ymbmg95's guide to tower defence, part three: mathematical ideas, tower range, damage, and stats

[ymbmg]

[REDACTED]

 

[KitKatkels17]

I Find this very usefull

 

[Emmy]

gj m8 :)

 

[KitKatkels17]

Congrats on finishing the guide

 

[Power Ranger]

Much math much wow. :P

(Will have to read later when I get the time) :P

 

[Mr_Aro]

holy son man, I have never in the history of cc, seen a post that long, gg and thanks a lot for all the new info :D

 

[betty's oldies]

I dont find this use full im sorry but i believe that turrets are much better

Cool pic, ymbmg. I recognize that image - it's combinational logic with logic gates and demuxes, isn't it? Now, on topic:

Turrets fire arrows every 0.5 seconds (level 3 = 3 arrows). This equates to 6 arrows per second. A single archer level 4 fires 4 arrows per second for roughly 10% of the turret's cost. Comparing the two at the same cost, archers deal at least 6 times more DPS than turrets. (40 archer arrows vs 6 turret arrows per second).

Edit: The fire effect from turret arrows is applied to any mob it hits. Fire arrows aren't worth it - pair a mage tower with a bunch of archers and you get the same effect but better.

 

[ymbmg]

holy son man, I have never in the history of cc, seen a post that long, gg and thanks a lot for all the new info :D

I've spent something close to five and a half hours on the part prior to this. As far as this part goes, I spent a few weeks in-game just getting all of the numbers for prices, and a few of those in-game pictures. It took eleven hours of just typing, rearranging pictures, and thinking about how I wanted to convey my information. Although, I did stop to eat and whatnot in that time... and I did take some time to find good music to play while organizing everything. This is the third part in a series of posts that i want to do. I plan to have around eight in total. It takes me awhile to come up with what all I want in a part of the guide. I think I'm typing about team-building (as in, making or joining a team) next. Some of my readers say that they team with random players often, and that other teammates are uncooperative. I'm certain that this is the case, I rarely run into other teams that stick together, and are organized. I'm not sure if my readers are looking for a team, or if they like going solo. Either way, I plan to make a part devoted to getting a team together, because the game changes a lot when you're with other good players.

 

[betty's oldies]

I've spent something close to five and a half hours on the part prior to this. As far as this part goes, I spent a few weeks in-game just getting all of the numbers for prices, and a few of those in-game pictures. It took eleven hours of just typing, rearranging pictures, and thinking about how I wanted to convey my information. Although, I did stop to eat and whatnot in that time... and I did take some time to find good music to play while organizing everything. This is the third part in a series of posts that i want to do. I plan to have around eight in total. It takes me awhile to come up with what all I want in a part of the guide. I think I'm typing about team-building (as in, making or joining a team) next. Some of my readers say that they team with random players often, and that other teammates are uncooperative. I'm certain that this is the case, I rarely run into other teams that stick together, and are organized. I'm not sure if my readers are looking for a team, or if they like going solo. Either way, I plan to make a part devoted to getting a team together, because the game changes a lot when you're with other good players.

Same here. The catch is, how do you deal with incompetent teammates? Even after giving advice and asking to not place mages/ice/poison next to each other, people still do it. Only once our team threatens to kick (and actually do kick if they fail to comply) and/or threaten to report for team griefing do they start listening or leave.

 

[Power Ranger]

Somebody should really pin a message with all of your guides they are so awesome :D

 

[betty's oldies]

Somebody should really pin a message with all of your guides they are so awesome :D

He announced that he'll make 5 more parts. Probably by then he'll put them all to a "master thread" with links to each part.

To be on the leaderboard is an achievement and it sure has to come with a lot of experience. I've never seen anyone on the leaderboards in any game write a guide like this. I'm well impressed.

 

[ymbmg]

Same here. The catch is, how do you deal with incompetent teammates? Even after giving advice and asking to not place mages/ice/poison next to each other, people still do it. Only once our team threatens to kick (and actually do kick if they fail to comply) and/or threaten to report for team griefing do they start listening or leave.

I simply don't play with "incompetent teammates" anymore. I only play with my team, and if they're not around, I don't play. The frustration of having bad teammates would've been enough to get me to quit long ago. I wait hours a day sometimes just to play around five matches... but when I get to play, it's worth it.

