What's something that both a gamer and a programmer can find extremely useful? My answer would be solid reasoning skills. I've always been fond of mathematics, logic, and intuitive thinking, so I'm dedicating a section of my guide to just that, building intuition with visualization and mathematics. I'll be talking about how to get the most out of your towers by discussing range, damage, stats of mobs and towers, and some mathematical ideas behind Tower Defense in general. I'll start by giving you the stats of all of the towers, excluding their damage, followed by the stats of all of the mobs in game. After I give you the stats, then we can discuss range in detail. Tower Stats I will say here that I do not have access to how much damage any of the towers do, except for quake towers. For this reason, I'll be leaving tower damage out of the stats section, along with any stat information that I don't have access to or haven't observed... this does include whether tower cool-down periods drop while there are no mobs within range. Artillery towers fire TNT at regular intervals. They do area of effect damage to ground mobs, meaning they can hit several mobs at once. Their cost to place them initially is 100 coins. At level 1, they have a range of 8 blocks, and they fire 1 TNT explosion once every 10 seconds. To upgrade an artillery tower to level 2 costs 100 coins. At level 2, they have a range of 10 blocks, and they fire 2 TNT explosions once every 8 seconds. To upgrade it from level 2 to level 3 costs 150 coins. At level 3, they have a range of 11 blocks, they fire 3 TNT instead, and they fire once every 6 seconds. To upgrade an artillery tower to level 4, it costs 200 coins. They fire 4 TNT once every 4 seconds, and they have a range of 12 blocks. It should be noted here that TNT explosions do indeed hit air mobs if ground mobs (with the exception of cave spiders) are under them. Total cost from initial placement to max level: 550 coins. Archer towers shoot arrows at regular intervals, and can target a number of air mobs up to the number of arrows they fire at a time. This means that if an archer tower shoots 4 arrows every second, the arrows will target 4 air mobs individually. An archer tower does not fire multiple arrows at a single target. If you have 4 blazes move passed a row of archer towers at level 4, or 1 blaze move passed the row, their health is generally the same by the time that they reach the end of the row of archer towers. Their cost to place initially is 70 coins. It'll fire 1 arrow every 2 seconds, and have a range of 10 blocks. To upgrade it from level 1 to 2, 2 to 3, and 3 to 4, costs 30, 46, then 60 coins respectively. At level 2, an archer tower fires up to 2 arrows every 1.5 seconds, and has a range of 12 blocks. At level 3, an archer tower fires up to 3 arrows per second, and has a range of 14 blocks. A level 4 archer tower fires up to 4 arrows per second, and has a range of 15 blocks. Total cost from initial placement to max level: 206 coins. Mage towers light all mobs in range (that aren't immune to fire damage) on fire, and are the only towers that target cave spiders directly, although leach towers can damage them and iron golems have been seen targeting them on rare occasions. Artillery towers can damage cave spiders as long as other ground mobs are close to the cave spiders. The number of seconds that a mob is on fire increases per level, but isn't specified on the tool-tip that tells you the tower information. Mage towers do not target mobs that are on fire or are immune to fire damage, nor do they add damage or prolong the damage of mobs that are on fire. A mage tower costs 200 coins to place initially, lights mobs on fire once every 5 seconds, and has a range of 10 blocks. Mage towers cost the same amount to upgrade them to levels 2-4 as artillery towers. A level 2 mage tower lights mobs on fire once every 4.5 seconds, and has a range of 12 blocks. A level 3 mage tower lights mobs on fire once every 4 seconds, and has a range of 14 blocks. A level 4 mage tower lights mobs on fire once every 3.5 seconds, and has a range of 15 blocks. Mage towers do not do more damage per second than a regeneration potion heals mobs. Total cost from initial placement to max level: 650 coins. Ice towers slow down mobs in range. Ice towers can naturally see cave spiders too. Contrary to a common belief, they do absolutely no damage to blazes or any other nether mobs. The percentage or raw drop in a mob's movement speed has not been given to me. However, I still find that the speed of mobs drop enough to consider them. An ice tower's initial cost is 80 coins, its initial cool-down is 5 seconds, and its range is 10 blocks. It costs 56 coins to upgrade an ice