“I had a very strange dream last night,” Tweedledum said to Tweedledee. “I dreamt that we were not twins but quintuplets.”

“What were the names of the others?” asked Tweedledee.

“One of them was called Tweedledoo. I don't remember the other two, but they were also Tweedle-something. We had done something that was only possible in a dream. We made the Queen of Hearts happy. She rewarded us with enough tarts to fill the entire pantry. It was still daytime then, and we were to share the tarts equally in the evening.”

“I wish I have dreams like that, even if it is only a dream. What happened next?”

“The pantry was guarded by the Duchess's Cook. Sometime during the day, she suggested to me that I should make sure that I got my fair share. She took me inside, and helped me divide the tarts into five equal piles. There was one left over. I gave that to her while I ate my pile. Then we put the rest of the tarts back together. During the course of the day, I noticed that you and the other three went into the pantry one at a time with the Duchess's Cook, and each time, she came out eating a tart.”

“So the Duchess's crook made the same crooked deal with all of us.”

“By the time we divided the tarts in the evening, there were a lot less than before. Nobody spoke up. So I assume that each of us was as guilty as the others. The tarts now came out in five equal shares.”

“How many tarts were there altogether?”

“I was too stuffed to count. Let us ask Alice and see if she can figure it out.”

“This problem belongs to the type called Diophantine problems,” said Alice, “named after the Greek mathematician Diophantus. Such problems lead to what are called Diophantine equations. They look just like ordinary algebraic equations, except that only integral solutions are accepted.”

“This is not unreasonable,” said Tweedledum. “The answers to most problems are supposed to be positive integers.”

“So how many tarts were there in my twin brother's dream?”

Bézoutian Algorithm

An important result in the last chapter is the following.