IBM Ponder This #238 · February 2018

This month's challenge was inspired by a suggestion from Hamidreza Bidar (thanks!).

Three people play a game. Each one of them has an independent fair die with n_i faces (n_1 faces for the first person, n_2 for the second, and n_3 for the third).
They play a game where each one of them tosses his die once every minute until he gets all the possible outcomes. They all play simultaneously and the game ends when all the players have finished the game.
Find possible n_i such that the expected time till the end of the game, rounded to the nearest second, is 2569.

Solution

Best opened after a real attempt

To be added.