Fallout Shelter

For a quick and free way to gain lunchboxes that are needed to unlock legendary characters, you can do the following. NOTE this will only work on Windows 10, not Xbox (because you can't adjust the date, only time zones on Xbox One).

1) Create a new vault
2) Build the minimum power, water and food resources
3) Have however many dwellers you wish inside the vault.
4) Minimise Fallout Shelter window
5) Open the "Adjust Date/Time" Setting in Windows 10
6) Turn off "Set time automatically" and click the change date and time button, adding exactly 7 days to the current date.
7) Alt tab back to the game, it should reload in about 1 second and display a completed daily report.
8) Click to collect all daily rewards including the lunchbox. Click off the report so it closes.
9) Alt tab back to the date time window and enable "Set time automatically"
10) Alt tab back to the game so the in-game time re-adjusts.
11) Alt Tab back to the date time window and repeat from step 6

This should allow you to get infinite lunchboxes, one every 10-20 seconds.

NOTE: Make sure to go back to the automatic date time, if you continue to advance 7 days each time the game seems to crash. Reverting back to the present each time stops this happening.

Update: It has taken me a grand total of 631 lunchboxes to get 20 legendary characters. Imagine how much that would have cost if you'd purchased them...

Edit: One final precaution, only open the lunchboxes when you have the present time set. Do not open them in the past or future. I found when I did this my achievement progress would stop updating. Maybe a coincidence, but I wouldn't risk it.

Welcome to the grind.

This achievement is completely luck based and as such will either come relatively quickly or be one of the last you unlock.
There are 23 legendary dwellers in the game, all of which are characters from the fallout series. While this may make the achievement seem worse, I have James twice and can confirm duplicates do add to your percentage towards this achievement!
As far as getting the legendary dwellers, it is entirely possible to have them show up at your vault door. The chances are incredibly slim, and hard numbers on chance are difficult to come by, but 3 merged radio rooms fully upgraded and staffed with high charisma dwellers will increase the odds. [EDIT] While this was possible in the early days of the mobile version, it is now confirmed that in all current versions of the game the only way to get legendary dwellers is lunchboxes.

The only way of getting legendary dwellers is through lunchboxes. I bought 40 as a test and ended up with 8 legendary dwellers. You may get more and you may get less, but either way youre looking at a fair amount of money if you intend to get this solely through purchasing lunchboxes.
(Note: it seems I got incredibly lucky with my first set of boxes, Ive since opened about 40 more and got no dwellers. While high numbers of legendary boxes are possible, the consensus from comments seems to be around 1 dweller per 20 boxes)
Luckily boxes can also be earned through quests and objectives, so keep hammering at those and hope to get lucky!

As a side note; almost all legendary dwellers come with legendary outfits, weapons or both. The weapons and outfits they carry will add to your percentage for their respective achievements.

EDIT: Sasha has mentioned in the comments that there are at least a few quests where legendary dwellers can join your vault upon completion. With this being the case and the fact you can earn lunchboxes for quests, it would seem if youre willling to spend money on the game then buying nuka cola quantum (for instant travel time) would be more beneficial than buying boxes!

As with many of the other achievements, all 20 dwellers must be in a single vault.

This guide is to compile scattered information into a more descriptive guide.
I sincerely recommend reading through the 'common questions/answers' section pronto!
This guide will be in two sections:
1) How to obtain Legendary Dwellers
- 1.1 The 'Freebie' Dweller
- 1.2 Quests: Which?
- 1.3 Lunchbox Method (Credit to British Legends)
- Update: Picture of all Legendary Dwellers + Names -

2) Common Questions/Answers
- 2.1 Do duplicates count?
- 2.2 My Lunchbox Method isn't working?
- 2.3 Have I glitched my achievement?
- 2.4 Can you get Legendary Dwellers from the Radio Station?
- 2.5 What about the Piper-Quest?

1) How to get Legendary Dwellers

1.1 - The Freebie Dweller
To begin with - http://imgur.com/a/daeix - , when you begin you may buy a 'starter pack'. It does cost real money but does include a legendary dweller for a relatively low price. It's not required, I didn't, you don't have to, but this is your time/money graph not mine.

1.2 - Quests
There are three legendary dwellers to be gotten from quests:
1) Three Dog - Horseman of the Post-Apocalypse Part 3 - Quest 3/6
Has been reported that some people have had the opportunity a bit later in the
quest-line than this, but he *will* be in this quest-line unless you turn him down.
2) Preston Garvey - A Settler Needs your Help! - Quest 5/5
Just remember to *accept* him into the vault at the end.
3) Sarah Lyons - Climbing the Ranks - Quest 4/4
Same as above.

