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POW 4: A Sticky Gum Problem

This particular POW we picked up as an introduction to probability. The exact text reads as so:

1. Ms. Hernandez comes across a gumball machine one day when she is out with her twins. of course, the twins each want a gumball. they also insist on having gumballs of the same color. they dont care what color it is as long as they're both the same.

Ms. Hernandez can see that there are only white gumballs and red gumballs in the machine. the gumballs cost a penny each, and there is no way to tell which color will come out next. Ms. Hernandez decides to keep putting pennies until she gets two gumballs of the same color.

Why is three cents the most she might have to spend?

I instantaneously picked this problem up, since I had heard of a similar probelm involving finding a matching pair of black and white socks in a drawer in the dark. The answer, in this case, is 3. Reason for this is such: either you A: get 3 socks of the same color, or B: you get two of one kind and one of the other. Either way, the requirement is fulfilled.

The problem expands, however, when the hypothetical 3rd gumball is introduced, or the hypothetical 3rd sister. For both problems in respective order, the answers would be 4 and 7. This is because of the maximum number of gumballs that it takes to produce necessary results. In the case of the introduction of the third sister, the highest amount of gumballs possible without success would result in 3 unique colors. Any more would result in a guaranteed match of two gumballs. Concerning three sisters and a three colord gumball machine, the maximum would be 7. If you multiplied the maximum number of non matching colors possible (2) by the number of girls, then the result would be 6 before a match would occur between all three of the siblings.

The function can even be represseded in graph form, as demonstrated below, with vertical values representing the number of sisters, and horizontal values representing gumball variety, from lower left to upper right.

4 7 10 13
3 5 7 9
2 3 4 5
1 1 1 1

A relatively plain pattern is revealed in this graph, that can be used to reveal further patterns. This is quite useful, as imagining the patterns for this problem in your head, even on a 4x4 standard, is incredibly difficult.