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 $$n=30\,693\,385\,322\,765\,657\,197\,397\,208.$$