Sharing Gumballs

[From the notes of Emily Moore (who notes the problem comes from p. 4 of "MG"). Modified by Sam Rebelsky.]

The Petersons wheel their twins past a gumball machine. The twins shout that they each want a gum ball. From past experience, the Petersons know that if they give the twins different color gumballs, unspeakable horrors will follow.

The Petersons look closely at the machine and observe that there are 6 red balls, 4 blue balls, 5 white balls, and 7 green balls. How many gumballs must they buy to ensure that they can give both twins the same color gumball?

The Tennesons have a more difficult problem. They have triplets and still need to ensure that each child has the same color gumball. If they get to the machine before the Petersons, how many gumballs will they have to buy to ensure that they have three of the same color?

The Tinkers have a related problem. They also have triplets, but the triplets object to having everything the same. How many gumballs must the Tinkers buy to ensure that they have three different colors?

The McCaughey's have an even worse problem, given that they have septuplets. If they get to the machine first, how many must they buy? What if there are 20 balls of each of four colors?

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