I see what you mean, that was my first thought, but why wouldn't the two LEDs just light with half the current of the single LED? I'm seeing a current divider here...
The current would be divided if the LEDs obeyed Ohm's Law, like resistors do. But they are semiconductors.
An LED has a minimum forward voltage, say 2V, below which there is negligible conduction. The forward voltage increases only modestly with current.
For LED1 and LED3 to conduct in series, the voltage across them would have to be 4V. But by that voltage, LED6 would be conducting a huge current. So the current in LED6, and resulting voltage drop in the pin resistors, will ensure that the voltage across the series LEDs is nearer to 2V than 4V, and neither of them will conduct at all.