 

[ymbmg]

Cool pic, ymbmg. I recognize that image - it's combinational logic with logic gates and demuxes, isn't it? Now, on topic:

Turrets fire arrows every 0.5 seconds (level 3 = 3 arrows). This equates to 6 arrows per second. A single archer level 4 fires 4 arrows per second for roughly 10% of the turret's cost. Comparing the two at the same cost, archers deal at least 6 times more DPS than turrets. (40 archer arrows vs 6 turret arrows per second).

Edit: The fire effect from turret arrows is applied to any mob it hits. Fire arrows aren't worth it - pair a mage tower with a bunch of archers and you get the same effect but better.

Yeah, it is. I figured since I was mentioning math and logic, that GIF would be relevant. That circuit in that picture was constructed by me. I learned about Boolean Algebra when I was in high school. If I would have included set theory in my guide, which I had originally intended to, then it would be even more relevant. Boolean Algebra and Set Theory are nearly one in the same. Their operators share just about every single property, so much so... that I'd venture to say that if you're good with set theory or Boolean algebra, surely you're good with the other.

 

[betty's oldies]

We got a bunch of math and circuit logic too as part of the game, haha. I actually learned boolean algebra and set theory in two separate engineering classes though. They're two different things - subset, powerset, disjoint, etc conpared to boolean identities and such.

 

[Trigger]

Great guide for understanding the game!

 

[Clooki]

Love this guide. Helped me a lot!! ty

 

[ymbmg]

We got a bunch of math and circuit logic too as part of the game, haha. I actually learned boolean algebra and set theory in two separate engineering classes though. They're two different things - subset, powerset, disjoint, etc conpared to boolean identities and such.

I was expecting an answer like this eventually. Let me present a problem of each, and then go into a bit of detail for what I meant about their relationship.

We'll look at a problem in boolean algebra and one in set theory, and afterwards I'll elaborate on their similarities. I can't get the forums to superscript my ∁, so I'll say here that A∁, is the absolute complement of set A. I'll use ∪ for set union, and ∩ for set intersections.

A ∪ (B ∩ A∁)
We'll start with a distributive law first, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) or in our case: A ∪ (B ∩ A∁) = (A ∪ B) ∩ (A ∪ A∁)
(A ∪ B) ∩ (A ∪ A∁)
Now we'll use an identity for the union of set A and its complement (A ∪ A∁) = U; U is the universal set
(A ∪ B) ∩ U
Any set held in intersection with the universal set, is identically itself. (A ∪ B) ∩ U = A ∪ B
A ∪ B

So here we started with A ∪ B ∩ A∁ and we simplified it to A ∪ B. Now we'll look at an example in boolean algebra. I'll use ∨ for logical inclusive disjunction, or the OR operator, ∧ for logical conjunction, or the AND operator, and ¬ for logical negation, or the NOT operator.

A ∨ (B ∧ ¬ A)
We'll start again with a distributive law in boolean algebra, namely A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) or in our case: A ∨ (B ∧ ¬ A) = (A ∨ B) ∧ (A ∨ ¬ A)
(A ∨ B) ∧ (A ∨ ¬ A)
Here we'll use an identity for anything held in logical inclusive disjunction with its complement. A ∨ ¬ A = 1
(A ∨ B) ∧ 1
Finally, anything held in logical conjunction with 1 is identically itself. (A ∨ B) ∧ 1 = A ∨ B
A ∨ B

The union operator in set theory has the same properties as the or operator in boolean algebra.
The intersection operator in set theory has the same properties as the and operator in boolean algebra.
Taking the absolute complement of a set can be thought of as the not operator in boolean algebra.
The universal set has the same properties as 1, or true in boolean algebra.


As I said before, there's a relationship between boolean algebra and the algebra of set theory. I first learned this when I tried looking up logical inclusive disjunction on Wolfram Alpha, and it kept redirecting me to the set theory union operator. I was confused at first, but after a lot of insight and looking over their properties, I came to realize that there was a relationship. I hope I've clarified what I meant before. You may check my work as well, or even take it to someone else to be checked.

 

[betty's oldies]

I was expecting an answer like this eventually. Let me present a problem of each, and then go into a bit of detail for what I meant about their relationship.

