logical deduction - The Illuminati and the Knights Templar

Some people in the American Senate are working for secret organizations. Some work for the Illuminati, some work for the Knights Templar, and some work for both. The are also innocent citizens who are unaware of these organizations. Both organizations meet in secret with masks and hoods respectively, so that the members cannot identify one another outside the headquarters.

One senator, Reginald, works for the Illuminati, and likes his colleagues, but prefers the Knights Templar headquarters. He finds an arcane rule in the ancient texts, that gives him a brilliant idea. He will turn each Templar into an Illuminati and each Illuminati into a Templar. That way, he becomes a Templar and gets to hang out in their headquarters, but with his old friends.

The rule goes as follows: If Reginald speaks the name of a senator in the senate and touches his left shoulder, that senator becomes a member of the Illuminati, if he touches his right shoulder, the senator becomes a member of the Knights Templar.

This mechanism is meant to be used by people who know to which group the senator in question belongs. To avoid abuse by people using it blindly, the following rules are added:

• If a citizen is named, he remains a citizen.


• If the senator was already in the group he is assigned to but not in the other, things stay as they are


• If the senator was not in the group he is assigned to, but he is in the other, he becomes a member of both groups.


• If the senator was already a member of both groups, he is removed from the group he was supposed to be assigned to.

Can Reginald, who does not know to which group any Senator belongs, implement his plan, and turn every Illuminati into a Templar Knight and vice versa? The shared members should end up shared, and the citizens must never, even temporarily, become part of either group.

Answer

For each member of the senate, say the name and touch the shoulders in LRL order (or RLR)

For Illuminati (L), first touch (L > L) will do nothing, second one (R > LR) will make senator both, then third (L > R) will remove it, turning an Illuminati (L) to Templar (R).

For Templars (R), (L > LR), (R > L), (L > L), turning a Templar (R) to Illuminati (L).

For both (LR), (L > R), (R > R), (L > LR), keeping as both.

Normal people will just wonder why all the shoulder tapping.