Version of Tuesday 19 February 2019.

Dave Barber's other pages.

**§1.** When a card game is to be played honestly, the purpose of shuffling a pack of cards is to place them in a sequence which is not predictable, and which is not biased in favor of one player or another. Of course, when shuffling is done in an ideal manner, each player will sometimes get desirable cards and will sometimes not, but in a long series of games the distributions are likely to approximately balance out.

When a card player is dishonest, or a person is performing card tricks, the purpose of shuffling is something else entirely, and that matter will not be addressed here.

A *packet* is any collection of cards less than the full *pack*.

**§2.** Many sources talk about the shuffling of playing cards, but their terminology can be inconsistent or vague, so two key definitions are introduced here:

Minoring ("shuffling minor"): A procedure that combines two packets of about 12 to 30 cards each. The two input packets should have approximately the same number of cards, and the output will be one packet containing all the cards, in a sequence intended to be unpredictable.
Among many choices, the best-known minoring method is the riffle, which interleaves cards from the two input packets approximately alternately. Whatever the method, it is usually beneficial to do several minorings in immediate succession to more reliably establish unpredictability. To minor a single packet is, naturally enough, to divide it in approximately half, and combine the two parts as above. |

Majoring ("shuffling major"): A multi-cycle procedure that splits a full pack, particularly a large one, into packets to be minored individually and combined in various ways. At each split, the packets should be approximately the same size. The aim is a random sequence, with all possible sequences being equally likely. |

In both minoring and majoring, small deviations from exactitude are desirable, as a perfect shuffle does not give randomness. This explains the emphasis on approximateness which, however, must not be permitted to deteriorate into carelessness.

**§3.** Most people who play cards can effectively minor the well-known pack of 52 cards, or anything smaller. Minoring such a pack several times in succession usually provides sufficient unpredictability, and there is no need to major.

On the other hand, people may find it difficult to minor a large pack such as that employing the 104 cards used in two-pack solitaires or the 108 cards of most Canastas, and that is where majoring becomes helpful. Majoring may also be desirable when playing 78-card Tarot or 80-card double-pack Pinochle.

There is a particular problem — a lack of *dispersion* — which must be avoided in shuffling large packs, and it is the reason for establishing specific majoring procedures. Here are two scenarios:

After a game of two-pack solitaire, the player casually gathers up the cards in preparation for the next game. Most of the black cards happen to end up in the top half of the as-yet unshuffled pack, leaving most of the red cards to be the bottom half. Because many solitaire games tend to put the pack in order, this kind of irregularity is not unlikely. A proper shuffle would correct this uneven distribution, but that is not what happens.
Instead, the player splits the pack in two, thorougly minoring the top half, and thoroughly minoring the bottom half; and then places the original top half on the original bottom half. With no further shuffling, the player begins the game, dealing cards. Because the two halves of the pack never got merged, black cards will be concentrated in the early part of the deal, and red cards in the latter part. This uneven distribution could make the game more or less difficult than it should be. |

A player opens up two factory-fresh packs of 52 cards, minors each individually, stacks one on top of the other without mixing them, and begins dealing. The result will be that none of the first 52 cards dealt can be duplicates; a shrewd card player can use this information to advantage. Had the two packs been properly merged, there would almost certainly be duplicates among the first 52 cards dealt. The same applies to the last 52 dealt. |

**§4.** The following is a majoring procedure for a 104-card pack, and it can be easily adapted to other packs of similar size.

As a preliminary, divide the full pack into *quasi-halves* — two packets of approximately equal size. In this case, each quasi-half will have ±52 cards. Call the packets #1 and #2, and begin a cycle:

- Step one:
- Take the top quasi-half (±26 cards) of #1 and minor it with the top quasi-half of #2. Call the resultant packet #3 (±52 cards).
- Take the remaining quasi-half of #1 and minor it with the remaining quasi-half of #2. Call the resultant packet #4.

- Step two:
- Minor the top quasi-half of #3 with the top quasi-half of #4, forming packet #5.
- Minor the remaining quasi-half of #3 with the remaining quasi-half of #4, forming packet #6.

- Step three:
- Minor the top quasi-half of #5 with the top quasi-half of #6, forming packet #7.
- Minor the remaining quasi-half of #5 with the remaining quasi-half of #6, forming packet #8.

Continue with analogous steps as desired. This is the fundamental *two-way* majoring scheme, and all the majoring procedures described below are adaptations of this one.

A key observation is this: A card in packet #1 for example has a roughly equal chance of ending up in #3 or #4; in #5 or #6; in #7 or #8; et cetera. This is *full* dispersion, which is desirable. The alternative, *partial* dispersion, will be encountered in **§5B** and **§6** below, and is sometimes acceptable when implemented carefully.

An open question is whether, if there is a risk of sloppy minoring, step one ought to be changed as follows:

- Minor the
*top*quasi-half of #1 with the*bottom*quasi-half of #2, producing #3. - Minor what had been the
*bottom*quasi-half of #1 with what had been the*top*quasi-half of #2, producing #4.

**§5A.** In a two-handed game, both players can help with the shuffling, saving time — and reducing the risk that someone will be accused of crooked shuffling. In this majoring plan, the players are named Aaron and Erin.

As before, the majoring procedure begins by dividing the full pack into quasi-halves #1 and #2.

- Step one:
- Aaron minors the top quasi-half of #1 with the top quasi-half of #2, forming packet #3.
- At the same time, Erin minors the remaining quasi-half of #1 with the remaining quasi-half of #2, forming packet #4.

