Strangely Consistent

Theory, practice, and languages, braided together

t4: Rain in a world of cubes

<flussence> as a minecraft player I figured out what t4 was asking pretty much instantly :)

This is me trying to emerge from the big strange writer's block that has inexplicably formed around the t4 blog post. Here goes.

The t4 task was my clear favorite this year. It has a certain William Gibson quality to it, with virtual rain falling inside a three-dimensional world where everything is made of cubes which mostly just hang there, suspended, in mid-air.

## Simulate rain in a world of cubes

Write a program that calculates the volume of rain water collected in the cube
world described below.

The cube world &mdash; given as input &mdash; consists of a finite set of cubes
on integer coordinates `(x, y, z)`. The positive `y` coordinate means "up".

An infinite amount of rain then falls from an infinite height. Both of these
infinities are taken to really mean "large enough as to make no difference".
As it lands on cubes, the water will follow predictable rules:

* Rain falls everywhere.

* Water falling will land on the first cube below it. It does not fall through

* Water will collect on levels where walls on all sides will keep it in.

* Water will produce vertical waterfalls where such walls are missing.

* Cubes are packed tightly enough that gaps between cubes sharing an edge will
  not let water through. However, the same gaps will readily let air through if
  water needs to displace air for some reason.

Waterfalls work in the simplest way imaginable: if water "escapes" from a      
structure of cubes, it will fall straight down along the first available
"chute" of cube-formed empty cells until it hits a cube. (Which it may not
necessarily do. A waterfall may go on to infinite depth.) As a waterfall hits a
cube, it behaves just like other kinds of water: it may spread, collect, and
form new waterfalls as needed.

People had different ideas how to solve this one:

I had fun guessing what solutions people would come up with. I correctly guessed the first two, but not the last one. I guess it's a bit too mutable for my FP brain to come up with these days.

Anyway, the mistakes! Oh, the mistakes. Not just one or two contestants for this one; all of them. Turns out simulating rain on cubes is hard!

Here follows a choice list of assumptions broken by the contestants, that make their programs return odd results.

Assuming that rain can reach where it can't


Let me explain the above picture. In order to test the four entrants against odd cases, I wrote a small program that builds a cube world from the above syntax. It only describes a cross-section; and so walls in the depth direction are automatically added. In other words, the above depicts a sealed box with no way in.

It should contain no rainwater, of course. One of the programs returns that it's full of water.

Oh, and by the way, the script that produces coordinates from pictures like the above turned out quite cute and simple, so let me share it:

my %coords =
    ' ' => [         ],
    'X' => [-1, 0, +1],
    '.' => [-1,    +1],
    '~' => [-1,    +1],

for lines.kv -> $y, $line {
    for $line.comb.kv -> $x, $char {
        for %coords{$char}.list -> $z {
            say "($x, {-$y}, $z)";

Assuming that the water can rise higher than its lowest outlet


It's for cases like this that I felt a need in the problem description to talk about gaps between cubes that "will readily let air through if water needs to displace air". In other words, if the above is a kind of barometer, then it's a completely useless one, because it leaks air and water find an equilibrium based only on itself.

...which means that the correct answer above is 7. That's the number of waterfilled cubes when the water level is the same "inside" the barometer and at its mouth.

One of the programs got 9, assuming that the barometer fills up completely. Two programs got 0, assuming no water can even enter.

Speaking of which...

Assuming that some vessels are unable to contain water


Two programs had trouble with this one. I don't know if it's because of the banana shape or the cover over one of the ends. But they got 0 cells of rainwater collecting in it, when the correct answer is that it fills up all 9 internal cells.

Underestimating the size of a vessel


A small vessel sitting in a bigger vessel. A naive program might reach the brim of the small vessel, figure "oh, ok, we're done here", and then not fill up the bigger vessel with water.

This happened with one of the programs.

Concreteness and TDD

I've mentioned it in previous posts, but the way I pick problems for the contest is I find problems where I myself go "oh, that's easy, I'll just..." and then a while later, I go "...oh wait." Problems that look easy on the surface, but then turn out to have hidden depths. (A bit like these vessels holding water can have hidden depts, tunnels, nooks and crannies.) One of my favorite feelings when I design something is having the model "break" for a certain case. It's like the floor falling out from under me, and I have to re-orient myself inside the solution space to accomodate the new rules.

All the failures above emphasize the need for having actual test cases to run the program against. The base tests I send with the problems are (intentionally) inadequate for this purpose. The contestant is meant to think up their own tests, consider edge cases, special cases, and pathological cases.

To me, that's where unit testing shines. Development suddenly becomes a back-and-forth discussion between you and the programming substrate over something very tangible: concrete cases.

Only one champion still standing

Only one of the programs passes all of the above tests with flying colors. Well, I do want to stress that all four contestants made brave efforts. But for one reason or another, one of the four programs ended up especially correct.

Check out the reviews for details., wait

Hm. What about this case?


Should be able to hold 19 cells of water, right? Well, wouldn't you know. Our so-far unblemished program fails this one, with the cryptic error message Merging non-balanced water masses. (Two other programs get the correct 19, and the last one gets 0.)

So I take it back. None of the programs are correct. Pity. But my points about deep model thinking and representative test cases still stands. Correctness is hard!

Next up: distributing weights evenly in bags.