Math problem 83 or something, Real mathematicians can't count Options

[Raijinili]

Say that when you take pistachios, you take ten per handful, you eat the unshelled ones, take off the shells of the shelled ones, and drop the newly unshelled ones back into the bowl.
If you had a bowl of 100 shelled pistachios, how many grabs would it take to eat them all?

[jcdietz03]

This is your claim:
It's not true that there will be less than 10 after 19.

The < vs <= is a technicality. Don't argue technicalities, just provide the correction.

If I change my statement to <=, then it's right. Anyway, I think I did prove it, just follow the logic. The worst case scenario (for maximum pistachios left) is provided. Any other scenario is either the same or better.

I guess I still need to prove that last part though:

If you mixed it up a bit more (left shelled ones in the bowl when they are available) you would have even less than 10 left after the 19th grab, but some would be shelled.

The number of operations left in the bowl is calculated by [# of unshelled]+2*[# of shelled]. Each time you make a grab, the number of operations goes down by 10. After 19 grabs, you will have 10 operations remaining. If there are 10 unshelled in the bowl, you can complete the task on the next grab. If there is 1 shelled in the bowl then there must be 8 unshelled because there are 10 ops left in the bowl, and each shelled represents 2 ops. Meaning nine total are left in the bowl.

Therefore, the quoted statement is proven.