The Puzzling Power of Simple Arithmetic

We solved our last Insights puzzle by performing some arithmetic on a simple version of a complex problem in order to discover its patterns. Often this approach can reveal hidden insights. You can also use simple arithmetic to confirm that a complex formula you derived does indeed work.

Perhaps more surprisingly, playing with arithmetic can lead us to unexpected and profound discoveries that point toward deeper mathematics and sometimes even deeper science. 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. Sure, it was already known, but replicating a significant scientific principle just by playing with small numbers is an immensely heady and joyful experience. It’s the sort of thing that will get you hooked on mathematics. 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.

In that spirit, I present some simple arithmetic problems that I hope will give you that feeling of discovery. You can do these puzzles with a pen and paper, or use a spreadsheet. If you are proficient at coding, don’t immediately jump to writing a computer program. I encourage you to play a little first. (And if you recognize where the examples are leading, please don’t post spoilers for the first week or so.)

Puzzle 1

1. Find the next three numbers in the sequence: 5634 (6543, 3456), 3087 (8730, 0378), 8352 …

   Take your time to find the pattern, but if you need a hint, click here:

   Hint: Figure out how the numbers in parentheses are related to the one to their left. Then use arithmetic to figure out the next number in the sequence.
2. Follow the same procedure to generate sequences starting with the following numbers. Keep generating numbers until something interesting happens:

   i. 6372
   ii. 8956
   iii. 5058
   iv. 7191
   v. 5355

3. By now you should have made a discovery. Now try the procedure with a few of your own four-digit numbers. Explore further: Why does this occur? Can you find exceptions? Can you find numbers that require more steps than any of our examples? (Now’s the time for the coders to fire up your programs!)
4. Here is a cryptarithm (digit substitution) puzzle that may help (incidentally, all these words are acceptable in Scrabble). How many solutions do you expect it to have?

   

5. If you could do part d in your sleep, here’s a longer version with some Z’s and E’s inserted. Letters that were in part d stand for the same digits.

   

   Can you add more (or fewer) Z’s — and corresponding E’s — to the EAST number? What does this mean in the context of the original discovery? A harder problem: Does this mean that eight-digit numbers have a similar property as four-digit numbers under our procedure?
   
   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.
6. Try applying the same procedure you followed in parts a and b of this puzzle to the following numbers. Keep going until something interesting happens:

   i. 53955
   ii. 62964
   iii. 420876

The cycling phenomenon seen in 1f above led, in a different context, to a fundamental scientific discovery, which we will explore in our second puzzle. In this one, we do iterative calculations using decimal numbers, so you might want to use a calculator with an “Ans” key that recalls the last answer, if you have one. Otherwise, use a spreadsheet. Look only at the first few digits of the numbers you obtain (use your spreadsheet to show only three decimal places).

Puzzle 2

1. In this puzzle, we start with the seed number (x) which is initially 0.5. First subtract it from 1. This gives 0.5 again. We multiply this new number by x and then multiply the product by a constant (k) such as 2.4 to evaluate the expression kx(1 − x). This gives 0.6. This is our new seed. Now we do the procedure again. Subtract 0.6 from 1 to get 0.4. Multiply 0.6 by 0.4 and multiply the product again by 2.4 to get a new seed and so on. Do the above starting with a seed of 0.5 for the following constants. Again, keep going until you see something interesting happen.

   i. 2.4
   ii. 3.3
   iii. 3.5
   iv. 3.55

2. As you may have observed, there is a change in behavior at each of the above steps, as the constant k increases from 2.4 to 3.55. Also notice that the change in behavior takes place more and more rapidly as k increases. In fact, these are the actual values of the constant when the behavior changes to what you observe above for cases i to iv:

   i. 1
   ii. 3
   iii. 3.44949
   iv. 3.54409

   And changes continue in a similar manner as k crosses:

   v. 3.5644043
   vi. 3.5687594
   vii. 3.5696916
   viii. 3.56989125

You can see that the sequence of values for k seems to be converging toward a limit. Also, the ratio between successive differences, such as (3 − 1)/(3.44949 − 3) and so on, also converges to a limit. It turns out that the limit of the sequence is specific to the expression we used. The limit of the ratio of successive differences, on the other hand, is a fundamental constant of mathematics, as basic as π and e. Let’s call it ∂ (delta). Amazingly, it was only discovered through numerical procedures on simple equations such as this in 1975. If you look at what we did, we just took a quadratic expression kx(1 − x), started with a seed value of 0.5, and replaced each x with the value we got in the next cycle. It turns out that ∂ arises for any quadratic expression that has a single maximum value over the entire range of x, such as in the expression k − x². Since these kinds of expressions abound in our models of the physical world, this constant finds applications in many different areas of mathematics and many branches of science.

This is my attempt at “guided discovery,” though you are certainly free to explore more on your own if you are inspired to do so. At this point, I will not reveal what the constants we have found are, because I do not want to give you the temptation to read about these discoveries. I want you to experience the sense of discovery and exploration yourself. In the first week or so, I only want to see comments and insights from true explorers who did not know anything about these numbers coming in. After that, we’ll post comments from people who were familiar with these subjects, and we will also share videos that will tell you more.

Happy exploring!

Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt or one of the two Quanta books, Alice and Bob Meet the Wall of Fire or The Prime Number Conspiracy (winner’s choice). And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION.” (It will not appear online, so solutions to the puzzle above should be submitted separately.) Update: The solution has been published here.