The solutions.

     I ordered everybody to gather in the dining-car. The time came for me to present the solution to them. Actually, two distinct solutions. The first one was:

     Mr. Ratchett, who was aware of his one particular enemy gave his description to Mr. Hardman. He was to be a small dark man with a womanish kind of voice. He most likely joined the train at Belgrade, or more likely at Vincovci by the door left unbolted by Colonel Arbuthnot and Mr. MacQueen. Over his ordinary clothes, he wore the conductor's uniform and was provided with the key unlocking every door on the train, so he could easily go into any of the compartments despite them being locked. He stabbed Ratchett, put the weapon in Mrs. Hubbard's sponge-bag, not realising he had lost a button, put the uniform into one of the valises in an empty compartment and left before the train took off again. 

     My companions did not agree at all with the solution above, so I presented them the alternative one.

     Everything was planned way beforehand. The small, dark man with a womanish voice was a fictional character who could apply as well to a man or a woman from outside, who could be blamed for the crime without having an alibi. The idea of stabbing perfectly fit the situation. It was a  silent method and a dagger could be used by anyone, strong or weak. All twelve murderers entered Ratchett's compartment one by one through the communicating door in Mrs. Hubbard's compartment. None of the murderers knew which blow had actually killed Cassetti, but with so many blows, at least one must had done deadly damage. The letter which Ratchett received from the actual murderers was burnt, as it contained the key word - Armstrong, without which none of the passengers could have been suspected. When they realised that a part of their plan was impossible, because of the train running into a snow-drift they left us two 'clues' in Ratchett's compartment to confuse our investigation even more so. One incriminating Colonel Arbuthnot who had the strongest alibi, and the other pointing to Princess Dragomiroff who had an alibi provided by her maid and seemed too weak to stab Cassetti. The Wagon Lit conductor, Mr. Michell must have been involved in the murder, but that gave us 13 persons, not 12. I came to the conclusion that the innocent person would be the one who was to be a prime suspect, someone closest to the Armstrongs, which was Countess Adrenyi, so her husband must have taken her place.

     Justice finally found the monster which Cassetti was. We agreed to present the Yugo-Slavian police with the first option, to avoid ruining the lives, that those people still have to live.

Having placed my solution before you, I have the honour to retire from the case. Farewell, dear friends...