Terminal hacking is a significant skill in that gives you access to a wide range of narrative/combat options. Often, accessing the right terminal and choosing the proper commands can help you bypass huge enemy swarms or turn the tide of battle in your favor. If you want to unlock sealed doors, understand life before the War, and access some of the biggest secrets in the game, you'll need to be a good hacker.
Hacking itself involves finding a terminal. Terminals are usually boxy and old-fashioned, fitting the 1950s aesthetic of the world, but they can also be circular or attached to the wall. If it has a keyboard, it's probably a terminal. While many terminals have freely accessible information, important info is locked behind passwords.
What makes hacking in Fallout 4 so fun is that it is not instant access to a terminal but rather a mini-game that requires you to search through lines of code for the correct password. When you boot up the terminal and start hacking, you'll get 32 lines of code split into two 16-line columns. The goal is to find the password among the many words randomly generated into the code. You only get a few guesses before the mini-game fails, and you have to try again.
To successfully hack into the terminal, you'll need to analyze the Likeness of your chosen word. Likeness represents how many letters match the password. For example, in the first image of the gallery above, RAIDER has a Likeness of 1. This means that one of the letters in RAIDER matches the password. TAVERN has a Likeness of 0, however. This means the password does not have a second letter of A, the only one shared with RAIDER. Therefore, the only remaining options for the password are SHOVEL and REMIND.
Likeness focuses on the order of letters in the word, not the letters themselves. This is why TAVERN's Likeness can be at 0 despite sharing letters with SHOVEL and REMIND; none of the letters are in the same place within the word. The options, meanwhile, share an E and an R
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