It can be difficult to picture In and Out shuffles when dealing with decks of large size. However, this can become significantly easier when dealing with a deck of N = 2k cards if we think of each of the cards in terms of their dyadic expansions. Consider a deck of N = 8 cards. We write the first digit of the dyadic (binary) expansion of each card so that cards 0 through 3 are assigned a zero and cards 4 through 7 are assigned a one. For the second digit, we assign cards 0, 1, 4, and 5 a zero, and cards 2, 3, 6, and 7 a one. The third digit is assigned where the even numbered cards each receive a zero and the odd numbered cards each receive a one. Thus, card in spot zero has a dyadic expansion of 000, and the card in spot five has a dyadic expansion of 101.
To express the results of an Out shuffle of a card in terms of its dyadic expansion, shift each digit one spot to the left, circling the first digit around to the last spot, where
For an In shuffle, circle the digits around as with the Out shuffle, but then switch the last digit so that if it is a 1 it becomes a 0, and if it is a 0 it becomes a 1, i.e.,O[000] = [000] and O[101] = [011].
I[000] = [001] and I[101] = [010].