The Bastille of Intellect puzzles in Nightingale are little treasure vaults scattered throughout the world. Working out how to solve these will net you some loot and potentially a crafting or building diagram to use when exploring the Faewilds. The weird thing about each Bastille of Intellect is that it contains an altar with space for a Puzzle Core, but this isn't actually required to solve it.
For those only just venturing into their own realm, you might want to know how to increase your gear score so you can take on your first Site of Power. There are lots of predators and Bound roaming each realm so it's a good idea to be well prepared with weapons and equipment. Either way, here's how the Bastille of Intellect puzzles work, plus some speculation about Puzzle Cores, their intended purpose, and why they're likely bugged right now.
How to solve the Bastille of Intellect
These puzzles are quite simple to solve, requiring you to copy a sequence to open the gate and get your reward. At the centre of each puzzle is an altar with an outlined empty space that says Puzzle Core—don't worry about this right now since you don't need it to complete the puzzle. Scattered around the nearby area you'll find little pillars labelled as Puzzle Pieces, which will periodically let out pulses of blue light and make a chiming noise.
The way to solve the puzzle is to watch for the sequence in which the Puzzle Pieces chime, and then copy the sequence back by activating each in order. If you get it wrong, some Bound enemies will spawn and attack you, so make sure you're ready to fight before attempting it. Luckily, these sequences are very simple to input.
Get it right and the gate will slide up, letting you loot the interior. For me, this contained a Favour of Endurance and once I released that, a Hope Echo appeared on the altar at the centre which gave me a building blueprint.
So what is a Puzzle Core?
The weird thing about Puzzle Cores is that you can actually find them in enemy-filled Occupations and other areas of Nightingale. These objects are little blue spikes with stone bases and look like they would fit perfectly into the outline at the centre of the altar in the Bastille of Intellect.
Here's the thing, though; you can't pick them up right now. My speculation is that inserting a Puzzle Core into the altar automatically solves the puzzle and gets you the reward. The fact that the Puzzle Core outline disappears from the altar once you finish the puzzle feels like further evidence of this.
Since this is an Early Access game, I think it's likely that the fact you can't pick up Puzzle Cores right now is a bug, since they're clearly intended to be used in each Bastille of Intellect. Either way, you don't need them to solve the puzzle and get the reward, so no harm done.