One of the most persistent math myths in popular culture is the idea that mathematics is primarily about getting right answers.
The truth is, the answer doesn’t matter that much in math. What really matters is how you explain that answer. An answer is “right” if the explanation makes sense.
And if you don’t give an explanation, then you really aren’t doing mathematics at all.
Try This Number Puzzle
Here is a short sequence of numbers. Can you figure out the rule and fill in the next three blanks?
2, 3, 5, 7, ___, ___, ___, …
Remember, what’s important is not which numbers you pick, but rather how you explain your answer.
Perhaps the sequence is the prime numbers?
2, 3, 5, 7, 11, 13, 17, …
The prime numbers make a wonderful sequence, though it isn’t the one I was thinking of.
Perhaps the sequence is the odd numbers?
2, 3, 5, 7, 9, 11, 13, …
But how do you explain that even number 2 at the beginning of the row?
This is the answer I thought of. As far as I know, no one else has described this exact sequence, so I get to name it. Letting kids name their math is a great motivator for creativity—everyone likes to name something after themselves.
Let me introduce you to the Gaskins Counting-By Sequence. The first number tells you what interval you are counting by, and the second number tells you where to start counting. From there on, it’s automatic.
The natural numbers 1, 2, 3, 4… form a basic Gaskins Counting-By Sequence. Other examples would include any list of multiples that starts with the n×1 term.
I liked my counting-by answer. Still, I wondered if I could come up with at least one more possibility, so I kept thinking. And then I found:
2, 3, 5, 7, 10, 13, 16, …
Again, I couldn’t find any reference online to this exact sequence. The OEIS had eleven almost-matches, but each list veered off after a few more numbers.
Hey, that means I got two bits of original mathematics from a single puzzle. Cool!
I named this Denise’s Differences Sequence. You count the differences between the numbers, and each difference is repeated according to its value: +1 once, +2 twice, +3 three times, etc. You can start with any two numbers, and then the difference between them determines the rest of your list.
Incidentally, the sequence of differences in my list (1, 2, 2, 3, 3, 3…) is called the Self-Counting Sequence.
What do you think? Can you come up with yet another rule to explain the numbers in this puzzle?
Or invent a new sequence. Give it a name. I’d love to hear how you explain it!
Please share your answers in the comment section below.