Sharing Scenarios
We can get even more complicated. Maybe the general and two colonels can launch the missile, but if the general is indisposed, it takes five colonels all together. Embed the message in a polynomial of degree four. Then, give each colonel an evaluation, and give the general three different evaluations. Put the general together with any two colonels, and they can solve for the five unknowns. Five colonels could also solve the polynomial. But a general and one colonel only have four equations. They can't reconstruct the message and neither can four colonels.
In fact, any sharing scenario you can imagine can be modeled using variations on this scheme. A message can be divided up among two delegations, so you need two people from the seven in Delegation A and three people from the 12 in Delegation B. Make a polynomial of degree four that's the product of a linear and quadratic polynomial. Give everyone from Delegation A an evaluation of the linear polynomial, and give everyone from Delegation B an evaluation of the quadratic polynomial. Grind through the math yourself; it will work.
As an enhancement to the scheme, instead of making the message the constant term of the polynomial, split it into pieces and make the secret the XOR of all the coefficients of the polynomial. Or make different coefficients different messages.
The important thing to remember with all of these schemes is that the random numbers have to be generated properly. When inventing a random polynomial, all the coefficients have to be random. If they are not, then the scheme is only as good as the random number generator that generated them. If your random number generator is good, you can divide messages up however you like without any fear of anyone stealing the recipe for your new and improve TasteLess burger sauce.
Bibliography
Shamir, A. How to Share a Secret.
Simmons, G.J., How to (Really) Share a Secret.
