Gandalf has assembled a fellowship of eight people to transport the One Ring to the fires of Mount Doom: four hobbits, two men, one elf, and one dwarf. The ring has great power that may be of use to the fellowship during their long and dangerous journey. Unfortunately, the use of its immense power will eventually corrupt the user, so it must not be used except in the most dire of circumstances. To safeguard against this possibility, Gandalf wishes to keep the instructions a secret from members of the fellowship. The secret must only be revealed if enough members of the fellowship are present and agree to use it. Requiring all eight members to agree is certainly a sufficient condition to know the instructions, but it seems excessive. However, we also know that the separate races (hobbits, men, elf, and dwarf) do not completely trust each other so instead we decide to require members from at least two races in order to use the ring. In particular, we will require a unanimous decision by all members of one race in addition to at least one member of a different race. That is, if only the four hobbits want to use the ring, then they alone should not have sufficient information to figure out the instructions. Same goes for the two men, the elf, and the dwarf. More explicitly, some examples: only four hobbits agreeing to use the ring is not enough to know the instructions. Only two men agreeing is not enough. Only the elf agreeing is not enough. Only the dwarf agreeing is not enough. All four hobbits and a man agreeing is enough. Both men and a dwarf agreeing is enough. Both the elf and the dwarf agreeing is enough. Gandalf has hired your services to help him come up with a secret sharing scheme that accomplishes this task, summarized by the following points: There is a party of four hobbits, two men, an elf, and a dwarf. There is a secret message that needs to be known if enough members of the party agree. The message must remain unknown to everyone (except Gandalf) if not enough members of the party agree. If only the members of one race agree, the message remains a secret. If all the members of one race agree plus at least one additional person, the message can be determined. Other combinations of members (e. g. two hobbits and a man) can either determine the message or keep it a secret (it is up to your discretion).