In 2018, I started doing research to try to create a mathematical card stacks. That lead to my post in Red Phish DeBruijn 52.

The result was the over 150 page book, Mathematical Stacks.

My favorite items in my book are The Bart Harding Stack, created by the late Arthur Bridgman, Cherry Stack, and the chapter on the Si Stebbins Stack.

The Introduction to the book states:


The book contains the following chapters and stacks:

• Suitcase Stack
  The Suitcase Stack can be used to perform the routine Suitability by William Larsen and T. Page Wright. The Suitcase Stack also contains some features of the Aronson Stack, specifically Spelling, A Ten Card Poker Deal, and Any Hand Called For. This stack has many of the cards in the same position as the Aronson stack.
  
  The routine Suitability is where knowing the suits of three consecutive cards allows identifying all the cards.
  
  
• Suitcase Faro Stack
  The Suitcase Faro Stack supports all the same routines as the Suitcase Stack. The deck can be cut and one Out-Faro shuffle done, and the routine Suitability is still supported.
  
  
• Suit Shuffle Stack
  The Suit Shuffle stack supports the routine Suitability, Spelling, and Ten Card Poker Deal. After one Out-Faro-shuffle, Suitability is still supported. After a simple cut, and another Out-Faro shuffle, Suitability is still supported.
  
  
• Suit Bracelet Stack
  The Suit Bracelet Stack supports the routine Suitability, and also has a red-black suit binary bracelet code, where the colors of the suit of six consecutive cards identifies the cards. This stack supports Spelling, Ten Card Poker Deal, and another the Suit Bracelet Poker Deal, which gives the dealer four Aces and other players good hands.
  
  
• Bracelet Faro Stack
  The Bracelet Faro Stack supports the routine where knowing the suit color of six consecutive cards identified the cards. After a Faro shuffle, the six card binary bracelet routine still can be performed. The Bracelet Faro Stack also supports, Spelling, A Ten Card Poker Deal, and Any Hand Called For.
  
  
• Two Suits Suitability Stack
  The Two Suits Suitability Stack supports the routine Suitability, and any two suits named form a binary bracelet code that allows any six consecutive cards identifying all the cards. The stack supports Spelling, Ten Card Poker Deal, and The Two Suits Suitability Stack Poker Deal, where the dealer gets an Ace-high straight.
  
  
• Ultimate Bracelet Stacks
  The three Ultimate Bracelet Stacks each support multiple bracelet codes.
  
  Selecting any two suits in any of the three stacks will form a binary bracelet code.
  
  If the Ace through King are numbered 1 to 13, then for all three stacks, the even or odd cards form a binary bracelet code.
  
  After that, each of the stacks supports value sequences that differ between the stacks. If the numbers from 1 to 13 are arranged in a circle, then starting on any number and going either six or seven cards form a group. One fo the three stacks will form a binary bracelet sequence for starting on any card and going 6 or 7 cards. For example, A, 2, 3, 4, 5, 6, 7 is one group. 10, Jack (11), Queen (12), King (13), Ace (1), 2, 3, is another group.
  
  There's a detailed section explaining more details and the stacks.
  
  
• Cherry Stack
  This is a method for an old routine where six cards consecutive cards are handed out, and the people with the cards with red suits, i.e. Hearts or Diamonds, step forward. There are very simple calculations to determine the suit and value of the first card handed out. With this full 52-card stack, there's no need to memorize or use a crib for any bracelet codes at all. Either the stack must be memorized or a crib will be needed to find the other five cards though. This is my preferred method for this type of routine.
  
  There are variants of this routine that use differnt ways to determine which of the card's suits are a Heart or a Diamond.
  
  
• Red Stack
  This is a method for the same routine as the Cherry Stack, however, the stack doesn't have any obvious patterns and unlike the Cherry stack, there are not six consecutive cards that all have red suits, nor are there six consecutive cards that all have black suits. The calculations aren't difficult, however, are both longer and more complicated than the Cherry stack. For that reason, I still prefer the Cherry Stack.
  
  
• Maroon Stack
  The Maroon Stack supports the same method to find the first card as the Red Stack, however, it has four consecutive cards that are the same value in the stack. The advantage of the Maroon Stack is that relatively simple shuffle sequences allow shuffling the stack to support any of the binary bracelet codes that the three Ultimate Bracelet Stacks support. These shuffle sequences are listed, along with tables with bracelet codes for each of the resulting deck orders.
  
