The 4th build is a troll build for Ryze, that has a lot of AD output as well as Spell power. I have gotten tired of missing kills because I was auto-attacking while waiting for CD.

The troll build will give you 304 AD (not including bonus damage from Trinity Proc), 110 AP, and 4710 mana. The downside is the lack of MR.

Here is a Damage Calculation based off of Builds 3, 4, and 5. I will update this once I do the calculations for build 1.

Build 3
Ability Damage Per Ability
Overload: 140 + 20% AP (70) + 8% Max Mana (391) = 601
Rune Prism: 200 + 60% AP (211) + 5% Max Mana (244) = 655
Spell Flux: 130 + 35% AP (123) = 253

Attack Damage per 5 seconds
Attack Damage: 447.6 (Base 104)
Attack Speed: 4.305 (Base .861)
Crit Hits/Chance: 0/0%


Build 4
Ability Damage Per Ability
Overload: 140 + 20% AP (22) + 8% Max Mana (376) = 538
Rune Prism: 200 + 60% AP (66) + 5% Max Mana (236) = 502
Spell Flux: 130 + 35% AP (39) = 169

Attack Damage per 5 seconds [Without Trinity Proc]
Attack Damage: 2,644.7 (Base 304)
Attack Speed: 5.242 (Base 1.048)
Crit Hits/Chance: 1.73/33%

Attack Damage per 5 seconds [With Trinity Proc]
Attack Damage: 3,939.8 (Base 456)
Attack Speed: 5.242 (Base 1.048)
Crit Hits/Chance: 1.73/33%

Build 5
Ability Damage Per Ability
Overload: 140 + 20% AP (101) + 8% Max Mana (333) = 574
Rune Prism: 200 + 60% AP (304) + 5% Max Mana (208) = 712
Spell Flux: 130 + 35% AP (177) = 307

Attack Damage per 5 seconds
Attack Damage: 447.6 (Base 104)
Attack Speed: 4.305 (Base .861)
Crit Hits/Chance: 0/0%


Total Ability Per Nuke Cycle (Roughly 5 seconds)
Ability Cycle: Q-W-E-Q-R-Q [Reason for this cycle is in the Skills Section)


Build 3 3,157
Build 4 2,623
Build 5 3,355

Total Damage Output per 5 seconds
Build 3 3,604.6

Build 4 5,267.7 Without Trinity Proc
Build 4 6,562.8 With Trinity Proc

Build 5 3,802.6

Build 3
Pros:
Tanky
High Max Mana
High Spell Damage Output
Spell Vamp for sustainability
High HP level

Cons:
Minimal MR
Reliant on casting spells
Can be shut down with silence
Slow


Build 4
Pros:
Great against AD carries
Can be silenced and have a high Damage output
High Max mana
Decent HP
Decent speed (426)

Cons:
Can be nuked by AP characters because of lack of MR
Susceptible to CC
Lower Spell Damage

Build 5
Pros:
EXTREMELY high Spell damage output
Spell Vamp

Cons:
Susceptible to CC
Silence WILL kill you
Glass cannon
lack of MR

I have 1 build for Ryze that I use, but I vary it depending on what type of team I'm playing against. To start with, Ryze is built VERY tanky and this build makes him very powerful as well.
You want to build a lot of mana on Ryze, and the items that help build this out are mainly tanky items to give you the benefits you will need to tank just about any team.

For Classic Masteries, I go 9/17/4. I don't use the new Offense Tree or Utility Tree like some others might use it because I prefer a much tankier Ryze.

I put 9 into Offense to reach the 10% Mpen.
17 points go into Defense to put 2 points into Honor Guard to reduce all damage taken by 1% as well as putting 2 points into Indomitable to reduce damage by 2 points. The main reason of going into the Defense Tree is to reach the CDR per level Mastery Enlightenment. This gives you 8.01 CDR at level 18 + the 4% from the Offense tree for a total of 12.01$. While you would only need 10% CDR on Ryze from masteries which you can get from going 9/0/21, you benefit more from going into the Defense tree.

For Dominion Masteries, I go 9/18/3. Since I use Ghost and Revive while taking bottom, I use the Offense and Defense Spell Masteries for the extra speed boosts associated with them.

I use the same core items when I play Classic or Dominion.

Tear of the Goddess
Sorcerer's Boots
Glacial Shroud
Negatron Cloak
Rod of Ages


I generally build these items in this order, but alternate Negatron Cloak and Glacial Shroud depending on how much mage/phyisical damage the player I'm laning against does to me.

By the end of an average Dominion match, I will generally have Tear of the Goddess at 700+ Bonus Mana, Sorcerer's Boots, Frozen Heart, Rod of Ages, Odin's Veil, and a situational item.

By the end of an average Classic match, I will have Archangel's Staff, Sorcerer's Boots, Rod of Ages, Frozen Heart, Banshee's Veil/Abyssal Scepter/Randuin's Omen, Void Staff/Rabadon's Deathcap/Force of Nature/ Will of the Ancients.

The last 2 Items are strictly situational. If I need more damage output, HP, and MR, I get BV. If I need more MPen, I'll get Abyssal Scepter or Void Staff depending on how AP heavy their team is. If their team is heavy AD, I'll get Randuin's and/or Thornmail in addition to my Frozen Heart to really tear them down (That would be +274 Armor).

One of the items that I am still testing is Will of the Ancients. The Spell Vamp on that, in addition to Ryze's Ult, makes your sustainability better in some cases, but that's something you'll have to gauge on your own.