I haven't and probably won't write up problems 2 through 5 and to be honest, don't know how many days I'll go on solving this year. I thought that today's problem was interesting enough for a write up so if you're interested, keep reading.

Here's a link to the problem: https://adventofcode.com/2023/day/6

And my solution(s) are here.

What I liked about today's problem is that there are various, let's say levels to the solution(s).

For part 1 you have a series of races. Each race has a time length and the record distance. You press the "accelerate" button for as long as you want and then release it which releases your toy boat.

In the example they give, the time is 7 and the record distance is 9.

So, if you press for 2, you only can travel for the remaining 5 seconds but your speed will be 2, hence 10 traveled. When you press for 3 your speed becomes 3 but you only have 4 seconds to travel so 12 distance. etc.

You have to figure out how many "press" options yield a distance greater than the distance they specify. In our case, they specified a distance of 9 so there are 4 times that are greater.

You're given a list of races, each with a time limit and a record distance you're trying to beat.

The first way a student might solve this is with straight simulation. Loop over each race and then for each race, loop over the distance to calculate the time. Straightforward and works and a "math phobic" student can get the job done.

The second way is by using a little math. One might notice that the distance traveled given the amount of time you press and the time of the race can be calculated using \(distance = press * (time - press)\)

Then for each race, you can loop over each time and use the equation. A small improvement and also a gentle introduction to how some basic math can clean things up.

Finally, one might notice that using that equation we're looking to solve this relationship:

Which we can write as:

Which is a quadratic equation. If you look at the chart you'll notice that the distance traveled start at 0, go up, and then return to 0. A parabola - the shape graphed by a quadratic.

We can map the above variable names to the more familiar \(ax^2+bx+c=0\) representation as follows:

If one knows the quadratic equation, they can solve for the two roots. The solutions to our problem will be in between them. Yes, you have to be careful on the floating point issues with solving but it means just plugging in to a formula for each race.

The nice thing about this problem is it can be attacked in a few different ways. It also relates the CS solution to HS math and is a really nice way of showing how a quadratic can be useful.

Here's the plan:

1. Have the students make a chart of time and distance like we did up top
2. Chart the graph
3. Note it's a parabola
4. Derive the first equation
5. Show how it's a quadratic
6. Hilarity ensues

Of course you could switch the order of a few steps, do it differently, or branch off afterwards.

Now, if a teacher doesn't know the quadratic formula or about quadratics they couldn't teach this way and that's a limitation and a problem. Should a CS teacher know quadratics? Maybe, maybe not. I'd personally say they should. Regardless how much and what a teacher should know should be a bigger deal than it currently is, particularly in CS where there is far too much teaching to the script.

That, however, is for another day.