Induction from Patterns

FIGURE 1.

I found the diagram above on a site called Puzzler's Paradise, and decided to adapt it to teach some spatial and inductive reasoning.I thought this problem was interesting because during my recent trip to Guatemala I bought the bracelets below that had this and similar patterns on them. Use the form at the end of this post to submit your answers to the Questions.



On Puzzler's Paradise, the question for the red white and black pattern in Figure 1 is:

Q1) How many outlines of triangles can be drawn in the above pattern?? (HINT -- triangles can be of different sizes)



This is a good question since it requires spatial reasoning on multiple spatial scales. As you can no doubt tell, there are multiple ways to form triangles in the above pattern, and each of those "ways" has many triangles of that type that can be formed.


The power of mathematics and inductive reasoning is to be able to take a pattern like the one above that is amenable to simply counting up all of the solutions, and make a broader conclusion about how the number of triangle outlines relates to the size and nature of the pattern itself. Such analysis represents an important tool in problem solving, namely using a smaller, more manageable circumstance to infer or induce the properties off larger problems of this type.

Before proceeding to our induction though, let's take a second to see how well you remember the definitions of different kinds of triangles.

Q2) Of the triangle outlines you identified above, how many are equilateral triangles? How about right triangles?

Now, what if we had a pattern like the one above which, instead of having 4 black diamonds had 25 black diamonds running across the center? Or 50 diamonds? In these cases, you wouldn't want to have to draw or make the pattern, and then count up all the triangles like you have just done. Instead, you might want to abstract your answer to the 4 diamond problem, and see if you can use that answer to solve the problem for ANY number of diamonds. This, mathematically, could be referred to as the N-diamond problem.

The process of preceding from a simple instance, like the diagram above, to the more general case of a pattern where we may not want to have to manually count all of the triangle outlines is an example of reasoning by induction. This type of reasoning is an important tool in the problem solver's arsenal. By seeing how the question can be solved for a simpler case (in the pattern above, for the 4 black diamond case), we might be able to arrive at a system that allows us to make conclusions for much more complicated cases (like for 25 black diamonds across).

To begin, we may want to either look at another simpler case, and determine if there is a pattern from the case of 4 diamonds to 5 diamonds. This is particularly important to make sure that the pattern is the same for every number.

Q3) Draw the pattern above for the case where there are 5 black diamonds across the center. Repeat Q2 for the case of 5 diamonds across.

So what did that lead you to believe? Is the pattern the same for every number? What do you think the answer to Q2 would be if there were 6 diamonds?


Q4) Determine the relationship between the number of black diamonds across the center of the pattern above, and

1. the number of triangle outlines,
2. the number of right triangle outlines, and
3. the number of equilateral triangle outlines.

Your answer can be in the form of a table that includes the number of each of these quantities for all multiples of 5 from 10 to 100.

Q5) Is this relationship linear in any of these cases? For the cases where it is linear, if x represents the number of black diamonds across, what is the equation of the line that represents the quantity to be calculated?

EXTENSION QUESTION

Can you use inductive reasoning to answer the question below?

Q6) Everyone at a party shook hands with everybody else exactly once. In all, 66 handshakes took place. How many people were at the party?