Time for another puzzle, to get the juices going again!

Instructions:

1. Deal out 2 full packs of cards in one long row, all face down (or in circles or loops or whatever, as long as you can count along them!).
2. Starting with the first card, turn over every card.
3. Starting with the second card, turn over every second card.
4. Starting with the third card, turn over every third card.
5. Starting with the fourth card, turn over every fourth card.
6. And so on, until every card has been used as a starting point.

Question: how many cards are now face up?
Bonus: what will be the effect of including the jokers?

Solution over the fold

The answer is: 10. They are the cards numbered 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the perfect squares).

Explanation:
The number of flips a card makes is determined by how many factors it has (including one and itself). An even number of factors means that the card will be face down ultimately. An odd number of factors would mean it will be face up. But of course a number can’t have an odd number of factors since each factor has its corresponding factor to form the product. However, the square numbers are unique in that one of the pairs of factors is the same number and thus does not get the double flip bestowed by an even number of factors. They are the only ones left face up.

Adding the jokers makes no difference. One would only see a difference if there were 121 cards in the mix at which point it would be 11 cards face up.