[pronoiac]

Funny thing: if that added number isn't relatively prime to 100 - and 60% of random numbers would share a factor - then they've just made a lot of smaller cycles.

And it's easy to break if you can swap two. Say the warden adds one. If you swap 50 and 100, you have two cycles of 50.

[Retric]

Again assuming the uniform random distribution. Select from (1, 3, 6, 7, 9, 11) is just as random as selecting from 1-100.

I would argue that not specifying the boxes have a specific order is a much larger problem, but saying 'random' does not nessiarly mean what you might think. As to moving the cards, they are specifically excluded from communication.