Chapter #157:

The Sheep Count Themselves

On a beautiful day, when the sunshine was exceptionally bright and warm, Bububunny decided to open the windows to let in some summer warmth. Kittyb@ was still sleeping, as it was time for her routine beauty sleep, and the Friendies were busy sweeping up the crumbs of the breakfast sandwiches.

Bububunny opened the windows and let the soft breeze blow through the room. The breeze blew the sandwich crumbs around the room again, and Kittyb@’s coat got a bit tousled, but she didn’t mind. However, a dandelion seed flew into the room with the wind, landing on Bububunny’s nose, and making her want to sneeze. She carefully removed the seed, because she was worried that if she would sneeze, Kittyb@ would wake up. Since Bububunny is interested in a wide variety of things, she gave the seed a closer look.

To her big surprise, she immediately noticed teeny-tiny letters on the seed! She quickly fetched her magnifying glass to see what was written. It was the following sentence: “Help us if you can! One is missing and we cannot find her! Sincerely, B@.”

“Hmmm,” Bububunny thought, “Somebody needs help here. But how do I find out where did the dandelion puff come from? Its real name is pappus, but puff sounds better.”

Bububunny started her computer and browsed the weather forecasts for the last previous days. “I have to look at the direction of wind and the duration of each such direction, and go from there. Here, dandelions stopped blooming a couple days ago, so it cannot have come from far.”

“Friendies, we have a mission!” Bububunny invited them to the computer.

“Looking from our home, the range of our search area is this widening cone, where the wind has blown toward us. But we shall widen the area a bit into each direction, because, sadly, we don’t know the exact direction of the wind,” she said, marking the search area on the computer screen.

The Friendies took a look at the marked area, memorized it, and set out immediately. First, they went directly upwind, reached a fence, turned a bit left, carefully walked across the street, then a bit to the right, and reached a nice meadow. On the meadow, sheep were bustling about, discussing something amongst themselves. The Friendies rushed up close, asked what was wrong, and if they had sent an emergency message using a dandelion puff.

It turned out that the sheep had indeed sent the message, because they knew that Bububunny lives quite nearby, and the wind was from a favorable direction. They had all expressed their opinion about which corner of the meadow would be the best for sending out the puff, then found the mean of all the locations, and after writing the message on the puff, sent it off with a light gust of wind.

So what was the problem about? In the morning, one of the sheep decided to count them all up, just in case. Allegedly, according to an old legend about counting, there should be 137 of them. But since they cannot count much more than up to 17, one of them asked them all to pair up. The number she got was still bigger than she had knowledge of. She then asked the sheep to form groups of 7, but the number she got was still very big. She then had an idea: grouping up in a way to form lines and rows. For example, lining up by 5, and forming 5 rows. After the first group was formed, the others followed. Altogether, there were 5 such groups, and…only 11 sheep were left ungrouped. This makes only 136 sheep! Maybe there was a fault in this method, maybe there was one too many in one of the groups?

The poor sheep ran around that big flock of sheep, looking around and checking carefully if maybe there is one that is not standing in a line of 5. But no matter how carefully she looked, nobody stood in a wrong place!

Then, another sheep walked out of the flock, saying, “Wait a minute, I know a better method for counting us up with less fear of mistakes!”

Of course, the other sheep had nothing against this.

And so they began from the start, “First, let’s take two sheep. You, the one that did the counting before, will be first, and you’ll be second. Let’s add those together, we get two sheep! Got it, now, let’s take two more sheep from the flock, and let’s put them stand here, next to the first two, or the first and the second. How many do we have now? Four sheep, or one sheep, one sheep, and two sheep. Now, let’s take a look at the last two numbers, 1 sheep and 2 sheep. Let’s add those together. We get 3 sheep. Very good—three sheep, would you please be so kind to come and stand here, next to those two sheep. You may graze, but don’t move! Now we have 1 and 1 and 2 and 3 sheep. Next, we shall again add two of the last numbers together. We get 5 sheep. Which means 1, 1, 2, 3, and 5. Now, would 3+5 sheep come here!”

They repeated that method until they reached the tenth addition, where they should have had 55, but got only 48!

“We tried everything, but there is still one sheep missing!” one of the sheep said to the Friendies.

“Hmm…” B@b@ was thinking, “I’ve seen this set somewhere. Of course, it’s 137, the value of the fine-structure constant, as Bububunny explained. It is the thirty third prime, an Einstein prime, a Stern prime, a Pythagorean prime, and so on. Bububunny could talk about the number 137 for hours on end.” But why did the sheep only get 136?

Fofefa quickly jumped by each of the sheep, and, upon coming back, she could only confirm, “Fofefa!” There were 137 sheep.

Then, B@b@ had an idea. She told the sheep to get in the following formation: the first row has 1 sheep, and the next has 2 sheep. Now this makes 3 sheep. The next row has 4 sheep, and those 4 add to the sum, which is now 10. Each next row gives one more sheep, until they reach row 16. Now, all the sheep were in formation, and according to the above described calculations, this big sheep triangle has 136 sheep! And one sheep was left, so this makes 137 sheep!

For the future, the Friendies left the sheep instructions to form a triangle, and when exactly one sheep is left after forming this triangle—the one who checks that everyone is in the right row—it means that there are 137 of them. The sheep were very happy and grateful for those instructions, and gifted the Friendies some wild strawberries—the ones that grew on that meadow—to eat on their way home. The Friendies did each sheep a b@, and happily went back home.

Bububunny giggled happily upon hearing about the problem and the solution, but she did it quietly, because Kittyb@ was taking her afternoon nap, and she didn’t want to wake her up.

One question still remained, how did they manage to produce such small letters on the dandelion puff? It’s a good thing that we still remembered. Namely, sheep are extremely short-sighted, and therefore their vision is very clear when looking at things up-close. Now taking a dandelion seed—a pappel, to be precise—and asking a honey bee’s help by telling her what to write, she will kindly scrape a message onto the seed using her tiny foot.


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