Termites have a caste society. This means that they have a king, a queen, ladies-in-waiting who want to be queen, soldiers, and workers (pawns).  A war with termites does not end if you kill the queen and corner the king. In fact, there are several reasons why attempting to destroy termites by killing the queen will not work:

The "Ladies-in-Waiting" Molt into Queen-dom

Termites always have back-up plans. If it was not for the destruction that they do to human abodes, you could almost respect their ability to plan ahead. There are several "ladies-in-waiting" who help tend the queen, and they are generally known as "secondary queens."

If the queen is killed via human extermination, or by a warring termite-dom or ant militia, a secondary queen (or two or three) will immediately begin the molting process to become the new queen. The secondary queen that gets through the molting process the fastest, wins. The others either have to strike off on their own or risk being killed by the new queen. In this way, the nest is NEVER without a queen, and will not die unless you exterminate them all.

The King Will Find New Mates or Concubines

King termites are very much like human kings. Their main duty is to the primary queen and mating. However, termite kings will go in search of concubines or a new queen whenever the first queen is killed. If the king remains alive in the nest, the nest will carry on with the king and the new queen.

Warring Termites Can Replace the Nest You Just Tried to Kill Off

Termites have warring "tribes" that will invade and attempt to take over another nest. Even if you are successful at killing off most (if not all) of the nest, you leave it open and vulnerable for an invading soldier termites from another nest. Then the process starts all over again.

The Only True Way to Kill Termite Kingdoms (and Win Your Chess Match Against Them)

Termite extermination by a professional exterminator is the only way you can win this war. Not only does the exterminator terminate the entire nest that is munching on your property, but he/she kills off future generations of termites and other places where other termites are nesting on your property. This may take a few visits, considering the survival techniques that termites are known for and utilize, but eventually, they will be obliterated.

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