The board, pin by pin
Four pins at one end, four bare pads at the other carrying the same four signals, and every part hidden under the display. No pad is square, so the names printed on the back are the only guide to which pin is which.
Four pins, in this order

GND and VCC are power. CLOCK and DATA are the two signal wires from the last article. The back prints them in lower case, clock and data; the library and the chip's datasheet call them CLK and DIO. Same wires, three habits.
The order is the same at both ends, GND at the top. That is worth reading off the back rather than guessing, because nothing else on the board marks it: all eight header pads are the same oval, so there is no square pin 1 to count from.
What is under the display
Everything. The TM1637 is a surface-mount chip on the display side, in the gap under the display's body, with two 10 kΩ resistors and two capacitors beside it.
The resistors are the pull-ups that hold CLOCK and DATA high, and they go to VCC. That is the most consequential fact on this board: the voltage you feed VCC is the voltage those two wires sit at, and so the voltage your board's pins see. 3.3 V or 5 V is about nothing else.
The capacitors, 100 nF and 10 µF, are across the supply. The chip lights one digit at a time, so the display takes its current in short bursts, and the capacitors keep those bursts from showing up as dips on the supply wire.
The pads at the other end
The four pads at the far end are the same four nets as the header: GND, VCC, CLOCK, DATA. No header is fitted there. They are not a pass-through to a second address (the protocol has none), just the same wires brought out again.
They are useful for passing power and clock along a row of displays. They are also how people build a row of displays that all show the same number, which more than one display goes through.

The enclosure in each six-pack wraps the board and leaves the pins out of one end. The smoked film goes between the display and the window, and it makes unlit segments disappear into the black instead of showing as grey bars.
When it does not work
Correct. The chip, the two resistors and the two capacitors are surface-mount parts on the display side, under the display itself. The back carries the silkscreen, the solder joints of the display's twelve pins, and nothing else.
None of them is: all eight header pads on this board are the same oval. Go by the names printed on the back. Seen from the display side with the header on the left, the top pin is GND and the bottom one is DATA.
The end with the pins. The four pads at the other end are the same four signals brought out again, so either end works electrically, but only one has a header, and fitting one to the other end means soldering.
Solder a 4-pin header into the spare pads and use that end instead. The order is identical, GND at the top, because the two connectors are wired in parallel to the same four nets.
Mounting. They are 4.6 mm, 56 mm apart along the board and 16 mm across it, which is the spacing the TinkerBlock base plate and the enclosure use. They are not plated and nothing is connected to them.
Four wires, one library and a sketch that puts 1234 on the display.
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