Seeds and Stalks

The grails of math activities, for me, are those that involve almost no special knowledge to get into, but have near-infinite depth. (Like this one.) We sometimes describe them as having a short barrier to entry, and no ceiling. (A common suggestion when we work with teachers is to “remove the ceiling,” that is, find ways to change the problem so that the learning doesn’t end when you get the answer…)

Here’s a spanking new puzzle I’ve been playing with, and it feels like a perfect example of a problem with virtually no barrier to entry, and no ceiling either. I call it Seeds and Stalks.

Here’s how it works. We’ll generate sequences (in a kind of Fibonacci-like way) by choosing a number and adding it’s digits to itself to get the next number. For example:

16 goes to 16 + 1 + 6 = 23.
23 goes to 23 + 2 + 3 = 28.
28 goes to 28 + 2 + 8 = 38. And so on.

So we have a sequence that goes 16, 23, 28, 38, … I call this Seeds and Stalks because there are two pieces here… the seed that starts the sequence (the seed) and the sequence that grows out of it (the stalk).

Of course, there might be a seed that leads to 16 in the stalk. And indeed, 8 leads to 16. The most primal seed we could pick for this stalk is 1, since having 1 as the seed leads to the stalk:

1, 2, 4, 8, 16, 23, 28, 38, …

All well and good. But as soon as I thought of the mechanism, I was besieged by questions. The first was:

What’s the smallest collection of seeds that you need to include every number in a stalk?

I can see I’ll need 3 as a seed, since 3 isn’t in the stalk.

3, 6, 12, 15, 21, 24, …

Now 5 isn’t in either stalk, so I’ll need that too. How many seeds do I need to get every number? Or will I need infinitely many seeds?

For me, this is a perfect storm. All I need to start this problem is addition. And yet, I have no idea what will happen. I can feel that there are all kinds of patterns to find. My instinct now is to turn it over to students and see what they can find.

But I’ll look to the internet first.

What questions can we ask about Seeds and Stalks?

What answers can we find?

Comments 3

  1. Kalid

    I like the exploratory nature of this question. It shows that math isn’t just about crunching well-understood paths, and even simple “I wonder…” situations can get pretty deep.

    I’m curious to write a program to investigate. My gut says we’ll need infinite seeds — as the number of digits increases, so does the average “jump” (expected value of the sum if digits). That leaves more chances for un filled gaps, in my eyes. (There may be a cool Benford’s law distribution around the crossover points like 100, 1000, etc.)

  2. Dave Radcliffe

    I agree with Kalid. Given any positive integer N, I can find N consecutive integers, each having digit sum greater than N. Each of these N numbers would have to come from a different seed.

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