Once upon a time, Rapunzel, Queen Dido, my current prince and I accompanied Smart Joe on a business trip to Switzerland. After Smart Joe had signed a contract with a clock factory, we still had plenty of time and decided to do some touristy stuff. We bought five bars of Swiss chocolate and went for a hike in the Alps.
After hiking for a couple of hours I got hungry, so we unpacked the chocolate (which was lucky for us since it was already half molten) and the others started to discuss the best way to divide the chocolate between the five of us.
"I know!" my Prince said triumphantly. "Since we are five people and we have five bars of chocolate, let's just give one bar to each person!"
"Oh, what a bore!" exclaimed Rapunzel. "We don't want to divide the chocolate up in the easiest way, we want to find the mathematically most interesting way." (And to the me she said, "Next time you have to take someone prisoner, be sure to take a genius and not a prince, OK?")
Now Smart Joe had an idea, brilliant as always: "We could just leave the first one whole, divide the second one into halves, the third one into thirds, etc. Then, in order, we assign each piece a number and each person gets a number too. For each piece we then take the piece number mod 5 and the answer will tell us which person gets that specific piece. I'll be number one..."
"No!" cried the prince, "I want to be number one!"
"I have an even better idea," came Queen Dido. "My ancestors, the Egyptians, were faced with a similar problem almost 4000 years ago. Their solution was to give each person one fourth of a bar more than the person before."
"But I want to be number one!" cried the prince.
"OK," was Dido's reply. "And I'll be number five. Rapunzel, you'll be number two, you Dragon will be number three and Joe'll take four. OK?"
"OK," I quickly replied to make an end to this whole thing. "In that case I get exactly one chocolate bar." And before there could be any further discussion, I ate my share so they had to be happy with theirs too.
Can you help the Prince find out how big his share is? Do you need a hint?