This is a famous problem in mathematics education. Suppose you lined up 1000 dolls in a row, all facing up. Then suppose you walked down the line and turned over every other doll. Now, one-half of the dolls would be facing up (those representing odd numbers) and one-half of the dolls would be facing down. Now suppose you walked down the line again, and this time turned over every third doll. That is, if it was facing up, you would turn it so that it was facing down; if it was facing down, you would turn it so that it was facing up. Continue in this manner 1000 times. At this point, which dolls will be facing up and which dolls will be facing down?
The following suggestions may help you with the problem: Some students find it helpful to make the problem smaller—for example, 50 dolls or 25 dolls. Some students find it helpful to model the problem. What other objects could you use to represent the dolls? Most students find it helpful to make a table to represent their findings.
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