Advent of Code 2023 - Day 6Wait For It

| Problem statement | Source code | Tags: Mathematics

Part 1

If I held the button for $x$ milliseconds to reach speed $x$, then the distance traveled in $t$ milliseconds is $(t - x)\cdot x$. I need this to be more than $d$. Rearranging gives $x^2 - tx + d < 0$, which means I can't win if $\Delta = t^2 - 4d < 0$. Otherwise, I need $(t - \sqrt{\Delta})/2 < x < (t + \sqrt{\Delta})/2$. I just need to count the integers in this range.

The number of integers in the exclusive range $(a, b)$ is $\max(0, \lceil b \rceil - \lfloor a \rfloor - 1)$. Why? Because $[\lceil b \rceil, \infty)$ and $(-\infty, \lfloor a \rfloor]$ are the integers outside the range.

winning_vals <- function(row) {
  t <- row[["times"]]
  d <- row[["distances"]]
  discriminant <- t^2 - 4 * d
  if (discriminant < 0) {
    return(0)
  }
  x1 <- (t - sqrt(discriminant)) / 2
  x2 <- (t + sqrt(discriminant)) / 2
  ceiling(x2) - floor(x1) - 1
}

Part 1 is the product of applying this function to each row.

Part 2

Because we aren't doing simulations, just calculations, making the input larger doesn't change anything. Previously it was wins <- apply(parsed_data, 1, winning_vals); now it's wins <- winning_vals(list(times = times, distances = distances)).