If you look at the topography no matter which pair you get rid of it will still be your schematic Wookie. There will be one common 555 with the other two feeding it. It may not look it, but the circuit with 6 LEDs is actually symmetrical, with each 555 feeding the other two.
You could do it with 2 555s. One of the things wrong with the design I put up is only two LEDs could be lit at a time. This idea wouldn't look as random, I suspect you could eventually spot the pattern, but it is simpler and might meet the OPs requirements, and 3 LEDs at a time could be lit.
Never mind, upon thinking about it two of the LEDs would be simple flashers. 3 555's it is.
You could do it with 2 555s. One of the things wrong with the design I put up is only two LEDs could be lit at a time. This idea wouldn't look as random, I suspect you could eventually spot the pattern, but it is simpler and might meet the OPs requirements, and 3 LEDs at a time could be lit.
Never mind, upon thinking about it two of the LEDs would be simple flashers. 3 555's it is.