tower to level 2, 70 coins to upgrade it to level 3, and 120 coins to upgrade it to level 4. At level 2, it has a cool-down period of 4 seconds, and a range of 12 blocks. At level 3, it has a cool-down period of 3 seconds, and a range of 14 blocks. At level 4, it has a cool-down period of 2 seconds, and a range of 15 blocks. Total cost from initial placement to max level: 326 coins. Poison towers do overtime damage to mobs just as mage towers do, but it seems to be surprisingly slower. Whether poison towers increase the damage of towers that hit a poisoned mobs remains a mystery to me still. I can verify that poison towers can have their range blocked by towers in front of them. They cost 800 coins to place initially, and their upgrade costs are the same as mage towers and artillery towers. They have a 10 second cool-down period at level 1, and a range of 10 blocks. At level 2, their cool-down period drops to 6 seconds, and their range goes up to 12 blocks. At level 3, their cool-down period is 4 seconds, and the range increases to 14 blocks. At level 4, they have a cool-down period of 2 seconds, and a range of 15 blocks. Total cost from initial placement to max level: 1250 coins. Quake towers stun mobs, and deal area of effect damage to them too. They can't target air mobs or skeletons, but they're great around artillery towers. Their tower damage is the only damage given in game. Tower level matters a lot for this tower. The stun lasts for roughly half of a second. Initially, they cost 200 coins to place, they have a range of 8 blocks, and they attack once every 5 seconds. The cool-down period drops 1 second for each level. Quake towers cost 150 coins to upgrade to level 2, 250 coins to upgrade to level 3, and 360 coins to upgrade to level 4. Quake towers increase in range by 2 blocks per level. Total cost from initial placement to max level: 960 coins. Zeus towers attack a single mob at a time, and deal slightly heavy damage. Only cave spiders are immune to this tower. I'd recommend them around ice towers, and maybe quake towers. They cost 1,000 coins to place initially. At level 1, they attack once every 5 seconds, and they have a range of 10 blocks. To upgrade a Zeus tower, it costs 300 for the first upgrade, 400 coins for the next upgrade, and 500 coins for the final upgrade. At level 2, it has a range of 12 blocks, and a cool-down period of 4.5 seconds. At level 3, its range is 14 blocks, and its cool-down period is 3.5 seconds. At level 4, its range goes up to 15 blocks, and it's cool-down period is 2.5 seconds. Total cost from initial placement to max level: 2,200 coins. Sorcerer towers summon passive mobs like chickens and cows to fight for you. Their damage output isn't really impressive, but they're great for blocking the wither's attacks. They cost 800 coins to place initially. At level 1, it summons 1 mob to fight for you, has a cool-down period of 3 seconds, and a range of 12 blocks. Its range increases by 1 block per level, Its cool-down decreases by half of a second per level. Its costs 400 coins for its first upgrade, 500 coins for the next upgrade, and finally, 550 coins for its last upgrade. Total cost from initial placement to max level: 2,250 coins. The next three towers can only be upgraded to level three. Necromancer towers summon an iron golem. If they're not near the right towers, they're practically useless, because they may not target the mobs that they need to. Necromancer towers can kill giants with ease if placed correctly. Iron golems can glitch and hit cave spiders, but this happens rarely. They can also target the wither from time to time, and whether this is a glitch or not remains unclear, as the only ground mob that they don't target normally is the cave spider. A necromancer tower costs 1,200 coins to place initially, and has a range of 10 blocks at first. Their cool-down never decreases, but how often the golem hits does decrease. The golem's stats basically increase with each level. It costs 500 coins to upgrade it the first time, then 600 coins the final time. The range of the tower is 15 blocks at level 2, and 18 blocks at level 3. Total cost from initial placement to max level: 2,300 coins. Turret towers shoot arrows just as archer towers do. At max level, they shoot fire aspect arrows. For reasons stated in part 1 of my guide, I recommend not placing them at all. They cost 800 coins to place, and fire 1 arrow per second at level 1. They start with a range of 10 blocks. To level a turret tower to levels 2 and 3, it takes 600 and 700 coins respectively. They fire an additional arrow per level. Their cool-down period is 0.8 seconds at level 2, and 0.5 seconds at level 3. Total cost from initial placement to max level: 2,100 