diff --git a/crates/border-wars/src/hex.rs b/crates/border-wars/src/hex.rs
index 627f48a..c7902ce 100644
--- a/crates/border-wars/src/hex.rs
+++ b/crates/border-wars/src/hex.rs
@@ -1,10 +1,10 @@
-//! All fonction related to calculations in hexagonal grid.
+//! All functions related to calculations in a hexagonal grid.
 
 use std::collections::HashSet;
 use std::ops;
 
-/// Hexagonal position (in a hexagonal grid).
-/// We are using the axial coordinate system explained in this
+/// Represents a position in a hexagonal grid.
+/// We use the axial coordinate system explained in this
 /// [documentation](https://www.redblobgames.com/grids/hexagons/#coordinates).
 #[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)]
 pub struct HexPosition {
@@ -16,19 +16,20 @@ pub struct HexPosition {
 }
 
 impl HexPosition {
-    /// Returns the distance between two [HexPosition]
+    /// Returns the distance between two [HexPosition]s.
     ///
-    /// # How does it work ?
+    /// # How it works
     ///
-    /// Hexagonal grid using the
+    /// In the hexagonal grid, using the
     /// [cube coordinate system](https://www.redblobgames.com/grids/hexagons/#coordinates),
-    /// is like a cube in 2D space.
-    /// The Manhattan distance between two positions is equal to:  the half of
-    /// the sum of abs(dx) + abs(dy) + abs(dz)
+    /// it's akin to a cube in 3D space.
+    /// The Manhattan distance between two positions is equal to half of
+    /// the sum of abs(dx) + abs(dy) + abs(dz).
+    /// However, in hexagonal grids, z is defined as -q - r.
     ///
-    /// # Example:
+    /// # Example
     ///
-    /// ```no_run
+    /// ```
     /// use border_wars::hex::HexPosition;
     ///
     /// let a = HexPosition { q: 0, r: 0 };
@@ -37,32 +38,31 @@ impl HexPosition {
     /// assert_eq!(a.distance_to(&b), 2.0);
     /// ```
     pub fn distance_to(&self, other: &Self) -> f32 {
-        // dx = |x1 - x2| and x = q
+        // dx = |x1 - x2| where x = q
         let dq = self.q - other.q;
 
-        // dy = |y1 - y2| and y = r
+        // dy = |y1 - y2| where y = r
         let dr = self.r - other.r;
 
-        // dz = |z1 - z2| and z = -q - r
-        let ds = self.q + self.r - other.q - other.r;
+        // dz = |z1 - z2| where z = -q - r
+        let dz = self.q + self.r - other.q - other.r;
 
         // Manhattan distance = (abs(dq) + abs(dr) + abs(ds)) / 2
-        (dq.abs() + dr.abs() + ds.abs()) as f32 / 2.
+        (dq.abs() + dr.abs() + dz.abs()) as f32 / 2.0
     }
 
     /// Returns all positions within a given `range` from the current
-    /// HexPosition.
+    /// `HexPosition`.
     ///
     /// This function iterates over the possible q and r values within the
-    /// specified range and inserts HexPositions into a HashSet, ensuring
-    /// that each generated position is within the given range from the
-    /// current position.
+    /// specified range.
+    /// Note that the original position is also returned.
     ///
-    /// for more details: https://www.redblobgames.com/grids/hexagons/#range
+    /// For more details, refer to: https://www.redblobgames.com/grids/hexagons/#range
     ///
-    /// # Example:
+    /// # Example
     ///
-    /// ```no_run
+    /// ```
     /// use border_wars::hex::HexPosition;
     ///
     /// let position = HexPosition { q: 0, r: 0 };