# Man of War: A Numbers Problem

The story is set in an alternate universe, where magic prevails.

A man breathes heavily, muttering inaudible words while hunched over a spellbook. This person is a castle mage, tasked with determining the amount of MoW (Man of War) an enemy has. If he doesn't, he'll lose his head, like many others before him. Or else why would they be left with only a third-tier seer? With his lack of ability to control his Sight, he can only access whatever information his grimoire gives him. Yet it has proven always to be sufficient.

And the grimoire starts speaking.

"Four in the front."

A pause.

"Four in the back."

A longer pause. But the man knows that there's more.

"Four on the Left."

"Four on the Right."

"Four in the Center, and Four Outside."

It shivers and is silent.

How many MoWs is the enemy sending?

My guess:

Eight arranged like so:

1234 in the front, 5678 in the back, 1256 on the left, 3478 on the right, 2367 in the center, 1458 outside.

• Nice. rot13(Gurl zvtug nyfb or va na K sbezngvba)... Commented Feb 10, 2023 at 1:22
• Thanks, this was the intended answer, so I'll accept it, but I guess the others make sense too. :) Commented Feb 11, 2023 at 1:10

The most likely solution:

8, arranged like an X:

1  2
34
56
7  8

Here

1234 are in the front (that is, looking from the front you see machines 2431 left to right), 5678 in the back. 1357 are left and 2468 are right. 1278 are outside and 3456 are in the center.

There are some other somewhat plausible possibilities with different number of machines:

16, in a 4x4 square. Sides are obvious, outside ones are those in corners (they are the furthest out) and center ones are the 4 in the middle.

4, arranged in a rotated square (by any degree that is not multiple of 45). Looking from any direction you see 4 machines, all of those 4 machines are both in the center (they have one corner in the center of formation) and on the outside (one of their corners is the outermost place of formation).

• Well, you did get the intended amount, but I was really aiming for a 2 by 4 rectangle. Still, this works too, so I'm gonna upvote this too :) Commented Feb 11, 2023 at 1:12

I'd found an answer but it turned out to be the same as the most popular one, so here's a different one (ignoring the weird capitalization choices):

Actually, it can be any number starting from 8: The front-back, left-right and center-outside contrasts leave out the MoWs in between. Let the number be $$8+n$$.

On the table below, let's give -1 to front/left/center, 0 to in-between and 1 to back/right/outside. As long as we have 4 -1s and 4 1s in all columns, we're good.

Front/Back.......Left/Right........Center/Outside
-1.............................-1.........................-1
-1.............................-1.........................-1
-1.............................-1.........................-1
-1.............................-1.........................-1
1.................................1...........................1
1.................................1...........................1
1.................................1...........................1
1.................................1...........................1
0.................................0...........................0

Funnily enough, in a 3D shape, the front/back state can be associated with the angle with the $$yz$$ plane. Left/right would be the $$xy$$ plane and center/outside would depend on how far the rays of the angle have to go. Everyone has a place in a 3D shape.

• I don't really understand, could you please explain? Commented Feb 11, 2023 at 1:12
• There can be more than 8, but it requires some MoWs to be neither on the left nor on the right (exactly aligned with the mage). Same for the front and the back. Only the "center" and "outside" are (partly) absolute terms based on a certain border/wall between them depending on the mage's position. Commented Feb 11, 2023 at 11:05