John and Gary are playing a game. John spins a spinner numbered with integers from 1 to 20. John and Gary are playing a game. John spins a spinner numbered with integers from 1 to 20. Gary then writes a list of all of the positive factors of the number spun except for the number itself. Gary then creates a new spinner with all of the numbers on his list. John then spins this spinner, and the process continues. The game is over when the spinner has no numbers on it. If John spins a 20 on his first spin, what is the maximum number of total spins (including the one he already made) that John can make before the game is over?
 A: So on the turn before the game is over, the number that comes up is "$1$", because any other number chosen will have a "$1$" on the next spinner. This can happen any turn, but must happen when the turn before that came up with a prime number. A prime number is the best result from a number with two prime factors, from the point of view of getting the maximum number of spins.
The best case then becomes clear for an initial spin of $20$ - at each spin we shed (at least) one prime factor of the original number. 20 has three prime factors therefore at most three spins later we have only "1" on the spinner. Thus the specific answer here is that maximum total spins are 4 - the original spin to get 20, then one per prime factor of 20. The three possible pathways for four spins are:

*

*$20,10,5,1$

*$20,10,2,1$

*$20,4,2,1$
For the absolute longest game, we need the value between $1$ and $20$ that has the most prime factors - $12=2\times 2\times 3,$ $18=2\times 3\times 3,$ $20=2\times 2\times 5$ but $16=2\times 2\times 2\times 2$ - four prime factors. Therefore, getting $16$, at most four spins later we have only "1" on the spinner. Thus the specific answer here is that maximum total spins are $5$ - the original spin to get $16$, then one per prime factor of $16$ as each is stripped out. The pathway for five spins is thus: $16,8,4,2,1$.
A: If John spins a 20, then Gary's list contains the numbers 1, 2, 4, 5, 10. Thus, these are the numbers on the second spinner.
If John spins a 1, then Gary's list will be empty because there are no positive factors of 1 besides itself. Thus, the game will be over. This yields a maximum of 1 additional spin.
If John spins a 2, then Gary's list will only contain the number 1. Then on John's next spin, we will have the same scenario as above. This yields a maximum of 2 additional spins.
If John spins a 4, then Gary's list will contain the numbers 1 and 2. As we have already found above, spinning a 2 yields more additional spins than a 1, so the maximum additional spins in this case is 3 spins.
If John spins a 5, then Gary's list will only contain the number 1. As above, this will yield a maximum of 2 additional spins.
Finally, if John spins a 10, then Gary's list will contain the numbers 1, 2, and 5. Of these numbers, 2 and 5 have the highest maximum number of additional spins, so this case has a maximum of 3 additional spins.
Thus, of all of the possibilities, spinning a 4 or 10 next could result in 3 additional spins, so the maximum total number of spins is $\boxed{4}$. These would be achieved by spinning 20, 10, 2, 1 or 20, 10, 5, 1 or 20, 4, 2, 1.

