The Citadel is known to house some complex puzzles that offer no clues to the puzzle solutions. You might have to do your investigations on your own to figure out how to solve these puzzles. There are 3 Coghearts scattered across the late-game areas, and you need to figure out where to use them. This guide will detail where to locate and how to use them in Hollow Knight: Silksong.
All Cogheart Locations in Hollow Knight Silksong
The three locations you need to visit and collect the Cogheart are the Choral Chambers, Whispering Vaults, and the Memorium. Before you start, it is best to collect the Faydown Cloak ability from Mount Fay, as some areas are hard to reach without it. Once ready, you can start hunting down the Coghearts and figure out their use.
Within each location where the Coghearts are, you need to solve a memory puzzle. The room is filled with bells, and these bells will chime in a particular order. You need to memorize the order and hit the bells accordingly. If done correctly, you will receive the Cogheart.
Cogheart #1 Location: Choral Chambers

Venture out to Choral Chambers and get close to Citadel Spa on the west. From here, you need to make your way up the western section of Choral Chambers to enter a segment on the western wall. It gets easier to get here if you have taken the route from The Slab, but this is also doable from High Halls.
Cogheart #2 Location: Whispering Vaults

The next location you need to go to is the Whispering Vaults. It is easier to get here either via Cogwork Core or Choral Chambers. Once you reach the Vaults, drop down to the lengthy central area and enter the southwestern section to find the puzzle room and the second Cogheart.
Cogheart #3 Location: Memorium

You could use the Songclave exit to get to the Memorium above, then go east and up to find the puzzle in the eastern room. Or, you could also use the Putrefied Ducts exit into the Memorium, head west, then go up and east. Whichever path you take should lead you to the puzzle room.
How to Use the Cogheart in Silksong

The Second Sentinel Automaton at Cogwork Core requires the completed Cogheart to make it function again. After acquiring all three Coghearts, make your way to the lower sections of Cogwork Core. There is a passage that you will come across just below the bench on the western wall. Enter this passage and go through the opening above to get to the sleeping Automaton.
Interact with the glass casing to break it open, then inspect the robot to attach the Coghearts. This activates the Second Sentinel, after which it promptly flies away. You will meet it a second time randomly during your exploration, and after the second encounter, visit the Songclave wish board and pick up a unique wish called Final Audience in Hollow Knight: Silksong. This one will have you defeating the Second Sentinel to get the Reserve Bind.
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