I just read an article by Rob Dougherty where he explained how the judges should have acted to catch Mike Long cheating.

The situation: Darwin Kastle and Mike Long are playing each other with a top 8 on the line. Darwin is 2-0 with his draft deck and needs a win or a draw to get into the top 8. Amongst the spectators were famous players like Rob Dougherty and Zvi Mowshowitz, and they both independently discover that Mike is doing an extremely weird shuffle: He first puts his deck in four piles, puts the third pile on top of the fourth and the first on top of the second pile, then takes both piles and does a"grip" shuffle - which looks like a riffle shuffle, but is nothing more than putting both piles on top of each other.

The judges are informed, and when Mike presents his deck for the second game they swoop over and do a deck check. Forty-five minutes later, the judges find nothing extraordinary about the presented deck and Mike is free to win the match and proceed to top 8.

So as a mathematician, I got intrigued and wondered about what happened there that day. After doing some research, I think I now know what Mike was doing that day.

Suppose you start with a deck of forty cards - like Mike's deck. Next, we will number the cards in the deck to keep track of what happens to them. Let's number them by the order in which they appear in the deck; let's call the first card 0, the second 1, and so on. So our starting deck configuration is 0, 1, 2, 3, and so on all the way to 40.

After dividing this deck into four piles, we get:

• A first pile with cards 0, 4, 8, 12... 32, 36 = 0+4*k.
• A second pile with cards 1, 5, 9, 13... 33, 37 = 1+4*k.
• A third pile with cards 2, 6, 10, 14... 34, 38 = 2+4*k.
• A fourth pile with cards 3, 7, 11, 15... 35, 39 = 3+4*k.

And when we put the first pile on top of the second on top of the third on top of the fourth, we get a new configuration:

0, 4, 8...32, 36, 1, 5... 33, 37, 2, 6... 34, 38, 3, 7... 35, 39.

Now we need to use the modulo. Some of you will remember this from high school, but I think it's best to explain again: When you take the modulo of a number X to a number Y, you need to subtract Y from X until X is smaller than Y.

For example: When we want to calculate 15 mod 6, we see that 15 is larger than 6, and thus we need to subtract 6. 15-6=9 is still too much, so we do it again and we get 9-6=3. So this calculation shows that 15 mod 6 = 3.

When we use this notation, we can see that the new deck can be written as

4*k mod 39 with the exception of the last card (card 39) - but this card will always stay on the bottom of the deck when we shuffle like this, so we can ignore card 39.

Doing another pile shuffle will give us the configuration:

• 4*4*k mod 39 = 16*k mod 39 = 0, 16, 32, 9, 25...
• Doing it a third time gives: 64*k mod 39 = 25*k mod 39 = 0, 25, 11, 36, 22...
• A fourth time gives: 100*k mod 39 = 22*k mod 39 = 0, 22, 5, 27, 10...
• And the fifth gives: 88*k mod 39 = 10 mod 39 = 0, 10, 20, 30, 1...
• And after the sixth pile shuffle we get: 40*k mod 39 = 1*k mod 39 = 0, 1, 2, 3, 4...

...which is exactly the same deck as we started with!

So Mike could just arrange the deck the way he wanted it to be. After installing a few land clumps and some spell clumps to prevent detection by a possible deck check, followed by pile shuffling his deck six times and acting like he really randomized the deck thoroughly, he could get the exact same deck back again.

To prove once more that pile shuffling can be abused by cheaters:

Suppose you do a seven-pile shuffle with a sixty-card deck. You put all cards in seven piles and then you take the first pile and put it on top of the fourth, put these two on top of the seventh, on top of the third, on top of the sixth, on top of the second, on top of the fifth.

Done correctly, you would get a new deck configured like this:

• 1st pile
• 4th pile
• 7th pile
• 3rd pile
• 6th pile
• 2nd pile
• 5th pile.

Doing this gives you a deck with configuration 7k mod 60.

When you do this four times, you get:

• 7*7*7*7k mod 60
• = 2401k mod 60
• = 1k mod 60.

...and thus, doing a seven-pile shuffle four times gives you exactly the same deck again! Just try it out once if you're not convinced.

So what does this mean? Is there no way to prevent your opponent from cheating this way? Of course there is; just shuffle his deck several times using different techniques (a pile shuffle, a riffle shuffle, a normal shuffle) before you present it. And change the way you shuffle in between games.

Most of your opponents probably aren't using some complicated way to get an advantage over you by shuffling... But these cheaters are out there, and when you get paired against one, you don't want him to mess with you. I think that shuffling every deck that is presented to you is a small price to pay in order to ban this kind of cheating from Magic.