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Cheat Engine
Cheat Engine is a program that will allow you to do various things with the game that you can't normally do. You can make yourself invincible, give yourself whatever items you want, make your own challenges, etc. It's safe and I've used it before to try out different things. There's basically no risk of being banned from Steam since Isaac has no multiplayer component. You'll also need a cheat table (unless you want to do your own programming with it), which you can find here. Make sure you have the latest one. You may have to go back a few pages to find one that's just for the base game.

FAQ

How do I open those doors with the sword image?
There are two types of these rooms. One has a plain sword icon. You just have to be at full health. But if you have enough soul hearts to make up the difference (IE: 2 hearts on 3 full, one soul heart) the door will open. If you have Wrath of the Lamb, another type will show up. The icon for this room is a bloody sword. To enter this room, you must have one heart or less health remaining. Soul Hearts don't count towards the total.

How do I get all these items behind pits?
There are lots of ways, but for most rooms you will see some rock tiles near the pits. Blow up a bomb on the side opposite the pit and it will fill the hole. You can't always do this, but there are items that let you fly, levitate, even one that attracts items.

There's a room with a ? on the map, how do I get there?
Bomb the center of any wall next to it will do it. Some items will teleport you there also.

What does * item do? and what is the bar next to it?
You can check the wiki for a fairly full item list, but there are a few item types:

Activated items - These are the ones you use with Space and can be anything from giving you a buff to healing you to FIRIN MAH LAZER.

Collectable items - These are the ones you just pick up and get the effect. Usually they give you a bonus to your stats but they can really do anything like letting you fly, granting you followers or changing the way your tears work.

Tarot Cards/Playing Cards - There are 22 cards that are a one-time use or last for one room for the base game. Wrath of the Lamb adds an additional 5 more cards that can be picked up. These new cards are named after playing cards (2 of hearts, Joker, etc) rather than having strange occult names. You may only carry one card at a time.

Pills - 17 pills that can either be a good or bad effect. Wrath of the Lamb adds an additional 4 pills. These are unidentified at first (unless you have the PHD item), but they'll be identified once you use them so you can recognize them. The colors are randomized at the start of every game. You will only encounter 6 of the 17 or 21 variations of the pills on any one game. Acquiring the PHD or the Lucky Foot will eliminate the possibility of finding pills with negative effects. The PHD will also change any currently acquired negative pills into their opposite positive effect (ie health down pills are now health up).

The bar is the recharge time. You need to clear rooms to recharge it (though there is a certain item...) and unfortunately the amount of bars doesn't represent the amount of rooms needed. Some are fast, some take a while.

Trinkets - Wrath of the Lamb introduces a new item category called trinkets. These items mostly offer passive benefits to Isaac. There are a few that function as followers. You can find them in chests as well as win them from the new Fortune Telling machine. They do not take up regular item space so you can hold one trinket and one active item at the same time. There is a collectable item pickup that allows you to carry two trinkets at once instead of the normal limit of one.

What is Luck?

[there was a luck write-up here but it's kinda outdated. check my pastebin. -color printer]

Tips

? Bomb rocks that are a different color for some pick ups.

? Bad Trip pills will normally damage you, however if you're at 1 heart or less they act as a full heal
? Slot machines can have some very nice rewards. It can be quite tedious though if you have a lot of money to burn.
? Blood Donation machines normally take you down to 1/2 heart before using soul hearts. However, in the Womb and beyond they take a full heart and will instantly kill you if you donate with 1 heart remaining regardless of soul hearts.
? Using an item that grants invulnerability allows you to donate to the Blood Donation machine without taking damage. If you have Wrath of the Lamb, this also applies to Sacrifice Rooms and Devil Beggars.
? Bombing machines like blood machines or arcades will give you some pick ups too, so if you're really hurting for a heart or something...
? You can blow open doors with explosives/bombs to bypass particularly tough rooms. This is especially useful in later levels where you take a full heart of damage if you get hit. Doing this doesn't clear the room, so if you have to backtrack be prepared to use another bomb or beat the room.
? The Devil's Shop lets you trade powerful items for hearts, you can randomly teleport there and sometimes it will appear after killing a boss.
? It can be a good idea to spend your money as much as possible. The farther you get in the game and the more money you have, the more likely it is you will encounter Greed/Super Greed instead of an Item Shop or in the Secret Room.
? During the final boss fight in the Depths, try to get the boss to stomp on its eye for massive damage
? The Bible will instantly kill the final boss of the Depths and the final boss of the Womb. However, using it on the final boss of Sheol, Cathedral and presumably The Chest will instantly kill you.
? Eternal hearts are white colored hearts added in Wrath of the Lamb, finishing the level with 1/2 eternal heart will grant you a new heart container. Also, acquiring a full eternal heart at any point in the level will instantly grant you a new heart container. Be aware that Blood Donation machines and Devil Beggars will take eternal hearts before regular ones. It's probably best to save picking up eternal hearts until you are ready to leave the level.

How do I use Cheat Engine?
First, you need to acquire the program and the cheat table as already mentioned. Then, run Binding of Isaac and Cheat Engine.

Occasionally, I've had some problems with stuff not wanting to be checked on or off. Reloading the program seems to fix the issue. It works pretty well for the most part.

Known Issues/Bugs
Achievements have been wonky since the release of Wrath of the Lamb. Sometimes achievements won't unlock despite meeting the conditions for them. Some people have been able to get the Steam achievements to register by simply restarting Isaac. Others have used Steam Achievement Manager to get the ones they acquired, but didn't get credit for.

Unlocked secrets in the game may not show up either in the stats page. In my case, the counter for found secrets keeps going up but my last 20 or so secrets don't actually appear there. The items themselves are showing up in game properly for me at least.

The secret room in The Chest will probably trap you if you enter it. It normally can't be entered, but you can teleport yourself in there if you get unlucky. You can't bomb your way in or out. Not even X-Ray glasses will let you enter or exit. Working properly in latest patch.

Trying to charge Brimstone while firing will cancel the beam.

Conquest can replace War occasionally even though the boss intro shows War.

As of Patch 1.4, secret rooms now appear to only border one room. Working properly in latest patch.