## \-\-- Day 13: Claw Contraption \-\--

Next up: the [lobby](/2020/day/24) of a resort on a tropical island. The
Historians take a moment to admire the hexagonal floor tiles before
spreading out.

Fortunately, it looks like the resort has a new
[arcade](https://en.wikipedia.org/wiki/Amusement_arcade)! Maybe you can
win some prizes from the [claw
machines](https://en.wikipedia.org/wiki/Claw_machine)?

The claw machines here are a little unusual. Instead of a joystick or
directional buttons to control the claw, these machines have two buttons
labeled `A` and `B`. Worse, you can\'t just put in a token and play; it
costs *3 tokens* to push the `A` button and *1 token* to push the `B`
button.

With a little experimentation, you figure out that each machine\'s
buttons are configured to move the claw a specific amount to the *right*
(along the `X` axis) and a specific amount *forward* (along the `Y`
axis) each time that button is pressed.

Each machine contains one *prize*; to win the prize, the claw must be
positioned *exactly* above the prize on both the `X` and `Y` axes.

You wonder: what is the smallest number of tokens you would have to
spend to win as many prizes as possible? You assemble a list of every
machine\'s button behavior and prize location (your puzzle input). For
example:

    Button A: X+94, Y+34
    Button B: X+22, Y+67
    Prize: X=8400, Y=5400

    Button A: X+26, Y+66
    Button B: X+67, Y+21
    Prize: X=12748, Y=12176

    Button A: X+17, Y+86
    Button B: X+84, Y+37
    Prize: X=7870, Y=6450

    Button A: X+69, Y+23
    Button B: X+27, Y+71
    Prize: X=18641, Y=10279

This list describes the button configuration and prize location of four
different claw machines.

For now, consider just the first claw machine in the list:

-   Pushing the machine\'s `A` button would move the claw `94` units
    along the `X` axis and `34` units along the `Y` axis.
-   Pushing the `B` button would move the claw `22` units along the `X`
    axis and `67` units along the `Y` axis.
-   The prize is located at `X=8400`, `Y=5400`; this means that from the
    claw\'s initial position, it would need to move exactly `8400` units
    along the `X` axis and exactly `5400` units along the `Y` axis to be
    perfectly aligned with the prize in this machine.

The cheapest way to win the prize is by pushing the `A` button `80`
times and the `B` button `40` times. This would line up the claw along
the `X` axis (because `80*94 + 40*22 = 8400`) and along the `Y` axis
(because `80*34 + 40*67 = 5400`). Doing this would cost `80*3` tokens
for the `A` presses and
`40*1` for the `B` presses, a total of `280` tokens.

For the second and fourth claw machines, there is no combination of A
and B presses that will ever win a prize.

For the third claw machine, the cheapest way to win the prize is by
pushing the `A` button `38` times and the `B` button `86` times. Doing
this would cost a total of `200` tokens.

So, the most prizes you could possibly win is two; the minimum tokens
you would have to spend to win all (two) prizes is `480`.

You estimate that each button would need to be pressed *no more than
`100` times* to win a prize. How else would someone be expected to play?

Figure out how to win as many prizes as possible. *What is the fewest
tokens you would have to spend to win all possible prizes?*

Your puzzle answer was `30973`.

The first half of this puzzle is complete! It provides one gold star: \*

## \-\-- Part Two \-\-- {#part2}

As you go to win the first prize, you discover that the claw is nowhere
near where you expected it would be. Due to a unit conversion error in
your measurements, the position of every prize is actually
`10000000000000` higher on both the `X` and `Y` axis!

Add `10000000000000` to the `X` and `Y` position of every prize. After
making this change, the example above would now look like this:

    Button A: X+94, Y+34
    Button B: X+22, Y+67
    Prize: X=10000000008400, Y=10000000005400

    Button A: X+26, Y+66
    Button B: X+67, Y+21
    Prize: X=10000000012748, Y=10000000012176

    Button A: X+17, Y+86
    Button B: X+84, Y+37
    Prize: X=10000000007870, Y=10000000006450

    Button A: X+69, Y+23
    Button B: X+27, Y+71
    Prize: X=10000000018641, Y=10000000010279

Now, it is only possible to win a prize on the second and fourth claw
machines. Unfortunately, it will take *many more than `100` presses* to
do so.

Using the corrected prize coordinates, figure out how to win as many
prizes as possible. *What is the fewest tokens you would have to spend
to win all possible prizes?*

Answer:

Although it hasn\'t changed, you can still [get your puzzle
input](13/input).