It turns out that the limit of the sequence is specific to the expression we used. Also, the ratio between successive differences, such as (3 − 1)/(3.44949 − 3) and so on, also converges to a limit. You can see that the sequence of values for k seems to be converging toward a limit. In fact, these are the actual values of the constant when the behavior changes to what you observe above for cases i to iv:Īnd changes continue in a similar manner as k crosses: Also notice that the change in behavior takes place more and more rapidly as k increases.
Keep going until something interesting happens: Try applying the same procedure you followed in parts a and b of this puzzle to the following numbers.People have been doing calculations for millennia, but it wasn’t until 1955 that this simple behavior of four-digit numbers was discovered by a mathematician who loved playing with numbers using pen and paper.What happens when we increase the number of digits? Well, something different, but still interesting. Take your time to find the pattern, but if you need a hint, click here: (And if you recognize where the examples are leading, please don’t post spoilers for the first week or so.) Puzzle 1 If you are proficient at coding, don’t immediately jump to writing a computer program. You can do these puzzles with a pen and paper, or use a spreadsheet. In that spirit, I present some simple arithmetic problems that I hope will give you that feeling of discovery. I daresay that if such examples of “guided discovery” were a regular part of math education at the lower and middle grades, there probably wouldn’t be nearly as many people who hate math. It’s the sort of thing that will get you hooked on mathematics. Sure, it was already known, but replicating a significant scientific principle just by playing with small numbers is an immensely heady and joyful experience. For instance, a few years ago in our puzzle “ Are Genes Selfish or Cooperative?” I posed a simple arithmetic problem that led some readers to rediscover a fundamental law of genetics. Perhaps more surprisingly, playing with arithmetic can lead us to unexpected and profound discoveries that point toward deeper mathematics and sometimes even deeper science. You can also use simple arithmetic to confirm that a complex formula you derived does indeed work. Often this approach can reveal hidden insights. We solved our last Insights puzzle by performing some arithmetic on a simple version of a complex problem in order to discover its patterns.