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why can't MENSA... have... teams that come up with solutions... Or inventions?
That is an interesting question. As a possible answer, some may find that approach to be exploitive; Issues of copyrights and intellectual property come to mind. It is my understanding that at one point these type of think-tank opportunities existed (facilitated by an outside firm, but were not televised) and there may have been some concern that others were benefitting financially for the work done and ideas generated by the synergy of think-tank participants.