1.3 - The Lunchbox Method (Windows 10 only)
This method is the method of the previous guide from 'British Legends', I give him full credit and recommend you read his guide for this part, but if not here's a brief summary.

Firstly, if you weren't aware you get a free Lunchbox once a week from your objectives interface (the thing that pops up when you enter the game each day).

While you're in-game: go into your time & language settings (sub-part of your general computer settings), then date & time, then turn your 'set time automatically' setting off, change the date to 7 days ahead, collect lunchbox and open the lunchbox before you change the settings back. Then, turn on the auto-update feature, turn off again and repeat. Here's a picture! http://imgur.com/a/JX0Yw

It does take a while, yet this is by far the quickest method.
It took me 492 Lunchboxes after getting the 3-quest dwellers: http://imgur.com/a/VAMbQ

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Update: Here's a pic of all the Legendary Dwellers:
http://imgur.com/a/kSQTd

And their names, in alphabetical order:
Abraham Washington, Allistair Tenpenny, Amata, Bittercup, Butch, Colonel Autumn, Confessor Cromwell, Dr. Li, Elder Lyons, Eulogy Jones, Harkness, James, Jericho, Lucas Simms, Moira Brown, Mr. Burke, Preston Garvey, Sarah Lyons, Scribe Rothchild, Star-Paladin Cross & Three-Dog.
---------------------------------------------------
2) Common Questions/Answers

2.1 - Do Duplicate Dwellers count?
Yes. Here's a screenshot of my duplicates after getting my achievement:
http://imgur.com/a/vsrnw
Moira is highlighted to prove it's her as she got pregnant before I snapped the pic'.

2.2 - My Lunchbox method isn't working... why?
The most common problem is that you are part of the way into your week and have set it 7 days in the future, which does not equal a lunchbox.
Turn your auto-set time back on to the correct time, then go into your objectives and check your current progress into the week; then set the time ahead however many days you need to get to the lunchbox.
THEN after that FIRST 'fix', set it 7 days ahead until your done your lunchbox boosting.

2.3 - Is my achievement glitched?
If you've been opening lunchboxes on a different day than you acquired them, possibly yes.
By this I mean, opening doing the time/date trick, obtaining the lunchbox but then switching back to the correct time before you open it.
This doesn't glitch it for certain, but it has been reported to.

2.4 - Can I get Legendary Dwellers from my Radio Station?
No. This has been confirmed by developers, it was possible for a short time but not now.

2.5 - I heard about a Legendary-Quest for Piper?
Yes, you did. However, this is an exclusive for the Apple version.
I know, upsetting. But we have the time/date method and they don't.

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I hope this guide has been helpful! If you have any negative critique or ways I can update this to be more helpful, feel free to inbox me!

You probably wont get this achievement without spending any money or cheating unless you are a very very dedicated person, I have opened over 150 lunchboxes now and only have 9 legendary dwellers getting 1 from the quest lines. Remember that duplicates do count towards the achievement. There is a way to forward time on your pc , it wont work on phones or consoles. Here is a video showing how to get unlimited lunchboxes.

As a small addition to all the other solutions already mentioned, I would like to share my experience with you.

First of all, you should know that I used the method where you adjust the date on the computer. If you don't know how this works in detail, take a look at the solution of British Legends.

This is what I found:
As I said before... I don't know if I was lucky, but if my solution doesn't help you please write me or show me an excessively bad rating and I delete the solution again

I just thought it wouldn't be wrong to share this information with you.

^{If you have any questions, suggestions, or critic, the best way to tell me is the comment-section :)}
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This solution in German:

For the most part, follow British Legend's guide. That's what I did. However, who wants to sit around for hours and sit while clicking back and forth on the time. I used macro recorder to record my clicks and keystrokes.

However, macro recorder has a demo version. Make sure the script repeats 9 times, not 10, or else you will get notifications about needing to purchase the full version. Bring the playback speed from 100 to 1000. This will speed up your clicks.

Now, download another regular auto clicker, and set it up to "play" the macro recorder every time it stops the script. I recorded the time and added a 2 second gap just to make sure it didn't mess up the script by clicking before it was finished.

I got 600 lunches boxes, and used 376, meaning I got an average rate of 18.8 legendary dwellers per lunch box opening. To be safe, I did 600 due to some other comments, but you're probably safe around 500~400. You can also use the auto clicker to open up your lunchboxes, however I would advise making sure rare vault dwellers don't clog up your waiting list outside the vault (If that can even happen).

You can also get the

&

while looking for this achievement if you didn't already get it through natural progression.