Rooting for NASA

Below you'll find everything I've discovered so far, regarding MathPickle's Rooting for NASA. If you have thoughts on the problem that you'd like to add, please post them to the Discord.


Lengths of segments connecting square grid points

Observation:

Every length connecting two points on a square grid, has one of two types:

  1. If the length is a whole number, then it's the distance between two horizontal or vertical points.
  2. If the length is irrational, then it's the diagonal length of some rectangle. for example, \(\sqrt{2}\) is the diagonal length of a \(1 \times 1\) rectangle (more specifically, a square, in this case).

Thus, we could enumerate the irrational lengths by enumerating rectangles with whole number side lengths. This amounts to enumerating pairs of whole numbers \((a, b),\) such that \(a \ge b.\)

Question:

Which irrational numbers can't you make, if any?

Question:

Can we make \(\sqrt{3}?\)

Answer:

No. To make \(\sqrt{3},\) we have a diagonal \(c = \sqrt{3}.\) But the Pythagorean theorem tells us that \(a^2 + b^2 = c^2,\) and so \(a^2 + b^2 = 3.\) But \(3\) cannot be written as the sum of two square numbers, and thus, \(\sqrt{3}\) is unachievable. Thus, a length of \(\sqrt{c}\) is unachievable if and only if \(c\) cannot be written as the sum of two squares. the sum of two squares theorem tells us exactly which \(\sqrt{c}\) are achievable.


Starbursts

Observation:

Because a starburst has exactly one vertical line of symmetry, we only need to draw half the starburst. Each starburst is comprised of a set of lengths. For example, the \(10\) starburst is made from the sequence of lengths: \(\sqrt{1},\) \(\sqrt{2},\) \(\sqrt{4},\) \(\sqrt{5},\) \(\sqrt{8},\) \(\sqrt{9}.\) We can thus notate this starburst as \((1, 2, 4, 5, 8, 9).\) If the square root of the first number is a whole number, it appears vertically in the diagram, and thus is not repeated. Same goes for the last number. Thus, the 1 and 9, in the aforementioned sequence, are counted once, while all others are counted twice.

Starbursts known so far:

Questions:

What's the lowest number starburst you can get? The highest?

How can we characterize the starburst numbers?

For each starburst number, in how many ways can it be made? We already know some starburst numbers can be made in more than one way (see the 10 starburst above, for example).


Falling into a black hole

Question:

Can the spiral continue indefinitely, or must it eventually crash into itself?


Rocks

Numbers that are the sum of 2 squares: OEIS entry A001481.

There is a polygon with vertices at grid points, with sides of length \(\sqrt{1},\) \(\sqrt{2},\) \(\sqrt{4},\) \(\sqrt{5}.\) The numbers 1, 2, 4, 5, are the first 4 positive numbers that can be expressed as the sum of 2 squares. Note that side lengths of \(\sqrt{1},\) \(\sqrt{2},\) \(\sqrt{4},\) could form a polygon, because \(\sqrt{1} + \sqrt{2} \gt \sqrt{4}\) (triangle inequality), but the vertices wouldn't occur at grid points, as required. We've used the first 4 numbers, and so they're now unavailable. From the numbers that remain: 8, 9, 10, 13, 16, ..., pick the first \(n\) that can form a polygon.

Questions:

Is there any pattern to picking these numbers? For example, can we always pick the first 4? Maybe the first 5? Can we bound \(n\) at all?


TODO

It would be lovely to have interactive programs for experimenting with these problems. Maybe these could be written using PaperJS or something similar.