# How you look at a problem can make it easier - AOC 2021 Day 7

Today's problem was similar to yesterday's in that it's ease or difficulty really depended on how you looked at the question.

For yesterday, the problem was hard if you approached it by modeling each and every lanternfish but it was much easier if you modeled the 8 days of the reproduction cycle. Sure, there was still work to be done but looking at the problem the right way made things much easier.

Same thing for today, at least for me.

We were given a list of crabs and their locations and the challenge was getting them all to the same spot. Fortunately, crabs in their submarines can only move in one dimension - they can only move horizontally. This gives us an input that's just a list of integers, each representing a crab location.

We want to move all the crabs to the same location and we want to do it as efficiently as possible. The crab submarines use one unit of fuel for each step taken so we want to minimize the overall amount of fuel used.

If we look at this from the crab point of view this could seem daunting - do we move the crabs one at a time? Do we have them converge? Will we have to do some crazy recursive movement thing?

It turns out, none of the above. As soon as we realize that we don't have to figure out actual moves or actually move the crabs the problem becomes much easier. It isn't about the crabs but rather about the locations. We want to figure out the minimal cost of moving all the crabs to each location and then we can pick the lowest overall cost.

This turns out to be rather simple.

For each possible location (looping from smallest crab location to larges):

1. calculate the distance from each crab to this location.
2. Add up all those distances.

We then just need to take the smallest distance and we're done.

On to part two.

For part two, there was a change to the fuel cost. Instead of just being the distance between the two locations, the cost of the fuel increased by one for each step. So, for moving 1 step the cost would be one. Two steps, 1+2, three 1+2+3 etc..

This was easy enough to model. I just changed the function that calculated the old cost with one that calculated the new cost. To do that, I took the difference between the two endpoints and calculated the sum from 1 up to and including that number.

This added an additional loop but I figured might as well try it. It was noticeably slower than part 1 - about 11 seconds as opposed to around 90 msec but fast enough to get an answer and finish the problem for the day. That said, it was easy enough to improve the solution if you know that you can calculate the sum from 1 to n using the formula (n*(n+1))/2. This got the speed back to just over 100 msec.

Looking over the adventofcode subreddit people were talking about using the median to more quickly solve the problem but what I did was good enough for me.

As always, the clojure code can be found here.

Enjoy.