1. First solution: Since the Rabbit ran at a speed twice that of Alice, by the time Alice arrived at the Duchess's, the Rabbit was again halfway. Since he was 10 minutes late, Alice spent 20 minutes on half the journey, and 40 minutes on the entire journey.
Second solution: Let the time it took for Alice to walk from the Rabbit's house to the Duchess's house be $t$ minutes. The Rabbit walked half the distance with Alice, which took him $\frac{t}{2}$ minutes. Then he ran a distance equal to $\frac{3}{2}$ of the distance from his house to the Duchess's house. Since he ran twice as fast, it took him half the time, i.e., $\frac{3 t}{4}$ minutes. In total, the Rabbit spent $\frac{5 t}{4}$ on the entire journey. The Rabbit was 10 minutes late, so $\frac{5}{4} t - t = 10, t = 40$ minutes.

Answer: The reception was scheduled for 12:40.