The Geiger counter from MightyOhm
includes a pulse output :
The PULSE connector (J6) has the following pinout:
1. VCC (nominally 3V)
2. pulse output – a short (100us) active high pulse every time the geiger tube fires
We need to connect the GND pin (the one on the right) with a ground pin on our arduino and the pulse output (middle) with a pin on the arduino so we can detect the detections.
Cut 3 sockets' worth of female header. (You can get away with two if you have to.)
Cut yourself two lengths of wire long enough to reach between your Geiger counter and your arduino. It's helpful to have two different colours, but not essential. I'm something of a traditionalist and have used black for ground and red for the pulse output...
Solder the ground wire to one of the outer pins and the pulse wire to the middle pin.
This assembly should fit nicely onto the male pins on the Geiger counter: