Lenny Conundrum - Round #419

I found this on http://lenny-conundrum-answers.blogspot.com/2011/09/lenny-conundrum-round-419.html:

Lawyerbot was attempting to organise his Usuki collection and came upon a box with 63 Deluxe Angel Usukis and 113 Devilish Usukis inside. He also had an additional 72 Devilish Usukis next to him in a bag. Then, his programming went a bit haywire.

Every time he'd reach into the box, he'd pull out two Usukis. If they were different, he'd toss a Deluxe Angel Usuki back in. If they were the same, he'd toss a Devilish Usuki back in. He kept doing this until there was only one Usuki left in the box. 

Which Usuki was it?

This sounds like a tricky question until you take it by cases. Let's look at the possibilities on drawing two Usukis:

So once you break it down you see that the only way of decreasing the number of Deluxe Usukis is to draw two at a time. In every other case you end up with a net change of 0.

Given that there is an odd number of Deluxe Usukis, you can never eliminate them entirely. Assume that you keep drawing them out two by two until one is left. No matter what you do past this point you can never draw two Deluxe and therefore the number of Devilish necessarily decreases until you're out.

So the last one remaining will have to be a Deluxe Usuki (whatever a Usuki is).