“Since most of you might have started to feel sick at the mention of Amentiae, I will discuss another topic today.”

Sighs of relief could be heard everywhere in the room as Yordar continued.

“It’s today only, however, and afterwards we will go back to our beloved figurines. Anyway! Today’s topic has to do with numbers.”

Mahon was between Zac and Allen and both men sat a little straighter at the mention of a truce in the Amentiae’s topic. The night before, the banquet had been uneventful, and the professors hadn’t shared any new information. Mahon had sat in the furthest seat like a week ago and the table had eaten in an eerie silence, both because of the weird atmosphere of the dinner and because Jorik had shut down every attempt at a discussion around the table.

This morning the training had been twice as hard as usual because, according to Slander, they had a full day of rest, so they had more energy to spend. Even with the lunch break, they were still exhausted, but professor Yordar’s words gave them hope for a better day.

“Lot of people know about the Colors but few know about the numbers. And yet, they were both brought by the Fada. Oh, you probably know about 6, 12 and 28, but there is much more to numbers than those three.”

Yordar went to the wide blackboard behind him and wrote 6, 12 and 28 with a white chalk that screeched every time it was in touch with the board.

“Fada were incredibly smart and knew everything about the different kinds of numbers that exist in our world. They shared some of their knowledge with us and everything we know today is because of them.”

The professor let his few sentences sink into the mind of everyone before pursuing.

“What is important when we talk about numbers is their divisors. A divisor is a number that divides into another number without a remainder. For example, 10 had three divisors apart from itself: 1, 2 and 5. Any other numbers can’t perfectly divide 10. If you try to divide 10 by… let’s say 3, it will not give you a whole number.”

While he explained, he wrote each number on the board to provide the visual display needed to follow his words with ease.

“That’s just simple math. I don’t think anyone has trouble with that. It becomes interesting, however, when you start to look at the divisors of all the numbers. And you may notice that two types of numbers appear.”

Yordar drew a tall vertical line that separated his board into two identical parts. He approached the left part and wrote the number 10.

“There are those like 10 which have divisors that add up to less than itself. Here, 1 plus 2 plus 5 gives 8, which is less than 10. Trivial, isn’t it?”

He moved to the right side of his board before writing 18 on the top of it.

“And there are those like 18 which have divisors that add up to more than itself. Here 9, 6, 3, 2 and 1 are its divisors and their sum equals 21, which is more than 18. Where are we going with this, you ask? Pay attention because it’s critical to the understanding of our army unit sizes.”

He went back to his desk and grabbed a red chalk with which he wrote some kind of titles on both parts of his board.

“Some numbers have few divisors and their sum doesn’t add up to the initial number, like 10.” With his chalk, he pointed to the left part of his board. “We call them lesser numbers.”

“On the other hand, some have a lot of divisors and their sum overpasses the initial number, like 18.” This time he indicated the other side. “Those numbers are the upper numbers.”

Yordar drew a big red circle exactly where he had drawn the line delimiting the two sides of the board.

“There also exists a special class of numbers in the middle. Those whose divisors sum equals exactly themselves. They are called perfect numbers and were worshiped by the Fada. Take 6 for example, its divisors are 1, 2 and 3 and their sum equals 6. The divisors of 28 are 14, 7, 4, 2 and 1 and again their sum equals 28. You see how this works?”

A student raised his hand at that time, and Yordar gestured for him to speak.

“What about 12 then? It’s not a perfect number.”

“Good question! However, I will not answer it, as we would need a much deeper understanding of the simple principles we saw here. Let’s just say for now that 12 is an even more beautiful number due to its relation with perfect numbers such as 6 or 28. Now onto practical teaching...”

Yordar then went to erase everything he had written on the board and noted instead a series of numbers. He pointed to the first of them while explaining.

“28 is the smallest size we use for a unit in the army. As we know, its divisors are 14, 7, 4, 2 and 1. Let’s ignore 1 for the moment but I will write the others here, like that.”

He added the divisors below 28, the first number of his list.

“A 28-men unit has 5 different ways to fight. One where the 28 soldiers fight together. A second where the unit is split into two 14-men units that can handle two things at once. A third where the unit is split into four teams of seven men. A fourth way is for seven teams of double duos to fight independently. And finally, the last one is to have the fourteen duos of the 28-men unit fight on their own.”

The professor drew small examples of warriors fighting for each of the cases he depicted.

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“See how it makes sense now? Divisors are a way to split a unit into independent subunits of equal size. You will learn how these different units can be used in different situations and what are their strengths and weaknesses. But before that, I would like to draw your attention to one crucial principle. The more divisors a number has, the more ways there are to split this number. And the more ways there are to split it, the more options a unit of the same size can pick from. With better options comes better efficiency to face any kind of situation.”

Yordar smiled and made a gesture like the conclusion was now obvious to everyone.

“Better efficiency needs more options, which means more divisors, which means a higher sum, and here we are, back to our upper numbers. See how we went from a practical need to mathematics?”

He paused just enough to make sure no question arose and continued.

“So to maximize our unit efficiency, we are looking for upper numbers. But not any upper numbers, no, in fact we are looking for the best upper number in a given range so we can have the best flexibility. That is, a number that has so many divisors that if chosen for a unit size, it can handle every single thing thrown at it.”

