Just click any piece on the right to pick it up, no need to hold your mouse button down and slide it to the left and click again once it's at the correct spot to click it into place.
You'll have some outlines and a faded image to help you. And that's it. Easy, right?

Do note, the hitboxes are a bit chunky so it may take you a few tries to get the piece to click into place, it's not you. It's the game.
There are also 3 missing achievements, which means that at the moment, you're unable to 100% it.

Below you can find all the images of the completed plates.

All 20 Anatolian plates are in order from top left to bottom right.

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20.

All 20 Kitty plates are in order from top left to bottom right.

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20.

All 20 Cyber plates are in order from top left to bottom right.

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8.

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20.

Thank you for visiting my guide and have fun!

With love,
Tchii