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Introduction

Lich is one of the easiest heroes to play, and is one of the first few heroes I started playing with. He is a good support, with a spammable nuke good for harassing, almost no mana problems, high armor because of Ice Armor, and a strong ult that can easily get double-triple kills if lucky.

Pros, cons

Pros
Fast base movement speed (315)
Spammable nuke - good for harassing
No mana problems
E skill denies your creeps while giving you mana
Easy double kills with ult
Extremely easy hero to play

Cons
Squishy

Lore

In life, the frost-mage Ethreain (not yet a Lich) had used the threat of destructive ice to enslave entire kingdoms. His subjects, aided by a few desperate magicians, eventually grew bold enough to ambush him. Armed with enough charmed rope to bind him forever, they tied the frost mage to adamant weights and dropped him in a pool known chiefly for being bottomless. It wasn't. He only fell for a year or so before an outcrop snagged him. There he rested, dead but undecaying, until the geomancer Anhil thought to verify the legend of the supposedly bottomless Black Pool. Anhil's plumbline snarled with the ropes that bound the drowned magician, and up he hauled an unexpected prize. Thinking that by rendering the dead undead, he could question the Lich about the properties of the pool, he removed the bindings and commenced a simple rite of resurrection. Even the descendants of Ethreain's enemies were long forgotten by time, so there were none to warn Anhil against imprudence. But he learned the error of his judgment almost immediately, as Lich threw off the shackles and consumed him.

Skills

Frost Blast

Lich's first skill. This should be the first skill you add, and this gives a slow and a good harass with a low cooldown (8 seconds).
Has a somewhat high mana cost, but this is easily solved by Sacrifice.

Ice Armor

A very useful skill which gives a large amount of armor.
At max level, this gives a whopping 9 armor, which is almost as much as a Platemail
(If you don't know how armor helps, visit this for info: Armor info )
Also, keep in mind that this stays for 40 seconds, which is quite long. (If you're following this guide and got a Mekansm, casting Ice armor on yourself gives 1 (base armor) + 3/5/7/9 (Depending on level of Ice Armor) + 5 (Mekansm passive armor) + 2 (if you used Mekansm within the past 25s), adding up to a total of 11/13/15/17, making you take 50.5% less damage.)

Sacrifice

This skill is what makes Lich have no mana problems. This skill gives 60% (at max level) of a creep's CURRENT hp to Lich as mana. Because this is current hp, this should be used on melee creeps with full health. (Which gives roughly 362 mana)
Alternatively, you can use this as a deny.
You should never use this on ranged creeps, unless you are trying to decrease the DPS of the creep wave without pulling (because you're **** at pulling like me).

Chain Frost

Lich's ultimate. This can easily get double or triple kills, if you're lucky. Keep in mind that this only bounces 7 times. The best situation to use this in is when you see two heroes very close to each other. Also, you can use this to interrupt channeled spells such as Enigma's Black Hole because of the ministun if you target him. (Note: This skill only ministuns the person you target, and only him.)
Something to note is that this will be blocked by Black King Bar and Linken's Sphere, but only to some extent. Black King Bar will not stop the ministun or the bounce, while Linken's Sphere blocks it, but still allows it to bounce.
You should use this when there are no creeps around for maximum effect.

Items

Starting Items

Get a Animal Courier first, remember that you're a support.
Next, go for a Tango and Healing Salve. You're squishy and can die easily if not careful.
Now, you can go for 4 Iron Branches
OR
2 Iron Branches and a set of Observer Wards if noone else is getting wards.
Lich's fast base move speed of 315 should make it easier for you to ward wherever you want and get back to your lane in time for the creep wave.

Early Game

Get a Magic Wand to clear up the inventory of Iron Branches first.
Next, get a Boots of Speed. No need to explain why.
Then get a Bracer to raise your strength/health.
Now get a Headdress to build your Mekansm. The additional health regen is useful too.

Core

Get a Flying Courier if no-one else has gotten one yet.
If not, get more Observer Wards
Build your Mekansm after that.
You can then turn your Bracer into a Drum of Endurance, or else get your Aghanim's Scepter first.
It upgrades the damage of Chain Frost and it's casting range, so you can rape 2 enemy heroes from a further distance.

Boots

Boot choice for Lich are either Phase Boots or Tranquil Boots
You should go Phase Boots if you are doing well in your lane and can use the extra mobility and damage. If not, the healing from Tranquil Boots is helpful too.
Do remember to always be using Phase Boots's active, because of the extra 16% move speed, if not Tranquil Boots give a better 75 MS instead of Phase's 55.

