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### Course: Algebra (all content)>Unit 18

Lesson 9: Deductive and inductive reasoning

# Using inductive reasoning (example 2)

Sal uses inductive reasoning in order to find the 50th element of a pattern toothpick shapes. Created by Sal Khan.

## Want to join the conversation?

• It has to be a personal mental "block" but the only way i can intuitively reach the solution of this problem is

6n - (n-1). There's something wrong?
• There's nothing wrong. You still get the same answer with your thinking.
• What if n is negative? Would I be correct to say that the equation could work, but you just couldn't express it using toothpick diagrams?
• The equation is intended to represent the pattern that is found in this real-life problem. One of the major keys to understand inductive reasoning is to know its boundaries. In this case, we start with the basic house shape and keep adding additions to it, so the formula only works for n=1. After this point, Sal found a way to make sense of the case where n=0, so the single "house wall" toothpick becomes the base case.

One of the fun things that happens in mathematics is that someone starts with a pattern like the one above for a restricted set (n>=1), and then figures out a way to extend the pattern into an area that was not originally envisioned. For example, the definition of the factorial where n! = n * (n-1) * ... * 1 started with only n>=1, since the terms were counting down to 1 and so would not make sense starting below 1. When we got to combinations and permutations, however, we saw another pattern where nPr = n!/(n-r)! and nCr = n!/[r!(n-r)!]. For this pattern, we needed to extend the definition of a factorial to include 0! = 1 so that nPn = n! and nCn = 1.

My point is that extending the equation into the realm of n<0 would require a different way of looking at the pattern that was not just adding a toothpick extension to a toothpick house.
• I don't understand how this is inductive reasoning. The question said it was a sequence, so it must have a pattern, and this is the only pattern that works. Didn't Sal deduct using logic the pattern rule, then apply it to the 50th element?
• I understand your thought process. It's because he didn't logically reason/prove all the way to the 50th figure, but it is 10000% going to turn out to be 251 toothpicks. It's just not seen.
It's just like how you know...Notebook A is made of paper, and since it's made of paper, Notebook B that's just a different color but has the same feel and look is made of paper too. You can prove it but you haven't. There's better examples of this concept.
• im confused with this question: mike must belong to the bartenders and beverage union local 165, since almost every Los Vegas bartender does. thank you
• @patrickb20:
Long answer: This isn't covered in the video or the videos before it but I can try to explain it.
Assuming Mike is a Los Vegas bartender. If it's "almost every Los Vegas bartender", that doesn't mean that he's part of that group. He could be that group of Los Vegas bartenders that aren't a part of the Bartenders and Beverage Union Local 165..if I'm seeing it correctly.
• in the second example of inductive reasoning, is that characteristic of deductive reasoning being a principle part of inductive reasoning? i'm confused on the mechanics of inductive reasoning... i understand the purpose of it, but not the act of it... it just seems like when you've run out of road but can infer somewhat plausibly beyond it, but it's still a matter of deductive reasoning to reach those conclusions, is that not correct?
i don't mind the math i can appreciate it as an effective method, but could a more rhetorical example be if i'm not sure of a road to go down where it is split on the highway i can look at clues around me to hope i'm about to choose the right direction? i think there's something missing in that analogy as to the end-orientation of inductive reasoning... i thought that inductive reasoning meant that i knew my end expectation (ie x=12) and from that i can inductively reason the question may have been 10+2 or how many letters are in 'catastrophes'... this being the sort of reasoning productive in developing creative solutions to unanswerable questions
• the title of this video is "inductive patterns" but the only reference to inductive or deductive is when Sal said "if we did our math or deduced correctly..." at . is this inductive b/c we're generalizing from this sample? or is it deductive b/c this is a formula that's easily demonstrated to be fact?
• The way I see it this is deductive reasoning if it is explicitly stated in the problem that this pattern is going to be consistent throughout until the fiftieth toothpick is reached. Otherwise it is inductive reasoning. Can someone clarify if this I am correct on this?
(1 vote)
• Inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true.
A classical example of an incorrect inductive argument was presented by John Vickers:
All of the swans we have seen are white.
Therefore, all swans are white.

We could apply that to this saying each group of toothpicks has five more toothpicks than the previous term. (All of the swans we have seen are white.)
Therefore all the toothpicks will follow this pattern. (Therefore, all swans are white.)

Good question! This is inductive reasoning. There is no guarantee that, for example, the toothpicks are stacked so the shape changes or the number of toothpicks added is changed, etc.

source:
http://en.wikipedia.org/wiki/Inductive_reasoning#Inductive_vs._deductive_reasoning
• What if the blocks were separated, and that they don't share a common wall. So, the 50th term will just be the multiple of 6( 6n ), right?
(1 vote)
• Yes, because they wouldn't share a common wall, it would be a multiple of 6. We know that each "house" requires 6 toothpicks, all we have to do is multiply the the number of houses, which I will assign a variable of n, by the number of toothpicks needed for each house, which is 6. So the formula would be 6n = the number of toothpicks.

I hope this helped : )
• How do I complete this conjecture: The number of segments determined by n points is ...
(1 vote)
• I think you are talking about line segments that can be determined by n points? Are these supposed to be collinear points or do they have to be non-collinear points?

Well, the first thing you should probably do if you don't have this filed away as geometric/mathematic trivia is to sketch out some tests to figure out how many segments you can draw if the number of points is 1, 2, 3, 4, 5, and maybe a point or so more, to see if you can see a pattern. A little table would help. It is kind of interesting that you cannot draw a line segment within one point, so the first result is zero. Two points give you one segment between them. Three give you three! What happens with 4 dots? Yikes, it jumps up to 6. Now what about 5 points?
1 0
2 1
3 3
4 6
5 ?
6 ?
Once you have figured out how the numbers of line segments are increasing, then you want to try to state the number of segments in terms of n, where n= the item number of the porcupines you are drawing, which happens to also be the number of points.

Next, see if your conjecture (your guess about how the number of segments is related to n the number of points) works for each of the examples you already have. This pattern isn't simple in the sense of number of sides = n/2

Hint, this is a famous factorial exercise.