Once upon a time in a fairy-tale palace there lived a king with his only daughter. The princess was a knock-out beauty. The king would have had an easy job finding her a partner among the richest and noblest princes, had in not been for an evil curse. Years earlier, the king had ordered a wicked witch to be burned at the stake. Just before she died in the flames, the witch cast a spell on the young princess, to the effect that the first man to kiss her would immediately turn into a frog. As even the mightiest of the king's wizards proved incapable of breaking the curse, the king could think of nothing better but to order his old butler to sacrifice himself and kiss the girl so she could be wed at last. The loyal servant protested loudly, begging to be beheaded instead, and also the princess was decidedly not pleased at the prospect of having the old man kiss her. She implored her father not to use force to resolve the matter. Instead, she promised to come up with a better idea the next day.

Luckily, the princess was not just good-looking; she had brains too. The next morning she came up with an idea that the king agreed to. He let it be announced in every town that the hand of his daughter and the future rule over the kingdom could be gained by any young man who would be willing to take a great risk. Soon, hundreds of adventurers started arriving at the gate. When a sufficient number of them had collected, they were let in to the central hall of the palace, where the princess explained the nature of her predicament. "Young men", she spoke, "there is nothing I would love to do dearer, than to take one of you as my husband. But first, I will need to be rid of my curse by being kissed for the first time. I cannot ask any of you to make this sacrifice for me, for such a person would be turned into a frog for the rest of his life, without the least chance of being released. Therefore, I will not ask any of you to kiss me for the first time. What I do ask of you, however, is to take the risk." After speaking these words, the princess paused for a while. An exited murmur spread through the hall and the young adventurers looked at each other uneasily.

"I hope to find two of you brave enough to take this great risk in order to free me of my curse and allow me to marry and continue the family line of my illustrous father. If I find two of you with sufficient courage, then things will go like this: these two men will spend the night in two rooms of our palace. Some time during those nights, probably close to the morning, I will visit each of them in his room and ask him to kiss me. The first who does will turn into a frog and the second becomes my husband. But I will not let them in any way know whether they are first or second. The walls of the rooms are thick and they are sufficiently separated for not a sound to pass from one to the other. In this way, each will have a fair chance to win my hand and the future rule over the kingdom. Should the first man, once I appear before him, lose his courage and refuse to kiss me, then I will replace him and repeat my game the next night. Should the second man refuse, because he does not know that he is second and I have already been released of my curse, then I will for sure not marry such a man but he will be sent off in schame, just like the first one, if he fails to kiss me. Think well, therefore, and only take on this task if you are really sure that you are brave enough for it."

A deep silence spread through the hall after the princess had spoken. Finally, a young and handsome prince, let us call him prince Know-it-all, dared to ask a question. "Most noble princesss, may I ask how you determine the time when you will appear before these young men?" The princess clearly thought about this before and answered immediately. "I will not come earlier than at eight in the evening nor later than at six in the morning. For me to be kissed and move from one room to the other, five minutes suffice. This morning I have thrown a dice to decide the times when I will visit the two rooms. I said to myself: if in my first throw a six comes up, then I will visit the first man at five to six and the other at six. If in my first throw something else comes up, then I will throw again. Now if my second throw yields a six, I will visit the the first man at ten to six and the other at five to six, and so on. I will throw my dice until the first six turns up, counting backwards in periods of five minutes, until I have chosen the times of my visits. So I did, this morning, and the dice has decided. I already know the times, but will not tell you."

Again, silence fell. When it appeared there were no further questions, the princess asked those courageous enough to take the challenge to come forward. Reluctantly, a few men stepped forward, then more followed. The princess chose two of the handsomest men, who suited her taste. A huge banquet was served to all those present. Then, shortly before eight in the evening, the young men who were selected by the princess were locked in their rooms, the keys of which were handed to the princess. One of them was prince Know-it-all.

As he heard the key turn and his room being locked, a shiver ran down his spine. He immediately wondered when the princess would appear. A clock over his bed showed two minutes before eight. With a shock, he realized she might come already in two minutes. For that to happen, as a little calculation revealed to him, she would have had to throw something else than six for fifty nine times in a row, and then a six. Even by fairy tale standards, that is a pretty unlikely event, but possible nonetheless. "If the princess shows up straight at eight", prince Know-it-all considered, "then I can be sure to be the first one, because she said she would not appear before eight." Immediately, he decided that in such a situation, he would not kiss her, since it was still better to be called a coward than to be turned in a frog dead sure.

In the meantime, the clock showed two minutes past eight. "She might come at five minutes past eight as well", he tought. "But in such a case, if I am the second one she visits, then she has visited my rival at eight. In such a case, he must have tought just as I did and not have kissed her. But if he refuses to kiss her, then she will not come to me at all but instead repeat her game the next night. Which means that if she appears here at five past eight, I can be sure to be the first one!" And again prince Know-it-all decided that should the princess arrive at five past eight, he would not kiss her.

When the term of five past eight had passed, he realized that his rival for sure reasoned in the same way as he did and would not have kissed the princess if she had come to him at five past eight. Which, again, meant that if she appeared at ten past eight, then he could be sure to be the first. So he decided against kissing at ten past eight as well. The same reasoning also forced him to resolve not to kiss at a quarter past eight and so on, the whole night long. He did not sleep a wink, endlessly trying to find the error in his reasoning but always arriving at the same conclusion: he could not kiss the princess at any time, since her arrival inescapably had to mean that he was the first one. Finally, dawn was coming and birds started to sing behind the window. Exactly at six o' clock, he heard the key being turned in the lock and in stepped the princess, more breathtaking than ever in her nightgown. In a corner of the room sat prince Know-it-all, his bed unslept on, his hair unkempt, staring emptily before him. Drained of all mind power, he did not realize that as she had promised not to come later than at six, her presence could only mean that he was second. Resolutely, he refused to kiss her and for that reason, as promised, he was sent away in shame.

Continue: read the mathematical epilogue or the fairy-tale epilogue.