- Step two:
- Erin minors the top quasi-half of #3 with the top quasi-half of #4, forming packet #5.
- At the same time, Aaron minors the remaining quasi-half of #3 with the remaining quasi-half of #4, forming packet #6.

Continue as desired.

**§5B.** Here is how majoring might be done in a three-handed game. First, each player takes a packet which is a quasi-third of the full pack, and minors it. After that, the players cycle through the following pair of steps several times:

- Each player passes about half their cards to the player on their left, and the remainder to the player on their right.
- Each player minors the cards they have received.

Under this plan, no player minors the same card in two successive cycles, improving the overall quality of the shuffle in case one player is not skillful at minoring. However, the dispersion is only *partial* at each cycle. Still, with each successive cycle the net dispersion approaches fullness. How to calculate this is shown in **§6**.

Full dispersion at every cycle can be achieved this way: After minoring, each player passes about one-third of their cards to the player on their left, and one-third to the right, keeping the remainder. Then each player stacks the three packets, divides that in quasi-half, and minors those cards.

**§5C.** With four players, an obvious partially-dispersive adaptation is for each player to take a quasi-fourth of the full pack and minor it. After that, each player passes about one-third of their cards to the player on their left, one-third to the player on their right, and one-third to the player sitting opposite. The cycle is repeated.

A slower dispersion is the result when each player passes about half of their cards to the player on their left, the remainder to the player on their right, and none to the player sitting opposite.

Full dispersion is achieved when each player passes about one-fourth of their cards to each of their three opponents, and keeps one-fourth.

**§5D.** In the four-player game a different plan might be preferred. For an example, assume Benny is sitting opposite Kenny, and Jenny is sitting opposite Penny.

To start, Benny and Kenny each take a quasi-half of the full pack and minor it. Meanwhile, Jenny and Penny do nothing. Then the following two-step cycle is repeated as desired:

- Benny and Kenny each passes about half of his cards to Jenny and the remainder to Penny, who respectively minor the cards they receive.
- Jenny and Penny each passes about half of her cards to Benny and the remainder to Kenny, who respectively minor the cards they receive.

This is fully dispersive.

**§6.** A more elaborate majoring procedure may be needed with even larger packs, such as with 120-card triple-pack Pinochle, 128-card Rubicon Bezique, or 162-card Samba. The following *three-way* procedure is very much a counterpart to the two-way method of **§4**, and assumes 162 cards.

Divide the full pack into quasi-thirds of ±54 cards. Call the packets #1, #2 and #3, and begin a cycle:

- Step one:
- Take the top quasi-half (±27 cards) of #1 and minor it with the top quasi-half of #2. Call the resultant packet #4 (±54 cards).
- The the remaining quasi-half of #2 and minor it with the top quasi-half of #3. Call the resultant packed #5.
- Take the remaining quasi-half of #3 and minor it with the remaining quasi-half of #1. Call the resultant packet #6.

- Step two:
- Minor the top quasi-half of #4 with the top quasi-half of #5, forming packet #7.
- Minor the remaining quasi-half of #5 with the top quasi-half of #6, forming packet #8.
- Minor the remaining quasi-half of #6 with the remaining quasi-half of #4, forming packet #9.

- Step three:
- Minor the top quasi-half of #7 with the top quasi-half of #8, forming packet #10.
- Minor the remaining quasi-half of #8 with the top quasi-half of #9, forming packet #11.
- Minor the remaining quasi-half of #9 with the remaining quasi-half of #7, forming packet #12.

Continue with analogous steps as desired.

This procedure is only partially dispersive at each step, but approaches fullness with multiple cycles. Some algebraic notation is convenient in order to provide hard numbers. Let *n* be a non-negative integer so that packet numbers can be calculated. Then in general, each step is to do this:

- Minor the top quasi-half of #(3
*n*+1) with the top quasi-half of #(3*n*+2), forming packet #(3*n*+4). - Minor the remaining quasi-half of #(3
*n*+2) with the top quasi-half of #(3*n*+3), forming packet #(3*n*+5). - Minor the remaining quasi-half of #(3
*n*+3) with the remaining quasi-half of #(3*n*+1), forming packet #(3*n*+6).

Here is how the cards of packet #1 for instance will be dispersed, on average:

step | dispersion | ratio | ||
---|---|---|---|---|

1 | ^{1}⁄_{2} to #4
| ^{1}⁄_{2} to #5
| ^{0}⁄_{2} to #6 | 1:1:0 |

2 | ^{1}⁄_{4} to #7
| ^{2}⁄_{4} to #8
| ^{1}⁄_{4} to #9 | 1:2:1 |

3 | ^{2}⁄_{8} to #10
| ^{3}⁄_{8} to #11
| ^{3}⁄_{8} to #12 | 2:3:3 |

4 | ^{5}⁄_{16} to #13
| ^{5}⁄_{16} to #14
| ^{6}⁄_{16} to #15 | 5:5:6 |

5 | ^{11}⁄_{32} to #16
| ^{10}⁄_{32} to #17
| ^{11}⁄_{32} to #18 | 11:10:11 |

6 | ^{22}⁄_{64} to #19
| ^{21}⁄_{64} to #20
| ^{21}⁄_{64} to #21 | 22:21:21 |

With additional steps, the ratio approaches 1:1:1, which would signify full dispersion.

The results for the majoring scheme of **§5B** correspond to these.