# July 2018 Puzzle Periodical - Marble Math

By Dr. Benjamin E., NSA Research Mathematician & Dr. Sean W., NSA's Super Genius Extraordinaire (ala Wile E. Coyote), NSA Research Computer Scientist

Creator Challenge Difficulty Rating: Easy - Hard

## Problem

1. (Easy Difficulty) You are given two bags of marbles, one contains only white marbles and the other contains only black marbles. You decide, for fun, to randomly take a number between zero and 20 marbles from each bag and put them in a bowl. Once you've done this, you mix the marbles up and pick two marbles from the bowl sight unseen. If the two marbles are both black, you put a black marble back in the bowl. However, if at least one of the two marbles is white you put a white marble back in the bowl. You repeat this process (drawing two marbles, putting one marble back based on this rule) until you are left with a single marble in the bowl, in which case you note the color of the final marble. You then put all the marbles back in the correct bags and repeat the experiment again with different numbers of white and black marbles. After doing this lots of times, you notice that sometimes there is a black marble left and sometimes a white marble left.
Can you figure out when the last marble will be black and when it will be white?

2. (Easy Difficulty) This is same as the first part, except this time when you draw two marbles, if at least one of the two marbles is black then you put a black marble back, while if both of the marbles are white you put a white marble back.
Can you figure out when the last marble is black and when it's white?

3. (Hard Difficulty) For the third experiment, you put a black marble back if and only if the two marbles you draw are different (i.e. one white and one black). If they are the same color (both black or both white) you put a white marble back.
Can you figure out when the last marble is black and when it's white?

Hint:

For all three parts, try working with small numbers first. For example, try all the cases where there are two marbles in the bowl, then three marbles, etc. When you find a pattern, see if that pattern must always hold.