diff --git a/2023/day6/src/main.rs b/2023/day6/src/main.rs
index 4ef05d3..4818c88 100644
--- a/2023/day6/src/main.rs
+++ b/2023/day6/src/main.rs
@@ -6,6 +6,33 @@ use nom::{
     sequence::{delimited, preceded, separated_pair, tuple},
     IResult,
 };
+use std::str::FromStr;
+
+#[derive(PartialEq, Eq, Clone, Copy)]
+#[allow(dead_code)]
+enum Approach {
+    BruteForce,
+    Math,
+}
+
+impl FromStr for Approach {
+    type Err = &'static str;
+
+    fn from_str(s: &str) -> Result<Self, Self::Err> {
+        Ok(match s {
+            "bruteforce" => Self::BruteForce,
+            "math" => Self::Math,
+            _ => return Err("unknown approach"),
+        })
+    }
+}
+
+#[allow(dead_code)]
+impl Approach {
+    fn values() -> Vec<Self> {
+        vec![Self::BruteForce, Self::Math]
+    }
+}
 
 fn number(i: &str) -> IResult<&str, usize> {
     map(digit1, |f: &str| f.parse::<usize>().unwrap())(i)
@@ -21,6 +48,79 @@ struct Race {
     distance: usize,
 }
 
+impl Race {
+    fn wins(&self, approach: Approach) -> usize {
+        match approach {
+            Approach::BruteForce => (0..=self.time)
+                .filter(|hold_time| {
+                    let time_travelled = self.time - hold_time;
+                    let speed = hold_time;
+                    let distance_travelled = time_travelled * speed;
+
+                    distance_travelled > self.distance
+                })
+                .collect::<Vec<_>>()
+                .len(),
+            _ => {
+                // the races form a quadratic function:
+                //
+                // T = the total time of the race
+                //
+                // t = charging time, in [0..=self.time]
+                // y = resulting distance
+                //
+                // y = (T - t) x t
+                // y = -t^2 + Tt
+                //
+                // By substracing the winning time, we know that all values of x *above*
+                // y = 0 are winings
+                //
+                // D = winning distance
+                //
+                // y = -t^2 + Tt - D
+                //
+                // We need to calculate the roots of that function using the quadratic formula,
+                // with
+                //
+                // a = -1
+                // b = T
+                // c = -D
+
+                // use isizes to enable negation and pow/sqrt
+                let a: f64 = -1.0;
+                let b: f64 = self.time as f64;
+                let c: f64 = -(self.distance as f64);
+
+                let roots: Option<(f64, f64)> = {
+                    let discriminant = (b.powi(2)) - 4.0 * a * c;
+                    if !discriminant.is_sign_positive() {
+                        None
+                    } else {
+                        Some((
+                            (-b + f64::sqrt(discriminant)) / (2.0 * a),
+                            (-b - f64::sqrt(discriminant)) / (2.0 * a),
+                        ))
+                    }
+                };
+
+                if let Some((x1, x2)) = roots {
+                    // Sort the roots by size. The order is determined by the sign of a, so it's
+                    // constant for our input, but this solution is more general
+                    let (x1, x2) = if x1 > x2 { (x2, x1) } else { (x1, x2) };
+                    assert!(x1 < x2);
+
+                    // We actually found two roots. All whole integers that lie *between* the
+                    // roots are races that we win. Exact matches are a tie and do not win.
+                    (x2.ceil() as usize - 1) - (x1.floor() as usize + 1) + 1
+                } else {
+                    // No roots, there is no way for us to win the race.
+                    0
+                }
+            }
+        }
+    }
+}
+
 #[derive(Debug)]
 struct RaceSheet {
     races: Vec<Race>,
@@ -54,7 +154,7 @@ impl RaceSheet {
     }
 }
 
-fn part1(input: &str) -> Result<usize, String> {
+fn part1(input: &str, approach: Approach) -> Result<usize, String> {
     let (rest, racesheet) = RaceSheet::parse(input).map_err(|e| e.to_string())?;
     if !rest.is_empty() {
         eprintln!("parsing rest found: {rest}");
@@ -63,23 +163,12 @@ fn part1(input: &str) -> Result<usize, String> {
     let result = racesheet
         .races
         .into_iter()
-        .map(|race| {
-            (0..=race.time)
-                .filter(|hold_time| {
-                    let time_travelled = race.time - hold_time;
-                    let speed = hold_time;
-                    let distance_travelled = time_travelled * speed;
-
-                    distance_travelled > race.distance
-                })
-                .collect::<Vec<_>>()
-                .len()
-        })
+        .map(|race| race.wins(approach))
         .product();
     Ok(result)
 }
 
-fn part2(input: &str) -> Result<usize, String> {
+fn part2(input: &str, approach: Approach) -> Result<usize, String> {
     let (rest, racesheet) = RaceSheet::parse(input).map_err(|e| e.to_string())?;
     if !rest.is_empty() {
         eprintln!("parsing rest found: {rest}");
@@ -102,18 +191,9 @@ fn part2(input: &str) -> Result<usize, String> {
         .parse::<usize>()
         .unwrap();
 
-    let result = (0..=time)
-        .filter(|hold_time| {
-            let time_travelled = time - hold_time;
-            let speed = hold_time;
-            let distance_travelled = time_travelled * speed;
+    let race = Race { time, distance };
 
-            distance_travelled > distance
-        })
-        .collect::<Vec<_>>()
-        .len();
-
-    Ok(result)
+    Ok(race.wins(approach))
 }
 
 fn main() -> Result<(), String> {
@@ -121,13 +201,14 @@ fn main() -> Result<(), String> {
 
     let args = std::env::args().skip(1).collect::<Vec<String>>();
     let part = args[0].parse::<usize>().unwrap();
+    let approach = args[1].parse()?;
 
     if part == 1 {
-        println!("Part 1 : {}", part1(input)?);
+        println!("Part 1 : {}", part1(input, approach)?);
     } else if part == 2 {
-        println!("Part 2 : {}", part2(input)?);
+        println!("Part 2 : {}", part2(input, approach)?);
     } else {
-        panic!("unknown part")
+        return Err("unknown part".into());
     }
 
     Ok(())
@@ -145,7 +226,9 @@ mod tests {
             Distance:  9  40  200
         "};
 
-        assert_eq!(part1(&input).unwrap(), 288);
+        for approach in Approach::values() {
+            assert_eq!(part1(&input, approach).unwrap(), 288);
+        }
     }
 
     #[test]
@@ -155,6 +238,8 @@ mod tests {
             Distance:  9  40  200
         "};
 
-        assert_eq!(part2(&input).unwrap(), 71503);
+        for approach in Approach::values() {
+            assert_eq!(part2(&input, approach).unwrap(), 71503);
+        }
     }
 }