Was the maniple checkerboard formation in republican legions adopted and used to defeat the phalanx . All armies ussually faced each other in a battle line . uninterupted .. but did the maniple frontage with gaps allow the phallanx to bulge ? .. stopping where their was resistance and bulging forward where there was no resistance ? Then could a missile and pilum attack into the sides of the bulges allow legionaries to break into the sides of the phalanx ?
According to Livy, the Romans had first used small round shields and operated as a phalanx "like the Macedonian phalanx", before adopting the manipular formation:
However, there is no visual evidence for this. All early depictions of roman warriors show them armed similarly to a Greek hoplite, not Sarissaphoi. These examples of Etruscan art show hoplite troops:
http://www.kingsacademy.com/mhodges/11_Western-Art/08_Roman/08_Roman.htm
Roman troops would have had similar equipment. My theory is that Livy mistook accounts of the older hoplite phalanx for the Macedonian phalanx.
Nonetheless, it establishes that the Romans reformed their army at some point. Livy states that this was done around 340 BC whilst allied with the Samnites against the Latins:
http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.02.0155%3Abook%3D8%3Achapter%3D8
Now, as for how the Manipular Legion fought, in regards to the gaps, there is a LOT of speculation. Brian Campbell in War and society in Imperial Rome 31 BC–AD 284 states that the gaps would be filled in by other troops before contact with the enemy, since each manipular had two centuries. The century in front would stay in position, the century behind would move into the void (p 51).
So, to be accurate and unhelpful, nobody knows for sure
As has been stated, we are not entirely sure how the maniples actually worked when they came in contact with the enemy.
I would say that the gaps in formation do not really hold much tactical benefit once engaged in combat. The formational gaps were more useful on the march as it allowed the army to maneuver over rough terrain without becoming hopelessly broken up. Each maniple could advance as it needed and the line would then be checked and reformed by the officers.
Gaps would also seem to help allow skirmishing lines to retreat back into the formation after the initial phases of a fight. When it came time to do battle it is most likely that the troops of the front row would become a single line while the rear troops remained with gaps to allow for easy reinforcing capability.
The true advantage of the checkerboard system was that it was a part of the Roman Triplex Acies. This system is unique because it keeps substantial units in reserve which is particularly Roman. These troops could remain rested and then committed to parts of the battle which needed it the most. Combine this with the maniples which had the disgression of acting independently under their centurion and the Roman army was much more flexible than other armies. This is what made it so powerful and not necessarily the gaps of the checkerboard formation.
I've got a video which describes this with animations which could help your understanding: http://youtu.be/OWAkNNWo920