coins. Leach towers charge by attacking nearby mobs, and releasing a shock-wave that has a massive range. While I don't have a good opinion of them, they may be the only tower with a good chance to kill 100+ blazes at once. They cost 2,000 coins just to place initially. Their cool-down period lasts 3 seconds for every level. Their range increases 2 blocks each level, and starts off at 10 blocks. The shock-wave that's released has a much larger range. The damage of the shock-wave is a bit overrated, it's not really that high. It costs 200 and 300 coins for its successive upgrades. Total cost from initial placement to max level: 2,500 coins. Castle guards are the last line of defense closest to the castle. You get 2 of them per match. They're not placed by players, and they get destroyed at armageddon mode. You can't upgrade them at all, and they don't have an associated level. They shoot fire aspect arrows, and only target up to 2 mobs at once. It's worth mentioning that there is a cool-down period between when their arrows hit a mob and when a player can hit a mob with a sword. What I mean is that when a mob gets hit with an arrow, you can't hit the mob with your sword for about half of a second. If you try to, you simply won't do any damage at all. While the castle guards are hitting a mob, you barely have time in between arrow shots to get any type of damage done by a sword. Mob stats Again, if I don't know something about a particular mob, I won't mention it here... that or I'll say that I'm not sure. I'm going to post each mob's stats in a bulleted list. and note anything about a mob that's worth mentioning. Exp return is how much experience you get for sending 1 mob of that type. Exp to coin ratio is the Exp return divided by the coin cost. Zombie Exp cost: 0 Coin cost: 10 Health: 40 Speed: 1.6 (Close to the walk-speed of a normal player) Exp return: 5 Exp to coin ratio: 50% Zombies are the mobs that you start with every match. They have the highest exp to coin ratio. Spider Exp cost: 80 Coin cost: 25 Health: 80 Speed: 2 Exp return: 10 Exp to coin ratio: 40% Cave spider Exp cost: 100 Coin cost: 35 Health: 70 Speed: 1.6 Exp return: 15 Exp to coin ratio: 42.857142% (857142 repeats indefinitely) Cave spiders can only be targeted directly by mage towers and ice towers. Creeper Exp cost: 200 Coin cost: 300 Health: 450 Speed: 1.3 Exp return: 75 Exp to coin ratio: 25% Creepers regenerate health over time slowly, but their regeneration speed isn't given. Zombie pigmen Exp cost: 150 Coin cost: 100 Health: 160 Speed: 2.5 Exp return: 30 Exp to coin ratio: 30% Zombie pigmen are the fastest mob in the game. They're immune to fire damage. Skeleton Exp cost: 150 Coin cost: 120 Health: 300 (per life) Speed: 1.6 Exp return: 30 Exp to coin ratio: 25% Skeletons start out as wither skeletons when sent, and come back to life as regular skeletons when killed. Wither skeletons are immune to mage towers and fire damage, but regular skeletons aren't. Skeletons can't be healed back into wither skeletons by regeneration potions or witches. Quake towers do not damage or stun skeletons at all, even with other mobs nearby. Silverfish Exp cost: 100 Coin cost: 80 Health: 180 Speed: 1.6 Exp return: 20 Exp to coin ratio: 25% Silverfish are flying mobs, and flying mobs can only be hit by archer, mage, ice, Zeus, turret, and leach towers. It's possible to get critical hits from a sword on flying mobs. Endermite Exp cost: 200 Coin cost: 350 Health: 250 Speed: 1.6 Exp return: 50 Exp to coin ratio: 14.285714% (285714 also repeats indefinitely) Endermites are flying mobs that are immune to fire damage. Blaze Exp cost: 500 Coin cost: 800 Health: 410 Speed: 1.6 Exp return: 70 Exp to coin ratio: 8.75% Blazes are flying mobs that are immune to fire damage also. Witch Exp cost: 250 Coin cost: 150 Health: 300 Speed: 1.6 Exp return: 25 Exp to coin ratio: 16.6% (6 repeats indefinitely) Witches heal all nearby mobs, but due to artillery being able to target them, they're detrimental to air mobs. They're immune to mage towers, but can be burned by the castle guards. Slime Exp cost: 800 Coin cost: 1,000 Health: 1,100 Speed: 0.8 Exp return: 80 Exp to coin ratio: 8% Just like skeletons, slimes come back to life too. The difference is that they come back as 2 smaller slimes, with less health. They split twice around, meaning that they start as 1 slime, split into 2, then each smaller slime splits into 2 more. Magma cube Exp cost: 1,000 Coin cost: 1,500 Health: 1,500 Speed: 0.8 Exp return: 100 Exp to coin ratio: 6.6% (6 repeats indefinitely) Magma cubes are immune to fire damage, and split 3 times around. Each smaller magma cube has less health than the previous incarnation. Giant Exp cost: 1,500 Coin cost: 6,000 Health: 