Situational

You can get a Blink Dagger or a Shadow Blade as an escape, depending on which you are better at using. Escape is important for Lich as he is squishy. Although if you're good enough you'll have no trouble without these. Personally I get a Shadow Blade because I don't like the 3 second cooldown thing for Blink Dagger.

Get a Town Portal Scroll anytime you want in the middle of the game.

I personally never use a Shiva's Guard, but I've seen other guides list it.

Get a Scythe of Vyse if you need to disable a hero such as a carry like Riki, and if no-one else gets one.

A Necronomimicon is good if you can micro well. Gives lots of strength and intelligence. I'm bad at micro so I have never used it.

Sentry Ward is used when you are up against invis heroes like Riki.

If you're doing EXTREMELY well and have tons of gold with nothing else to buy, get a Refresher Orb. (Really, if you are able to get this, the game is probably ending already unless you're just camping spawn.)

You can go get those Boots of Travel if you got the money. It allows you to teleport to creeps, as well as some heroes like Undying's zombies and Brewmaster's panda spirits.

A Force Staff can be used both as an escape and a way to catch up to escaping prey. It gives decent stats too.

A Eul's Scepter of Divinity can help you to stop escaping enemies.

A Veil of Discord increases the Magical damage taken by enemies in the area, allowing for extra damage from Chain Frost. Use this in teamfights.

A Pipe of Insight will help in teamfights.

Gameplay

Laning

Lich is a support hero, but it should be noted that he can possibly go mid as well. However, if someone else will benefit more from it, such a an actual carry (Lich is listed as a semi-carry on dotafire). In short, Lich can go wherever he damn well wants to go.

Early Game

Skill Frost Blast first and use it sparingly until level 2 when you skill Sacrifice. As Chain Frost is important for kills, skill it when you can at levels 6,11,16.
Ice Armor is a good skill, but Frost Blast and Sacrifice are way better and much more important, so add this last.
Spam Sacrifice all the time to reduce the power of your creep wave, thus denying the person you are laning against of one creep without having to get the timing right.

Tips

When you get

Ice armor, use it on all your teammates, especially your carry.
Ice Armor works on Roshan. Cast it on your teammates to reduce his AS, reducing the amount of damage your team takes.
When you get ganked, just use Chain Frost You're probably going to die, so at least take 2 with you. If you can, however, juke and run away.
At level 16, when you've maxed Chain Frost, you have a really low cooldown of only 60 seconds.
If you can, get them Boots of Travel! It gives you a huge 100 MS along with infinite TP scrolls that tp you to creeps!

Teamfights

Cast

Ice Armor on your allies before it happens.
Make sure everyone in the team is buffed with it.
Stay back. Your spells have a good range, and if you're right in front of the teamfight, you'll die and make it a 4v5 because you're squishy.
When you can, use Chain Frost on the enemies.
Use your Mekansm when needed. This heals a considerable amount of health while giving a bonus to armor.
Spam Frost Blast. If they're trying to escape, use that to slow them for your teammates to get the kill.
Use your items. If you've got a drum, use it. If you have Scythe of Vyse, hex the one who does all the damage.

Friends and Foes

Friends

Lich's ultimate can easily be countered by the enemies moving away from each other. Because of this, it is best if you have someone who can keep enemies together, such as Enigma's Black Hole or Dark Seer's Vacuum. Faceless Void's Chronosphere can both help and hinder you. If Void is on your team and he chronos the whole enemy team and you're outside, good for you! If Void's on the enemy team and he gets just 1 of your teammates, cast Ice armor on that one teammate. The attack slow will stop Void from doing as much damage as he would normally, and could possibly save your friend from dying. If you're in the chrono, gg!... unless he's focused on someone else. If he's a total failure and deserves to be on fails of the week and just chronos his own team, CHAIN FROST!
Another kind of hero that would go well with Lich are people that can clear creep waves quickly, such as Queen of Pain so that Chain Frost doesn't bounce to the creeps.

Foes

Lich's ultimate can also be countered by having a lot of creeps around. Because of this, if an enemy can somehow get creeps into the fight, such as Chen or Enchantress, Chain Frost will have little effect.
Also, Lich is squishy and will die easily to carries like Drow Ranger. Though Ice Armor counters DPS heroes to some extent through its attack and movement speed slow, this only applies to MELEE heroes. That means that this does nothing to Drow Ranger and other ranged DPS heroes.
If the enemy team has someone with a Blink such as Anti-Mage or Queen of Pain, they can blink out of the Chain Frost's range.
Nyx Assassin's mana burn will kill you easily. Your intelligence gain is 3.25 points per level. Which is quite high.