Early in Chapter 2: The Quarantine Zone, you'll run into the Half-Life: Alyx button puzzle. The puzzle is a big box with a 4x4 grid of buttons, connected to a gate by a cable. To raise the gate, you've got to push the right buttons.
But which are the right buttons? We're here to help. The short video above (and here on YouTube) contains the solution, or you can just keep reading.
(Note: If you're really impatient, just scroll down to the final gif.)
As you approach the area with the puzzle, you'll see a bunch of cardboard start hovering to form the shape of a giant eye. That's... weird. Then you'll enter the canal where you'll find the button puzzle.
You'll also see there are a number of mysterious drawings all over the walls under a bridge. You may notice a lot of those drawings are 4x4 grids of dots, just like the button puzzle. (Along with some very cool artwork depicting the history of Half-Life. Take some time to check it out.) But which grid shows the correct solution?
In the middle of this chamber there's a metal cage containing more cardboard. Pull open the cage by the red handle, and the cardboard will begin to float in the air. At first it's hard to tell what shape it's forming, but if you stand in the correct spot by climbing the ramp on the other side of the cage, you'll see the cardboard has formed into another giant eye.
And what's the eye looking at? The correct solution.
Push the buttons shown in the drawing—third from the left in the top row, second from the left in the bottom row—and the gate will open.