in the absence of any specified context or instruction on how to interpret the given sequence, many answers may fit the puzzle.

In addition, the manifold complexity of arithmetic itself easily mocks attempts to generate, in a strictly arithmetic manner, a unique number-sequence from just three or four given numbers. For example, consider the following part-sequence:

{1, 2, 3, ?, ?, ? …}

How very natural it is to continue the series {… 4, 5, 6 …} on the assumption that the given numbers are part of the most natural series of all: the series of integers that mathematicians call the natural numbers.¹ But suppose someone coming across such a part-sequence in an IQ test continues it {… 5, 8, 4 …} Should this be construed as cognitive perversity and marked wrong? If the examining psychologist should think so, they could well be making a grave mistake. For the numbers can be construed as a series in which the sum of the preceding two numbers is reduced, each time, to its digital root. (The digital root of a number is the sum of the digits of that number, further reduced, if necessary, to a single digit by continual digit summing. Hence, the digital root of 35 is 3 + 5 = 8, and the digital root of 178 = 1 + 7 + 8 = 16 = 1 + 6 = 7.) Consequently, {… 5, 8, 4 …} is a legitimate answer to the number-sequence puzzle { 1, ?, ?, ? …}.

And so, too, is {… 5, 7, 11 …} if the sequence is thought of as the ordered set of numbers that are divisible only by themselves and 1 (in other words, the prime numbers including 1).

And so, too, is {… 5, 9, 16 …}, for the simple polynomial

yields, for consecutive values of x starting with zero, the arithmetic series {1, 2, 3, 5, 9, 16, 27 …}.

And so too is {… 5, 8, 13 …} if the numbers are construed as a Fibonacci series with starting numbers 1 and 2 rather than 1 and 1. In other words, with the exception of 1 and 2, each number is the sum of the preceding two numbers.

Here, then, are five ways of continuing the sequence {1, 2, 3 …}, and a little imagination should yield considerably more.

Of course, offering just three starting numbers in a number-sequence puzzle is inviting a multiplicity of answers. Perhaps offering four will cut out the multiplicity. Well, it depends. Take {8, 16, 24, 32, ?, ?}, for example. One solution that might seem obvious is {8, 16, 24, 32, 40, 48 …}. Here each digit is 8 times its place in the sequence. But what is wrong with {8, 16, 24, 32, 42, 50 …}? Each number in the sequence (N) is obtained from the formula N = p + 6n + 1, where p is the value of an odd prime number (including 1) and n is the place of that prime number in the ordered set of all odd numbers.² Thus we have:

8	=	1 +	(6 × 1) + 1
16	=	3 +	(6 × 2) + 1
24	=	5 +	(6 × 3) + 1
32	=	7 +	(6 × 4) + 1
42	=	11 +	(6 × 5) + 1
50	=	13 +	(6 × 6) + 1

and so on.

Even the seemingly simple puzzle {?, 4, 6, 8, 10, 12, 14 …} admits to more than one answer. You might swiftly decide that 2 is the right answer. But what is wrong with 0? For the sequence {0, 4, 6, 8, 10, 12, 14 …} is simply the sequence of non-negative even integers that are not also prime numbers.

The elements in the mathematical universe seem, then, to be richly connected. For any number, there are multiple ways of getting to it from any other number (or sequence of numbers). If only to prevent themselves from misjudging a truly creative intelligence, intelligence-testing psychologists might, then, do well to require an explanation to be given for why a number sequence that forms part of an IQ test is continued in a particular way. They don’t do this. Instead they try to limit the number of possible answers to one by making the test item a multiple-choice question.

But the richness of the mathematical universe can thwart even the most diligent of test designers in their quest to ensure that only one answer is possible. To demonstrate this, consider the following question from a sample IQ test designed by IQ Test Labs.³

The answer given is B and the reasoning provided is this: “The difference between the numbers follows the series 1, 3, 9, 27, 81”. Well, that’s one reason for choosing B, but another is that each figure is three times the previous figure less 7. For example, 5 = (4 × 3) – 7; 8 = (5 × 3) – 7, 17 = (8 × 3) – 7; and so on. Another reason is that only B continues the sub-sequence created by subjecting each number to a mod 2 calculation, namely, {0, 1, 0, 1, 0 …}.⁴ In other words, only B continues the alternation of even and odd numbers. Another reason for choosing B is that the differences between the even numbers in the sequence are 4 and 36 and 36 is 4 × 9. The differences between the odd numbers are 12 and … we’re not sure yet, but if we assume the same factor between differences as for the even numbers (12), the next difference would be 9 × 12 = 108 and 17 + 108 just so happens to be 125. Another reason: form two sub-sequences, one for the even numbers, the other for the odd. The differences between corresponding numbers in the two sub-sequences create the sequence 1² + 0, 2³ + 1, 3⁴ + 0, 4⁵ + 1 and so on.

Thus the richness of the mathematical universe is clearly exposed. But can that richness yield answers other than B in the list of possible answers? It surely can. Let’s start with answer C: 112. One pattern that can easily be spotted is that (4 × 8) + (4 + 8) = 44, thereby linking the first, third...