2,200 Speed: 1.4 Exp return: 200 Exp to coin ratio: 3.3+% (3 repeats indefinitely) Giants regenerate just as creepers do, and their regeneration is not listed in game either. If a giant dies before it's close (about 5 blocks away from damaging the castle) to the castle, it splits into 2 magma cubes, 3 slimes, and around 10 zombies. (I've honestly never counted.) If a giant dies at the castle, it does not split into other mobs, and simply dies there. A giant occasionally let's out zombies as it walks. Mobs that come from the giant, whether while it's alive or after it dies, are called trailings by my team. Trailings give exp when they die. Wither The wither's health is unknown for sure, but I'd venture to say that it's close to 800. The wither regenerates overtime but can't be sped or slowed down by potions. The wither occasionally fires a bullet (or wither head) as it walks by. A wither head destroys the tower that it hits regardless of the type of tower, the level of the tower, or the price of the tower. Wither bullets target towers at random, but can be blocked by sorcerer tower mobs. Tower Damage Mathematics This section really applies to both offense and defense, so regardless of which you choose, you should read this. (The previous sections apply to both too, in all actuality.) Even though this section is about the mathematics of tower damage, we won't be working with that many known constants... and by that, I mean numbers like 4 or 2. We'll be talking about more arbitrary values such as time, and tower range. This may sound complicated for my younger or less mathematically-inclined readers, but it's really not too difficult to understand. Let me introduce what computer scientists and mathematicians call a function. A function is basically a way to calculate things based on given values, much like a formula. How far something moves given its speed and how long it's moving is a good example of a function. We'd write it looking like this: d(r, t) = r * t I've used the letter r for rate, (rate means speed, but "s" looks too much like a "5", so we use "r" instead) t for time, and d for distance. So we'll look at an example, before we talk about how this matters in Tower Defense. We won't worry about units as much here, as I'm looking to keep things as simple as I can in my example. So we'll say a car is moving at 5 miles per hour, for 4 hours. Its speed (or rate) is 5, and the time it's taken is 4. So in that time, it'll travel 20 miles. You get that by multiplying the speed and the time together. If we were to use functions to represent this, we'd say: d(5, 4) = 5 * 4 d(5, 4) = 20 So there you have it. In order to find distance, we needed to know speed and time. We would say that distance is a function of speed and time. Now that we've covered functions, we'll look at functions in Tower Defense. We'll think about how many times a tower hits a mob while it walks by first, assuming that no other mobs are in range for now... just 1. We'll think of it as a function. So we need to know how many many things a tower fires or shoots at a time, the cool-down period of a tower, the cool-down time left for the tower to hit again, and the time that it's in range. so our function would look like this: Hits(items_fired, cool-down, current_cool-down, time) Hits is how many times it will hit while in range, items_fired is how many items it's firing at once, cool-down is how long the tower's cool-down period is, and current_cool-down is how long the tower has until it's ready to hit again. Let's look at an example with a level 4 artillery tower. It has a cool-down of 4 seconds, and it shoots 4 TNT. We'll say that it hit a mob 3 seconds ago, so it needs 1 more second before it's ready to shoot again. We'll say that the mob is in range for 7 seconds. So to find out how many times it hit, we'll put those numbers in the function: Hits(4 TNT, 4 seconds, 1 second left before it's ready again, mob's in range for 7 seconds) Now we'll briefly talk about how you find how many times it hits. we'll say 1 second passes by, it'll be ready to hit with 4 TNT, then 4 more seconds will pass, and it'll hit again with another 4 TNT. So by 5 seconds in, it's hit 8 times. (Technically it's hit twice, but with 4 TNT each time... but the number 8 will matter more to us soon.) Now it needs 4 seconds to fire again, but the mob will only be in range for 2 more seconds, so it won't get to fire a third time. So it'll hit the mob 8 times. Now that we've discussed that, let's look at how we figure out how much damage a tower will do, then we'll mention what we can do to make our tower do more damage. We know that it hit with 8 TNT, but if we know how much damage it does per TNT hit, then we can find out how much