Week 1. Introduction

Look at the bottom card of the packet and make a note of it.

Put the top card on the bottom of the packet.

Turn the current top card face-up and place it back on top.

Now, give the packet a cut.

Next, spread off the top two cards, keeping them together…

…turn them over, placing them back on top.

Cut the cards at random again and then turn the top two over.

Give them another cut and turn two over.

Give them a final cut.

Turn the top card over and put it on the bottom of the packet.

Put the current top card on the bottom of the packet.

Finally, turn the top card over and place it back on top.

Spread the cards. What do you see?

Theorem. Let a deck of \(2n\) cards start all face-down.

After any number of turn two and cut at random operations, the following regularity is forced:

The number of face-up cards at even positions equals the number of face-up cards at odd positions.