They may then hypothesize that the number entered will equal the number of generations required to get to 4. The notion of limits is a powerful mathematical concept. Students often test huge numbers before realizing that they need to be more deliberate in their experimentation. Eleven generations may seem like a long time until a group of kids tests the number, 27. Gasps and a chorus of wows can be heard when 27 takes 109 generations.
Students often list some of the following hypotheses.
The number is greater than 25 We then test each of the hypotheses and discard most of them. The cubed number hypothesis is worthy of further investigation. One student may suggest that only odd perfect cubes take a long time. I then suggest that the other students find a way to disprove this hypothesis by finding either an odd perfect cube that doesn't take a long time or an even cube that does. The more you play with this problem, the more questions emerge.