Using Binary to Warn of Snow Beasts

Binary is a number system where each digit can only take one of two values: zero or one. Binary is used within computers, because it allows the computer to encode information in a series of switches that are either on (1) or off (0). Each digit of binary represents a power of two. The first (right-most) digit represents the 1's place, the second digit represents the 2's place, the third represents the 4's place, and so forth. For example, the binary number 10110 = 1*2^4 + 0*2^3 + 1*2^2 + 1*2^1 + 0*2^0 = 22 in decimal.

The outpost of Iceton stood on the northern edge of the kingdom. It was little more than a scrawny barracks surrounded by a short fence and miles of tundra. Snow fell during most of the year, except in the winter months when it was too cold for snow. Few soldiers were stationed there by choice.

Despite the outpost's persistent recruiting problems, it was absolutely vital to the safety of the kingdom. North of Iceton roamed dangerous snow beasts. The beasts resembled elephant-sized polar bears with sharp antlers. They had nasty tempers and a particular fondness for attacking the kingdom's outposts. Iceton watched for these incoming threats and warned the cities to the south.

Since few pigeon messengers could survive Iceton's wintery conditions, giant fires signaled any impending danger. A fire atop one of Iceton's large signal towers indicated an incoming snow beast. The lack of a fire indicated the comparative safety of frostbite, hypothermia, and tundra goblins. Iceton's neighbor to the south, Garroow, watched for signals of trouble from Iceton and relayed the warning further south.

One day, a Garroow guard noticed a troubling sight: two fires burning on the Iceton signal tower. Being that Garrow's entire platoon was new to the outpost, no one knew what two fires meant. After much discussion, Garroow's commander dispatched a rider to investigate. The unlucky soldier, Mavis Yates, arrived at Iceton a few hours later.

"What news do you bring us from Garroow?" asked Iceton's commander with a note of worry.

"I came to investigate the fires," answered Mavis. "What do they mean?"

Commander Dorn looked confused. "They mean the same thing they always have -- snow beasts are heading toward Garroow."

"Why are there two of them?" asked Mavis.

"There are two snow beasts," answered the commander as though the answer was obvious; which, in retrospect, it was.

Moreover, this confirmation was certainly not worth a two hour trek through the tundra. Mavis, desperately wishing to avoid any future journeys to Iceton, decided to clarify. "And three fires would mean three snow beasts?"

"Well…" stalled the commander, "Three or more. We only have space for three signal fires. If you see three fires, you will have to come investigation. There could be five snow beasts heading in your direction."

Mavis had no intention of repeating this journey. Especially not if there were three or more snow beasts heading toward her.

"Oh, no." Mavis protested. "There has to be a better way."

Commander Dorn shook his head. "I assure you that this is how we have communicated for the last ten years. It is quite effective. We only see packs of more than two snow beasts a few times a year."

Mavis shivered involuntarily. "We can do better with three fires. Why not use binary?"

"Binary?" asked the commander.

"Binary," confirmed Mavis. "Each signal fire can indicate a power of two."

"Power of two?

"From East to West they will represent powers of 0, 1, and 2. That is 2^0=1, then 2^1=2, and then 2^2=4. We can encode a lot more information that way."

"That will only tell you that there are 1, 2, or 4 snow beasts coming. How does that help?" protested the commander.

"You add the digits that have fires," explained Mavis. "Let's say the first and last fires are lit. That means there are 1 + 4 = 5 snow beasts."

Mavis walked to the wall, grabbed her hunting knife, and began to carve the following code into the wall:

NO-FIRE, NO-FIRE, NO-FIRE = 0 snow beasts

NO-FIRE, NO-FIRE, FIRE = 1 snow beasts

NO-FIRE, FIRE, NO-FIRE = 2 snow beasts

NO-FIRE, FIRE, FIRE = 3 snow beasts

FIRE, NO-FIRE, NO-FIRE = 4 snow beasts

FIRE, NO-FIRE, FIRE = 5 snow beasts

FIRE, FIRE, NO-FIRE = 6 snow beasts

FIRE, FIRE, FIRE = 7 snow beasts

She stepped back and examined her work.

"Binary," she stated.

The commander stared at the code. "I think it will work," he remarked finally.

"What happens if there are eight snow beasts?" asked one of the commander's aids. "Should someone from Garroow come and investigate?"

"I do not think it is worth worrying about that," the commander answered without looking away from the wall. Mavis smiled. She had found a way to avoid future treks altogether.

"After six, it does not matter much," continued the commander. "When there are that many, they skip Iceton and head straight for Garroow. It is not worth their trouble here. And all that Garroow can do is warn the other cities and flee south."

Mavis's smile vanished. "Wait… what?" she asked.

"Oh… right… you are from Garroow." responded the commander. "Umm.. Good luck with that."

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To learn more about binary, also read Unhappy Magic Flowers and Binary. Or read about Ann's visit to Garroow in The Important of Variable Names.

Unhappy Magic Flowers and Binary

Binary is a number system where each digit can only take one of two values: zero or one. Binary is used within computers, because it allows the computer to encode information in a series of switches that are either on (1) or off (0). Each digit of binary represents a power of two. The first (right-most) digit represents the 1's place, the second digit represents the 2's place, the third represents the 4's place, and so forth. For example, the binary number 10110 = 1*2^4 + 0*2^3 + 1*2^2 + 1*2^1 + 0*2^0 = 22 in decimal.

The deliveryman paused outside of Marcus's house. He stood transfixed, staring at the flowers. After two minutes, Marcus finally went outside to see what the problem was.