We'll look at a problem in boolean algebra and one in set theory, and afterwards I'll elaborate on their similarities. I can't get the forums to superscript my ∁, so I'll say here that A∁, is the absolute complement of set A. I'll use ∪ for set union, and ∩ for set intersections.

A ∪ (B ∩ A∁)
We'll start with a distributive law first, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) or in our case: A ∪ (B ∩ A∁) = (A ∪ B) ∩ (A ∪ A∁)
(A ∪ B) ∩ (A ∪ A∁)
Now we'll use an identity for the union of set A and its complement (A ∪ A∁) = U; U is the universal set
(A ∪ B) ∩ U
Any set held in intersection with the universal set, is identically itself. (A ∪ B) ∩ U = A ∪ B
A ∪ B

So here we started with A ∪ B ∩ A∁ and we simplified it to A ∪ B. Now we'll look at an example in boolean algebra. I'll use ∨ for logical inclusive disjunction, or the OR operator, ∧ for logical conjunction, or the AND operator, and ¬ for logical negation, or the NOT operator.

A ∨ (B ∧ ¬ A)
We'll start again with a distributive law in boolean algebra, namely A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) or in our case: A ∨ (B ∧ ¬ A) = (A ∨ B) ∧ (A ∨ ¬ A)
(A ∨ B) ∧ (A ∨ ¬ A)
Here we'll use an identity for anything held in logical inclusive disjunction with its complement. A ∨ ¬ A = 1
(A ∨ B) ∧ 1
Finally, anything held in logical conjunction with 1 is identically itself. (A ∨ B) ∧ 1 = A ∨ B
A ∨ B

The union operator in set theory has the same properties as the or operator in boolean algebra.
The intersection operator in set theory has the same properties as the and operator in boolean algebra.
Taking the absolute complement of a set can be thought of as the not operator in boolean algebra.
The universal set has the same properties as 1, or true in boolean algebra.


As I said before, there's a relationship between boolean algebra and the algebra of set theory. I first learned this when I tried looking up logical inclusive disjunction on Wolfram Alpha, and it kept redirecting me to the set theory union operator. I was confused at first, but after a lot of insight and looking over their properties, I came to realize that there was a relationship. I hope I've clarified what I meant before. You may check my work as well, or even take it to someone else to be checked.

Yep, I remember that from my discrete math class and I use a lot of booleans in programming. I haven't gotten far enough to see any discrete math applied to software in any engineering classes so far (save for boolean algebra, but that's for logic circuits).

Intersecting is just like the && boolean operator in programming. Same goes for || for or and ! for not.

Also, about the thread. Does that mean the radius is not circular? That means if we have a range of 16 horizontal or vertical blocks max and we make an isosceles right triangle, does that mean the diagonal range is 16*sqrt(2)? If that's so, then that could be an oversight with the range. Also, it is possible to have a circular range since vanilla mobs are programmed to spawn in a circular range around the player.

 

[ymbmg]

I can verify for sure that the range of a quake tower is indeed not circular or spherical, and though I can't be absolutely sure without extensive testing, I'm almost certain that this is the case for the other towers as well. I had thought about calculating range much in the same way. Out of curiosity... did you picture their range to work something like this?

Where we would have the magnitude of the hypotenuse of this triangle be equal to the range of the tower. If so, then we think alike. I had originally thought that all of the towers in this game had their range worked out like this, until I saw the range of the quake tower, as out-lined on the ground in the pictures.

 

[Pixelated_PI]

Ok, my first question would be your current age, to be bringing this sort of mathematics into a tower defense guide. It seems like the majority of cubecraft players are not taking high enough math classes to have learned any of this, possibly not even algrebra. And why did you choose to use function notation? It seems this wouldn't have been too hard to write with slightly more simple algebraic equations, making it far easier for most people to understand.
If you can follow it though, I definitely like the content, very thorough, and full of useful statistics I would eventually have bothered looking into myself in game, but now don't have to.
You also mentioned making a total of 8 parts. Afterwards, you should get the important parts of each one and put it all together in one guide. I was thinking about making my own guide, and you mentioned not making one on playing with random people, so I will probably look into writing about that. I play most of my games with random people, averaging about 10% rate of decent players. Roughly 1/10 vote kicks are successful, and they seem far better used as threats.
Anyway, thanks for the great guide, I can't wait to read all 8 parts.