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I am constructing a learning sequence for middle grade students designed to convince them that empirical arguments (arguments by example) are not sufficient in mathematics. To motivate this, I am using Stylianides' three-task sequence. However, I believe Stylianides' narrative of a pattern which falls apart at a large $n$ is a bit sophisticated for middle school learners. I'm wondering if others can help me think of another narrative showing a conjecture that falls apart at some unexpected $n$. I'm including Stylianides' narrative below:

A group of mathematicians was exploring outputs for the expression $1+1141n^2$ when $n$ is a natural number. After evaluating the expression for several natural numbers and looking for any patterns that seemed to be emerging, the mathematicians made the conjecture that this expression never returns a square number. To test their conjecture, they wrote a computer program to evaluate the expression for all natural number inputs and to stop when the output tested as a perfect square. Then they left the computer to its work.

They kept checking in with the computer’s work, and as $n$ grew bigger and bigger without an output of a square number, they became more and more confident that their conjecture was true. Imagine their surprise when they returned to the computer and found that it had stopped running the program, meaning that the program had an output of a square number. They were even more surprised to find that this expression does return a square number, but not until $$n=30\,693\,385\,322\,765\,657\,197\,397\,208.$$

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    $\begingroup$ See Examples of eventual counterexamples. $\endgroup$ – Joel Reyes Noche Jan 31 at 5:04
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    $\begingroup$ Thanks for sharing. I think these are still a bit out of reach for middle grade learners. I'm wondering if there are eventual counterexamples with more kid friendly language. And perhaps less complicated operations. $\endgroup$ – MathGuy Jan 31 at 5:38
  • $\begingroup$ You should definitely check out patterns that eventually fail too, some great answers there. $\endgroup$ – Shai Mar 7 at 16:48
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It might work better, with this age level, not to be concerned about how large n is when the apparent pattern falls apart.

My favorite example is the problem of making all possible straight-line segments between n points on a circle, and then counting the regions. 2 points makes 2 regions, 3 makes 4, 4 makes 8, and 5 makes 16. It sure looks like doubling...

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    $\begingroup$ This is actually the second task in Stylianides' sequence: the circle and spots problem. I am planning on doing this problem, but I was hoping to include an extra narrative to support the understanding that a pattern can fall apart at a large n. Students may believe they just need to try more examples after the circle and spots problem. $\endgroup$ – MathGuy Feb 2 at 5:06
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    $\begingroup$ There are some other questions on this here. I will try to find them. $\endgroup$ – Sue VanHattum Feb 3 at 5:58
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    $\begingroup$ I really enjoyed this task as a student. I was very surprised that 1 (for 1 point), 2, 4, 8, 16 wasn't enough of a pattern. @MathGuy It didn't occur to me that I could always try more examples but learned that you can't rely on a clear pattern. I definitely think this would make more of an impression then those with larger n where the students can't grasp the whole picture. $\endgroup$ – Amy B Feb 9 at 20:43
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I second Sue Van Hattum's suggestion that you should not be so concerned with how large the $n$ is where the pattern eventually fails. I'll go one step further and recommend an example where that $n$ is not only fairly "small" but also such a situation where the students can see why that $n$ makes the pattern fail.

Consider the function $f(n) = n^2+n+41$. Which values of $n$ make the output $f(n)$ a prime number?

You and the students could complete a group task to check $f(1)=43$ and $f(2)=47$ and $f(3)=53$ and $f(4)=61$ and $f(5)=71$ are all prime. You can then ask whether they think this will keep going. You can then "wow" them with a much larger value, showing that $f(20)=461$, for example, is also prime. (This could lead you into some tangential discussions about primality testing and factorization methods and so on.)

But after all that, I believe you could easily convince the students (or Socratically guide them into realizing on their own!) that $f(41)=1763$ is not prime. You can get them to verify that $41\times 43=1763$, and you can do the easy algebra with the function: $f(41)=41^2+41+41=41(41+1+1)=41\times 43$.

(At that point, you could even work with them to realize $f(40)=40^2+40+41 = 40(40+1)+41 = 40(41)+41=41(40+1)=41^2$ is not prime, either!)

I think this will be more instructive, at least as a first example of such a situation. Afterwards, you could use more "exotic" examples like the one you mentioned. But I believe the students will feel more comfortable with the ideas after having seen this example.

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This happened to me once on a topic that should be accessible to middle-school students. As an undergraduate, I formed the following conjecture:

An antiprime (also called a highly composite number) is a positive integer that has more divisors than any number less than it. The first few antiprimes are $$1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360,...$$ Conjecture: For every antiprime $n>1$, there is a prime $p$ such that $p\mid n$ and $n/p$ is an antiprime.

This would be a really useful fact, because you could make a list of all antiprimes recursively by just multiplying the ones you know times all of their prime factors plus their first unused prime factors and just check those instead of having to check every single number. But I was never able to find a proof for my conjecture.

Many years later, I asked on the xkcd math forums if anyone could help me with my proof. That website is gone now and my post wasn't archived, but someone came up with a counterexample instead of a proof! Thanks to kind people in MSE, that counterexample can be recreated:

$$362279431624673937974303738230488502933082643722886373107941760000$$

So, yeah, just trusting that it works out for small numbers wasn't a winning strategy for me here.

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Perhaps it is a good idea to start with truly simple examples.

The simplest example accessible to children is the notion that every odd number is prime.

Another is to calculate $\cos(\pi/n)$. For $n = 1, 2, 3, 4$ it works.

It is simple in principle to construct a (degree $N-1$) polynomial that takes "nice" values at the first $N$ integers by solving an interpolation problem. Of course the polynomial that results may not be "nice", but with some fiddling it should be possible to find a serviceable example this way.

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Suppose you put n distinct points on the circumference of a circle then draw segments connecting each point to every other point. What's the maximum number of regions into which the circle is divided?

1 point has nothing to connect to so you get 1 region, i.e. the whole circle.

2 points make a chord which divides the circle into two regions.

3 points make a triangle which gives you four regions, the interior of the triangle and three curved sections outside the triangle.

4 points make a quadrilateral with its two diagonals for a total of eight regions.

Now look for a pattern

1 point => 1 region

2 points => 2 regions

3 points => 4 regions

4 points => 8 regions

Inductive reasoning could lead you to conclude that the number of regions is equal to 2^{n-1}. However, the pattern breaks down when n = 6 and you only get 31 regions rather than the expected 32.

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    $\begingroup$ This is identical to the answer above yours. $\endgroup$ – Steven Gubkin Jan 31 at 15:30

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