  
• Red-Black Stack Hack
  The Red-Black Stack Hack chapter gives shuffling sequences to shuffle some well-known stacks to have a binary bracelet sequence. However, because some of these stacks are proprietary, the stacks, and tables with bracelet codes, are not listed, just the shuffle sequences.
  
  The stacks listed at Si Stebbins Stack, or any stack with alternating suit colors, the Nikola stack, the McCaffrey stack, the Joyal stack, the Bart Harding stack, the Marlo stack, the Aronson stack, the Osterlind Breakthrough Card System stack, the Mnemonica stack, Doug Dyment's QuickerStack, Patrick Redford's Redford stack, my own Easy stack, and finally, New Deck Order.
  
  
• The Si Stebbins Stack
  This chapter includes a very detailed method for card-to-position and position-to-card for any Si-Stebbins-like stack with any increment and any suit order with the deck cut in any place. There are lots of examples of calculations listed. Unless someone has discalculia, they can learn this.
  
  
• The Bart Harding Stack
  Bart Harding was the stage name of Arthur Bridgman. He created an algorithmic stack that makes card-to-position and position-to-card calculations relatively easy, and the deck can be spread and there are no easily discernable patterns. This is probably the best item in the book. Bart Harding was a genius. Some have criticized his stack because it has two exceptions, however, if the method is understood, then those exceptions are obvious.
  
  
• Easy Stack
  Easy Stack is an algorithmic stack I came up with that, quite frankly, isn't as good as Bart Harding's stack. The calculations are a bit more difficult than the Bart Harding stack, and there is a pattern in the deck, althought his isn't seen easily in a spread. It does allow getting all four Aces together easily though.
  
  
• Creating Stacks With Bracelet Sequence
  This final chapter is mostly about how I created the Ultimate Bracelet stacks. It includes a link to the program code in the C language. Modifications of this code found some of the other stacks.
  

My primary impetus to write this book was a book by Persi Diaconis and Ron Graham titled, Magical Mathematics. However, posts by glowball, TomasB, and Scott Cram, also made me want to write this book. If glowball, TomasB, or Scott Cram want a copy of the book for free, as long as they promise not to share with anyone else, I'll send it to them. They can contact me in a private message here in the Magic Cafe.

The Harding stack, while unquestionably important in its day (1962), is no longer seen as a particularly good solution to the algorithmic deck challenge. Not because of its rule exceptions, but because (1) it requires a level of mental arithmetic [adding and subtracting multiples of thirteen, for example] that is difficult for many to accomplish "under fire", and (B) it has detectable sequences [e.g., aces, twos, & threes are always followed by Jacks, Queens, & Kings, respectively).

Its core idea, though (starting with an easily calculated stack and then applying simple conversion rules to create a second stack), is an excellent one. New deck order is, unfortunately, not easily calculated by most people (quickly, what is the 37th card?). Considerably better solutions for algorithmic full-deck stacks have been developed in the 60+ years since Mr. Bridgman's proposal.

I like and use your Quickerstack too. It's a brilliant creation. It has even easier card-to-position and position-to-card calculations than the Bart Harding stack.

What I like about the Bart Harding stack isn't just related to the calculations. I do find those calculations relatively easy though. Adding or subtracting 0, 13, 26, and 39 is easy. That's required for most algorithmic stack position calculations. The calculations where 5 is added or subtracted can be treated as single digit calculations that never have any carries or borrows. So those also aren't difficult.

Also, because the suits fall in ranges of numbers in the base stack, after the calculations, I find the determining the suit of a card for Bart Harding stack really easy. There's no need to consider any remainders to determine the suit of a card.

The Bart Harding stack has the patterns you mentioned, although it is more examinable than many other algorithmic stacks I've seen. That's rarely an issue in performing though. There are other reasons I like the Bart Harding stack. By moving one or two cards, the Bart Harding stack supports a specific routine. It also can be easily shuffled to support two other routines.

"QuickerStack" and its successor, "QuickStack 3.0", are designed to be tetradistic, thus are not carefully examinable, due to the "banks of thirteen" issue (although they offer shortcut ways around the associated arithmetic). But they enable a whole host of published effects, which is why tetradistic stacks are so popular ("the strongest of all stacks", to quote cardman Alan Ackerman).

On the other hand, an algorithmic stack such as my "Q Stack" (for example) is thoroughly and extensively examinable, has no exceptions to the algorithm, has no "thirteen" issue, no repeated anything, and requires only trivial arithmetic (no multiplication, division, remainders; no addition or subtraction of values greater than three).