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The Standard Achievements - 20 of 50
There are 14 for single player:
Wake Up Call, You Monster, Undiscouraged, SaBOTour, Stalemate Associate, Bridge Over Troubling Water, Tater Tote, Vertically Unchallenged, Stranger Than Fiction, White Out, Tunnel of Funnel, Dual Pit Experiment, The Part Where He Kills You and Lunacy will all be given to you just for playing the game.
If you need some help with getting these, you can follow my Portal 2 walkthrough (This links starts you out with The Courtesy Call, but you should be able to just follow them through to the end, or just search for the one you want by chapter).
There are also 6 for co-op:
High Five, Team Building, Confidence Building, Bridge Building, Obstacle Building and You Saved Science will also be earned in the course of the campaign. Just pair up with someone and enjoy.
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Single Player - Skill
Drop Box - You need to get a box onto a button without actually touching the box. For this, you should be able to find a button in Test Chamber 6 that lets you send a cube through a portal and onto the button for the door.
Overclocker - Just finish Test Chamber 10 in 70 seconds. I won't say that it's easy, but the concept is clear. For the most part, the trick revolves around using the same portals much as possible. Do the room a few times to familiarize yourself with the layout, and then just try to map out your portals ahead of time.
Pturretdactyl - This is fairly easy to do, just knock out a turret, and then drop it onto a faith plate.You'll get a few chances to do this.
Schrodinger's Catch - In the old portion of the base, you'll get a few chances to cover cubes with blue gel. There isn't any secret to this. You just need to move fast and grab it before it hits the ground.
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Single Player - Hidden Goals
Final Transmission - One of the most hidden ones in my opinion. In Test Chamber 6, there will be a radio that comes past as part of the cloud of garbage that GLaDOS throws at you. Save the radio and then carry it up with you into the Rat Man's hidden den. It's close to the exit, behind some panels. If you bring the radio in with you, then you'll hear his transmission and get the achievement.
Preservation of Mass - You get this one by saving the companion cube in Test Chamber 7. Just use a portal to bring it up with you after you open the exit.
Good Listener - Another decent joke one. As you escape from GLaDOS with Wheatley, GLaDOS will ask that you please go into the final chamber. If you use a portal to leave the ramp, or get to the final room in any other way, she'll seal you in and gas you, which nets the achievement.
No Hard Feelings - During the escape with Wheatley, you'll find some turrets on the "turret redemption line." There's one that's begging you to save it, noted that "I'm different." You can pick this turret up and take it off of the line to get a few more lines and an achievement.
Scanned Alone - When you remove the template turret during your sabotage efforts with Wheatley, you can just walk into the scanner for an achievement.
Ship Overboard - When you pick up GLaDOS in her potato form, you should be able to find a door hidden behind some random objects. Pull them aside to find the dock for the Borealis.
Door Prize -You'll find several large steel doors along the way marked as "vitrified." if you examine the door or the intercom, you'll get more Cave Johnson lines. I'm going to guess that most people miss the three in the Borealis docking area though. Make sure that you check the dry dock during your run to get them all.
Portrait of a Lady - For a look at the path, just keep your eye open for the locked off room as you leave the first test of the orange gel. If you go inside, you can find a portrait of Cave Johnson and Caroline.
You Made Your Point - For your first test under Wheatley, just refuse to press teh button and solve it. It seems to take a little less than a minute for the achievement to register.
Smash TV - You can smash the monitors Wheatley uses to communicate with you. If you crash into them, hold a turret in front of them or point a laser at them, they will be destroyed. You need 11 in total.
Pit Boss -To get the pit boss achievement, you need to wait for the beginning of the final chapter, "The Part Where He Kills You." When Wheatley calls for you to please come back to the pit that you started on, just be cooperative and run back. Wheatley will be surprised, and then go ahead and crush you.
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Co-Op Challenges - Skill
Iron Grip - I have a feeling that many friendships will be lost over this. You need to never lose the cube in Chamber 6 of the Mass and Velocity Chamber, which is much easier said then done. I suggest doing this once to get the timing down between the two of you, then a repeated effort to actually score the achievement. You'll want the catcher to start their jump at about the same time as the cube drops, for a successful catch.
Air Show - While it's more impressive to do this early in the course, you'll receive plenty of opportunities to do chains of air gestures while you're in excursion funnels later in the course. Just put two together to get this.
Portal Conservation Society - This is probably the hardest. In Test Chamber 3 of the Hard Ligth Bridge course, you may only place five portals. Don't try this on your first go. Get a friend to work it out with you, and then map out the portals carefully. Make sure that you know who's doing what too. It's five portals between the two of you, so it's possible for a missed click to ruin this achievement.
Four Ring Circus - This actually isn't as hard as you might think. You need to go through four portals, but it basically ust takes a cooperative partner. In just about any course, you can put two sets of portals in an "N" setup, where you fall through a partner's portal, exit their's on the ceiling, enter your portal on the floor and exit above the original portal just to fall through the setup again. You then just need your partner to swap out the portals below to keep the chain going and get the achievement.
Triple Crown - Another one that's at least easy in concept. Just get through the 3 chambers in the Mass and Velocity Chamber quickly. You have a minute for each one. If you've run the course before with a friend, it shouldn't be hard to do it on a repeat.
Still Alive - This is one last one that should be easy for repeat players. You just can't die during the Excursion Funnel Course. If you've done it before and remember the steps, it should just be an exercise in caution.
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Co-Op - Hidden Goals
Professor Portal - This is probably the hardest to legitimately do. You have to have a friend who has never played co-op, invite them to a game and lead them through calibration. A random newbie who's not your friend won't do it.
Friends List With Benefits - Theoretically easier than Professor Portal, you just need some people on your friends list who play Portal 2. Give 3 of them hugs and call it a day.
Gesticul-8 - Just go through all 8 gestures once you have them.
Rock Portal Scissors - You need to just play Rock Paper Scissors until you get 3 victories in a row. Just have a patient friend.
Can't Touch This - This is an easy one. At one of the many points during the tests wehere you have to block turrets with a hard light bridge, have someone get into their range and use the dance emote.
Asking for Trouble - Use the taunt gesture in front of the camera on a test course, for each of the five courses.
Empty Gesture - This is really just a timing challenge. While it's funnier as a prank, for the achievement you just need to take turns. Get a light bridge extended over some deadly water, and then water for them to start their gesture. Use the countdown if you need it. Then just move the portals to dump them into the water.
Narbacular Drop - This is just an easier version of Empty Gesture. Just get your party to do a gesture, then put a portal under them and one on a side wall to toss them around.
Party of Three - You can find the hidden companion cube on the last course of the Excursion Funnel course. Look for the building with the desk and put a funnel past it to carry yourself to the secret area and find the cube. Note that you will die from this.