1.2. INDUCTIVE REASONING

Much mathematical discovery starts with inductive reasoning – the process of reaching general conclusions, called conjectures, through the examination of particular cases and the recognition of patterns. These conjectures are then more formally proved using deductive methods, which will be discussed in the next section. Below we look at three examples that use inductive reasoning.

Number Patterns

Suppose you had to predict the sixth number in the following sequence:

1, -3, 6, -10, 15, ?

How would you proceed with such a question? The trick, it seems, is to discern a specific pattern in the given sequence of numbers. This kind of approach is a classic example of inductive reasoning. By identifying a rule that generates all five numbers in this sequence, your hope is to establish a pattern and be in a position to predict the sixth number with confidence.

So can you do it? Try to work out this number before continuing.

One fact is immediately clear: the numbers alternate sign from positive to negative. We thus expect the answer to be a negative number since it follows 15, a positive number.

On closer inspection, we also realize that the difference between the magnitude (or absolute value) of successive numbers increases by 1 each time:

3 – 1 = 2  6 – 3 = 3  10 – 6 = 4  15 – 10 = 5  …

We have then found the rule we sought for generating the next number in this sequence. The sixth number should be -21 since the difference between 15 and 21 is 6 and we had already determined that the answer should be negative. Based on this rule, we could go on further and predict the seventh, eighth, or any subsequent number in this sequence.

But is -21 really a definite answer? In other words, can we unequivocally dismiss any other number as wrong? As it turns out, we cannot. Using inductive reasoning we identified a rule that generated the five numbers in the sequence and used it to conjecture the sixth. Surely, many other people would have also spotted the same pattern and provided the same answer (these type of questions are, in fact, commonly used in cognitive tests). However, this pattern is not unique. We could, for example, think of the following sequence:

1, -3, 6, -10, 15, 101, -103, 106, -110, …. , where the first 5 numbers follow our initial pattern but actually constitute just a group within a larger pattern. Here the sixth number jumps rather arbitrarily to 101. The situation now looks pretty bleak, as it can easily be shown that there are infinitely many other patterns that could work for these numbers (albeit, not as obvious as the first one we found to get the answer -21).

What this example illustrates is that inductive reasoning does not lead to conclusive and definite answers in theory. Hence it does not provide sufficient proof for a mathematical claim. In practice, however, this approach is immensely useful to scientists studying natural phenomena. Their observations of specific patterns and behaviors in the natural world lead them to theories, conjectures and hypotheses that can then be carefully confirmed in a controlled environment, such as a laboratory. This is precisely how scientists can validate their claims and explain occurrences in the natural world. For this reason, inductive reasoning is often called the scientific method.

Polygonal Numbers

In Ancient Greece, the Pythagoreans (ca. 550 B.C.E.) were famous for their obsession and beliefs regarding the natural, or counting, numbers: 1, 2, 3, 4, 5, ... One class of natural numbers they studied in particular were the so-called polygonal numbers that can be arranged with pebbles (or points) as regular geometrical shapes: triangles, squares, pentagons, and so on. Figure 1 below shows the first five triangular and square numbers.

Figure 1: The First Five Triangular Numbers (Top Row) and Square Numbers (Bottom Row).

After closely examining the polygonal numbers in Figure 1, some interesting patterns start to emerge. It seems that these numbers exhibit the following three properties:

1. Triangular numbers are sums of successive naturals: 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4 etc.

2. Square numbers are sums of successive triangular numbers: 4 = 1 + 3, 9 = 3 + 6, 16 = 6 + 10, etc.

3. Square numbers are sums of successive odd naturals: 4 = 1 + 3, 9 = 1 + 3 + 5, 16 = 1 + 3 + 5 +, etc.

Are you convinced of these properties?

Unless you are an entrenched skeptic, you would have no reason to doubt the validity of these patterns. Despite only looking at a few numbers, the numerical evidence is already compelling. After all, what are the chances that these results are just an incredible coincidence? If the properties above are true for the first five triangular and square numbers, then why would they be false for some bigger triangular or square number? You have no reason to suspect that, say, 4,068,289, the square of 2017, is not the sum of the first 2017 odd naturals (following the third property).

The truth, however, is that we never truly justified why these three properties should hold for all triangular and square numbers. Even though there are infinitely many of these polygonal numbers, we should still explain why these properties apply to any one of these numbers, regardless of their size. In other words, we need a convincing deductive argument that encompasses all cases.