He raised his hands and started counting on his fingers as if to simulate the cumbersome task of going through every number to find the best upper numbers.

“Eh, don’t worry. We don’t need to do this ourselves. Once again, the Fada came to our help. They already found and documented a lot of this kind of super upper number. And we use them today, in the army.”

Yordar pointed to the following numbers of the series he had previously written.

“After 28, the sizes of warrior units are 120, 360 and 2520. 120 can be split in 15 different ways, 360 can be split in 23 ways and 2520 can be split in 47 ways!”

He wrote every single divisor below each of the three unit sizes as he had done before for 28.

“That’s a lot of combinations. However, you can see there,” he circled a divisor in the list below 360, “that 120 is a divisor of 360, which is itself a divisor of 2520. This means the warrior formations used in a 120-men unit are part of the formations used in a 360-men unit which are also part of the formations used in a 2520-men unit.”

He highlighted in different colors how the divisors from one number appeared on the list of another.

“So all in all, there are only 47 warrior formations to learn. There are 47 different ways of facing situations just because we selected a unit of a specific size. We did not choose a size of 2500 because it’s nice and round and neither 2521 to satisfy someone’s stupid ego. We chose 2520 because it’s the number that split the most in this range.”

As he said so, Yordar underlined multiple times the arrow he had previously drawn between the words ‘practical needs’ and ‘upper numbers’.

“Having said that, you might find it strange that the first unit size is 28, which is a perfect number and not, say 24, the super upper number of this range. 24 splits in seven different ways, while 28 only gives five different ways. Why this apparently drop in efficiency? Are we satisfying someone’s stupid ego?”

Yordar wrote the number 24 with all its divisors and no one was shocked anymore at the speed with which he listed the numbers. Was it from sheer memory, or did he compute it again each time? They didn’t know, but it was impressive anyway.

“There is a secret to perfect numbers. Remember what we said before. Its divisors add up to itself. Meaning, there is a sixth formation hidden behind 28.”

He went back to the list of drawn warriors below 28 and started to sketch another one.

“We can split it into one 14-men unit, one 7-men unit, one double-duos, one duo and one single men. One formation of each of the available formations. And you don’t need to learn anything in addition because they are formations you already know. This split… How to say… it’s... It’s invaluable.”

Just now, professor Yordar had shining eyes and you could see how elated he was at the way every piece fit together with perfection.

“Eh... You’re still inexperienced. You may not understand how incredible this is. Let’s just say that the now 6 ways of splitting 28 is much better than the 7 ways of splitting 24 and that’s why we use the 28-men unit.”

On the board, Yordar crossed out the number 24 and circled 28.

“However, as the numbers grow larger, the gap between perfect numbers and super upper numbers becomes too big to overcome, and that’s why we use super upper numbers in larger units.”

He went to write another number beside 360 while continuing his explanation.

“Take 496, for example, the next perfect number after 28. There are only 9 ways of splitting it, which goes up to 10 because it’s a perfect number. It’s nowhere close to the 23 ways we can split 360. That’s why we choose 360 over 496 this time.”

Yordar took a small break and concluded his exposé with his usual question.

“Good. Does anyone have any questions regarding this little bit of math?”

The room was silent as the explanation, coupled with the drawings on the board, made it a child’s play to follow.

“Perfect. That was the hardest mathematical part of your entire study, so congrats. Now let’s talk about the 47 warrior formations on how they can be used against Amentiae...”

Mahon was flabbergasted. He had known unit sizes for years, but he had himself thought it was just a random chosen number. The numbers were used in Ratho’s army, so they had simply used the same for the Nightmare’s army, but somehow the real knowledge behind it had been lost in the way.

He remembered clearly his first years where he had been taught unit fights. They knew no more than ten formations where there were more than forty-seven available! Although there was no school or proper training ground for Nightmare soldiers so it was perhaps understandable that with so little practice, they couldn't do all the formations. But even if they only could do half of it, it would still be double the amount they knew!

Mahon had not even realized he could split a 28-men unit in a perfect way. With one formation of each size, he could do so much more and with such ease. He could be on the side leading the unit while a duo protected him. The 14-men unit could be the main fighting component while the team of seven could harass a flank or impede a counterattack. The double duos could pierce through the ranks to take down an Amentiae sergeant and reduce the enemy's overall strength.

The thoughts exploded in Mahon’s mind while he was unveiling a new realm of possibilities he had never seen. He was so excited that he forgot to listen to the remainder of the lesson and just lost himself in theories and simulations. He didn’t know the missing formations, but he could easily count the divisors of 2520 and find their sizes. From a size, he could uncover parts of the formation’s purpose by relying on his own extensive experience.

Grabbing a paper, he furiously noted the different combinations and only when he had found the 47 formations did he put down his pen and lay back in relief. Of course, he had not been able to decipher everything, but even with what little he discovered, his mind was full of new potentialities and he felt like discovering strategy again for the first time.

It was not a complex concept. In fact, it was so blatant that now that he knew it, he wondered how he had not seen it before. It was so simple and yet so strong.

Just split it in more ways and you can do more things. Simple. Obvious. Brillant.