"Your flowers have changed since yesterday." observed the deliveryman. "I am sure that the one on the right was red yesterday. Today it is blue."

"No. Those are the same flowers." responded Marcus. "Some of them are just sulking today. Stupid flowers."

"Sulking? Do flowers sulk?" the delivery man asked.

"These flowers do. They are, of course, magic. If there is not enough rain, they sulk. It is quite aggravating really. It is not as though they are not getting water. I water them every day. Yet, they still insist on sulking when there is no rain. It appears that the water I give them is not good enough for them."

"Huh?" The deliveryman looked most confused. He looked back and forth between Marcus and the flowers.

"They sulk by changing to blue. Roses are supposed to be red. That is why so many poems start out that way. But, they insist on telling me how long it has been since it rained." Marcus gestured angrily at the roses as he spoke.

"They talk to you?" The deliveryman took a step back away from Marcus. He looked as though he was ready to run.

"Of course not. They simply change color." Marcus sighed.

"And, that tells you how long it has been without rain?"

"Well…" started Marcus. "They used to all change color together after three days without rain. A sort of mass protest. But, then they started to organize. They want to let me know exactly how unhappy they are. So, now they count the days."

"But… only two are blue. It has not rained all week." observed the deliveryman.

"Nine days to be exact." corrected Marcus after looking at the flowers. "They use binary."

"Huh?"

"Binary. Red flowers are zero and blue flowers are one." Marcus added.

The deliveryman did not look as though he was following. But, at least he did not look like he was about to run away anymore.

"Binary?" prompted Marcus. "Each flower represents a different digit, and thus a different power of two. The rightmost flower means one (2^0), the one next to it means two (2^1), the one next to that means four (2^2), and so forth. Add up the numbers represented by the blue flowers and you get the total number of days. Right now only first (2^0 = 1) and fourth (2^3 = 8) flowers are blue, so it has been 2^0 + 2^3 = 1 + 8 = 9 days."

The deliveryman looked. Sure enough, the five flowers across Marcus's porch were: Red Blue Red Red Blue (or 01001).

"Why do they use binary?" asked the deliveryman.

"They tried to spell out the numbers on their petals and they got too confused. So they had to settle for each flower being either all red or all blue. It turns out that flowers are not that smart. Binary is a simple enough system for them. If they were smart enough for anything else, do you think they would be complaining to me about the rain? There is nothing I can do!" Marcus shouted the last part directly at the flowers.

"But, how do they work together?" The more absurd the story got, the more interested the deliveryman became. He took a step forward towards the flowers.

"It is really quite simple for them to count in binary. When it rains, they are all happy and turn red. It is like reseting the counter to 00000. I like those days a lot." started Marcus.

"Then, each morning all of the flowers wake up and decide what color they are going to be for the whole day. If it has not rained, they increase the count.

After 1 day they are: Red Red Red Red Blue (00001 = 1)

After 2 days they are: Red Red Red Blue Red (00010 = 2)

After 3 days they are: Red Red Red Blue Blue (00011 = 2 + 1 = 3)

After 4 days they are: Red Red Blue Red Red (00100 = 4)

After 5 days they are: Red Red Blue Red Blue (00101 = 4 + 1 = 5)

And so on.

"You see, each flower looks to its right in order to decide what to do. If its right-hand neighbor changes from blue to red (1 to 0), then the flower flips its own color. I like to think of it as passing along the unhappiness. Basically, if its neighbor is changing from blue to red then the sullenness is rolling over to the current flower. If it is red, then it becomes blue. And if it is blue, it becomes red and passes the sullenness further down the line. This keeps happening until one of the flowers does not change from blue to red."

The deliveryman thought about it. "What if a flower changes from red to blue (0 to 1)?"

"Its neighbor does nothing. Didn't I just say that?" Marcus was impatient. He felt like he was talking to one of his apprentices. "Think about it the way you would counting with numbers 0-9. When you hit 9, you roll the first digit back to 0 and increment the next digit to 1. That next digit stays where it is until its right-hand neighbor switches from 9 to 0 again. You don't count 10, 11, 12, 23. Do you? NO! You wait until the digit to your right rolls over. Only here there are exactly two options 0 and 1, so things roll over more frequently."

"What about the right-most flower?" asked the deliveryman. "How does it know what to do?"

"Ah. THAT ONE IS THE INSTIGATOR! I am sure that he is the one that started it. Every morning there is no rain, he flips. He is the one that starts the process off. Red to blue to red to blue to..."

"And that system always works?" interrupted the deliveryman.

Marcus tore his attention away from the right-most flower. He suddenly wondered how the discussion had gone from a rant about magic flowers to counting in binary. Did the deliveryman not have any other deliveries? For that matter, where was the delivery for Marcus?

"Yes. They already counted out nine days, haven't they?" answered Marcus flippantly.

Then, seeing the look of interest on the deliveryman's face, Marcus returned to his teaching tone. "Imagine that these three flowers were blue." explained Marcus as he pointed to the first three flowers. "That would be 1 + 2 + 4 = 7 days without rain. The next day the first flower (1) switches to red, so the second flower (2) switches to red, so the third flower (4) switches to red, so the fourth flower (8) switches to blue. Then they would show 8 days without rain. The count effectively gets pushed down into the fourth flower."

The deliveryman looked impressed. Marcus could not understand why. The flowers were really very annoying.

"So tomorrow they will be: Red Blue Red Blue Red (01010)?" asked the deliveryman.

"Probably. But, I really hope that it just rains." responded Marcus with a sigh.