One way to validate these properties is by construction. For the first property, simply look at how the triangular arrangements of pebbles are formed from previous ones. Starting with one pebble, add a row of 2 pebbles at the bottom to get the next triangular configuration with 3 pebbles. Then again, add a row of 3 pebbles at the bottom to get to the next triangular configuration with 6 pebbles. Each time a new row with one more pebble gets added to the bottom of the previous triangular formation. The pattern now becomes clear. The third property can be similarly understood using square pebble configurations instead. By construction, each square formation is made from the previous one by adding an additional layer of pebbles to the right and to the top with an extra pebble in the top-right corner. It is easy to check then that these extra layers of pebbles added to the smaller squares are always successive odd naturals. Figure 2 illustrates this process: exactly two pebbles (one to the top and one to the right) are added each time to these extra layers, thus producing the sequence of consecutive odd naturals: 1, 3, 5, 7, 9, ...

Figure 2: The First Five Square Numbers as Sums of Consecutive Odd Naturals.

What about the second property? There is a constructive argument that works here as well (add the pebbles of the smaller triangle to the larger one in a way that configures as the square formation), but it is a bit more difficult to visualize. Lacking a simple constructive method, we could point to the numerical pattern with the first five square numbers seen in Figure 1 to suggest that this property is always true. Once again, we are reasoning inductively. However, the property would still need to be validated deductively.

These three properties of polygonal numbers (along with many others not mentioned here) are proved more formally using algebraic means, but this tends to be more involved. Another way to prove them is through the use of mathematical induction (see Appendix A for more on this powerful method).

Regions of a Circle

With the previous example, it seemed that we could always check properties of polygonal numbers inductively. All that was needed were a few numerical examples. Better yet, we managed to provide convincing constructive arguments for at least two of the three properties. So doesn’t that suffice as an adequate proof? In other words, couldn’t we always check mathematical statements inductively with some examples and assume that they must be true for all other cases? The next example illustrates – in spectacular fashion – why the answer to this last question is a resounding no.

Figure 3 shows five circles with an increasing number of points on its boundary. Thus the first circle has one point on its boundary, the second circle has two points on its boundary, the third circle has three points on its boundary, and so on. If all the possible lines joining these boundary points are drawn, as shown in the figure, they will divide these circles into different regions. The question we ask now is the following: how many regions can you expect to see in the next circle (not shown in the figure) with 6 boundary points?

Figure 3: Regions of Circles with 1, 2, 3, 4, and 5 Boundary Points

Counting different regions in each of the circles in Figure 3 we get the following numbers: 1, 2, 4, 8, 16. Do you recognize this sequence? It seems the number of regions is just doubling with each additional point on the boundary. Based on this observation, we would then conclude (inductively) that the circle with 6 boundary points should have 26 = 32 regions. However, as it turns out, this answer is incorrect!

Look at the circle with 6 boundary points shown in Figure 4. Here, similar regions formed by all the lines joining these points are colored yellow, red, black, green and orange to make the counting easier. There are 6 yellow circular segments outside the larger inscribed hexagon; 18 red, green and black triangular regions between the smaller inscribed hexagon and the larger inscribed hexagon; 6 orange pentagons and quadrilaterals inside the smaller inscribed hexagon; and finally one last triangular region in the center. So there are, in total, 31 different regions!

Assuming no three lines intersect at the same point, the actual number of different regions within a circle bounded by all the possible lines joining 1, 2, 3, 4, … boundary points is given by the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, … The explicit formula that generates this sequence of numbers is the following:

푛 (푛3 − 6푛2 + 23푛 − 18) + 1 24 , where 푛 is the number of boundary points. You can check that this formula does indeed return the correct number of regions for the first few values of 푛. For example, 3 27−54+69−18 if 푛 = 3, then (33 − 6 ∙ 32 + 23 ∙ 3 − 18) = + 1 = 4, as expected. The 24 8 combinatorial proof that leads to this formula is skipped here as it goes beyond the scope of our discussion1.

1 For a detailed explanation of this solution, check, for example, the following Wikipedia link: https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas

Figure 4: Regions of a Circle with 6 Boundary Points.

Thinking about this problem inductively led us directly to a mistake. This example shows, rather spectacularly, the dangers of relying solely on inductive reasoning to draw general conclusions. In retrospect, checking a few computations might have easily convinced you that the doubling formula for the number of regions had to be incorrect. This so-called exponential formula is given by 2푛−1, where 푛 is the number of boundary points. With only 25 points on the boundary, this formula returns over 33 million different regions within the circle. With 30 points, this number balloons to over a billion! Think about drawing a large circle on a field (roughly a few yards in diameter), attaching 25 or 30 pegs all around it, and then joining the pegs with taut strings to define the different regions. Would you ever expect to count such an extravagant number of regions? Of course not. The result seems rather far-fetched now.