# The Princess and the Pea (Soup)

Once Upon a Time, there was an unbearably snobby queen-mother with a son of marriageable age. The son was smart, charming, and funny, but not one of the princesses that he brought home was good enough for his mother. One day, while daydreaming in his calculus lecture, the prince met his true love. He was giddy with emotion, but realized that convincing his mother would take some work.

When the prince and his intended arrived at the palace for spring break, he found his mother uncharacteristically in the kitchen. “I've made your favorite,” cooed his mother unconvincingly, “split pea soup. My very own recipe. 9 million very small peas per liter. I'm sure your little friend will love it. But if not, well, not everyone is made for royalty.”

The prince looked at the mushy glutinous mass in his mother's cauldron with dismay. He knew that his relationship was doomed if he didn't find a way out. He racked his brain for solutions, but his courseload of calculus, microbiology, organic chemistry, and analysis of Swiss yodeling had left little room for culinary improvement. (In other words, he could barely boil an egg). Frantically he considered derivatives, limit theorems, integrals and natural logarithms, but they all seemed a bit, well, complicated, given that dinner was to be served in six minutes and his mother was out adding a fifth spoon to every place setting.

Just then his younger sister entered the kitchen and saw his miserable face. When he explained the predicament, she just smiled and said “all you need is plain old multiplication”. In vain the prince tried to explain that he needed to dilute the soup, not multiply it. “Exactly!" said his sister. "Anyone can eat a soup that has 90 peas per liter. All we have to do is figure out how to get it diluted to that level. So... "

“That's a dilution factor of 100 to 1. That way, each liter of soup will have only 90,000 peas instead of 9 million!”

“Yeah, but we'll also have 100 liters of soup! There's not enough tupperware here for all that food.”

“OK, that's a good point. I didn't think of that ... Well, we don't need to mix up all 100 liters. All Mumsy expects is a single pot. And all that matters is that we get the ratio right ... hmm ... let's say we started with a hundredth of a pot of the old soup and added 99 hundredths of a pot of water.”

“What do we do with the rest of the soup?”

“Um, keep it for your next girlfriend?”

Luckily, the kingdom measured in metric only. One soup pot held 1 liter of soup, so the sister removed 10 mLs and put it into a clean pot. She then filled the pot to the brim with water. Voila, soup with only 90,000 peas per liter.

The prince gave it a try and promptly spit it back out again. “Blaaah, still too much! Its nasty” he said, just as his second sister came in. A quick conference between sisters brought the second sister up to speed. “Let's do the same thing again!” said the second sister enthusiastically.

“Great, it didn't work the first time, so now we're going to do it again?”

“Yes, we already got it partway diluted, now we'll just dilute it some more.”

So Sister #2 took 10 mL of the diluted soup, put it in a third pot, and filled that pot to the brim with water.

The prince gave the double-diluted soup a try. “Still too strong,” he muttered. Cue Sister #3, another explanation of soup, mothers, and true love, and ... you should be able to see it coming ... a third dilution. But this time, going from 900 to 90 peas per liter required only a ten to 1 dilution. Hence the third sister put 100 mLs of soup (one tenth of the total) into a new pot, and filled the pot full to the liter mark:

The prince tried the triple-diluted soup. “Wow! Fantastic! It tastes just like water!”

At dinner the prince's girlfriend slurped down the watery soup like a trooper.

Sadly, the prince's mother was not blind or totally stupid. She quickly caught on to the soup-dilution scheme, found the leftover stock, dumped it over the prince's head, and sent the girlfriend packing. The prince passed calculus, graduated with slight honors, and lived out his life in bitter isolation. Moral of the story: don't mess with mumsy.

OK, there might be a few other morals:

1. Serial dilution means you do a series of dilutions, where each dilution gets you closer to your goal.
2. Each dilution can be seen as a conversion of a concentrated stock into a diluted stock. This is pretty obvious when you're dealing with factors of 10 or 100, but could be trickier if you had to use less comfortable numbers.
3. It doesn't really matter how much stock you start with, the important thing is to take out the right fraction of it (a tenth, a hundredth) and then add enough water to get back to the initial volume.