damage our artillery tower has done. total_damage_of_artillery(Hits, tnt_damage) = Hits(items_fired, cool-down, current_cool-down, time) * tnt_damage We don't know how much damage a single TNT explosion does, so we'll leave that alone for a moment, let's just look at how it would look if we knew: total_damage_of_artillery(8, tnt_damage) = 8 * tnt_damage Alright, now let's finish this section by talking about which values we can change. Well, we can't change the fact that a max artillery fires 4 TNT every 4 seconds, and we can't change the fact that it needs another second to fire, but we can change the time that the mob's in range by placing something that slows down or stuns the mob. Freeze potions, quake towers, and ice towers would do that for us. Let's say that we keep the mob in range for an extra 4 seconds, making it be in range for a total of 11 seconds. Let's re-evaluate the 2 functions. We'll do Hits first. So we need 1 second for it to fire 4 TNT, another 4 seconds to fire 4 TNT, then another 4 seconds to fire TNT again... by this point 9 seconds have passed and it's hit with 12 TNT, but it needs 4 seconds before it can fire again, and since 9 has passed and the mob will only be in range for 2 more seconds, it's out of time and won't fire again. So now that we've added 4 seconds on, we've fired 12 TNT, instead of 8 TNT. We have: Hits(4, 4, 1, 11) = 12 total_damage_of_artillery(12, tnt_damage) = 12 * tnt_damage While we may not know how much damage each TNT will do, we do know that 12 TNT explosions is better than 8. That's a 50% increase in damage overall. Tower Ranges Let's take a final look at tower ranges, and finally finish talking about optimization.So we've mentioned critical spots in the previous sections of my guide, but let's really get into what optimal tower placement is, and what it looks like... for all of my visual learners out there. We need a few pictures to show the range of a tower first. For this, I'm going to choose a quake tower at the back of Original, which is a level (map, field, place) in Tower Defense. The range of a quake tower is 14 blocks, that's 14 forward, left, right, back, and vertically-upward. I've highlighted the range of a quake tower above in green, and I'll verify that the range does extend that far, and that this picture is accurate. Now let's consider where mobs walk along a path. As of the time of this post, mobs are limited to walking within (or partly on, which is enough for a tower to hit a mob) the middle 3 blocks along a path. We'll highlight this area in red, then we'll look for the maximum area coverage, while still being in grid, in the next couple of pictures after that. First Picture, Red Range Highlighted: Second Picture, Quake Tower Moved 3 Blocks Rightward: Third Picture, Quake Moved Rightward 3 More Blocks: ...and there we have it! The quake tower's maintaining corner coverage, while maximizing its rightward area. Another 3 blocks to the right, and we'd only lose the red area on the left, which matters to us, and gain more green on the right, which doesn't matter. Right now, we're seeking to gain as much red area as possible. Now, we'll take a look at an artillery tower's range, which I'll show in blue, followed by both of their ranges overlapping. I'll make their overlapping ranges purple in general, and focus on where their red ranges overlap. Then I'll show in-game pictures of a quake tower, which is placed in the same spot as it is in the third picture above. It'll be hitting mobs so that you can see its range. Artillery Range Picture: Quake Overlapping (Intersecting) Artillery Picture: Again, the red range is where mobs can get hit by both the artillery tower and the quake tower. This placement will maximize the amount of time that a mob is in both tower's range... and hence will maximize the damage done to the mobs that pass by these towers. I used this spot, and not the two rows above because, we usually place archer towers there later on. As for the in-game pictures where the tower range can be shown... here they are: Top-right area of the quake range: Left area of the quake range: Well, that's it for part 3, I hope you've enjoyed reading my guides so far, and I look forward to seeing you all in game!

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

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.

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.

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.

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.

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.

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.

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.

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.