Card Trick Dynamics

• CARD is a discrete dynamical system where a card’s position is iteratively updated to converge to a unique fixed point.
• The methodology uses simple linear functions to track card positions, revealing that finite iterations lead to convergence.
• Generalizations to p×3 and p×q layouts illustrate the robustness of the fixed point mechanism and its broad applicability.

Searching arXiv for the specified paper and closely related work. arXiv search query: (Champanerkar et al., 2013) Stable Fixed Points of Card Trick Functions The 21-card trick can be formalized as a discrete dynamical system in which the only state variable is the position of the selected card in the deck. In "Stable Fixed Points of Card Trick Functions" (Champanerkar et al., 2013), the dealing, column identification, and recollection step is modeled by a simple linear discrete function whose repeated iteration sends every initial position to a single stable fixed point. The paper first analyzes the classical $21=7\times 3$ case, then generalizes the construction to $p\times 3$ and finally to $p\times q$ card layouts for odd integers $p,q\ge 3$. In this formulation, the apparent “magic” of the trick is the finite-time convergence of all possible starting positions to the midpoint of the deck.

1. Discrete-dynamical formulation of the trick

The paper treats the classic 21-card trick as a very simple discrete dynamical system: a selected card is tracked only by its position number, and each “deal, identify the column, and recollect with that column in the middle” step defines a function on positions (Champanerkar et al., 2013). This replaces any appeal to sleight or hidden information with an explicit state-transition rule.

For the original arrangement, the 21 cards are placed in a $7\times 3$ grid. If the selected card is in position $n$, the position is written uniquely as

$n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$

where $k$ is the row number and $l$ encodes the column. Specifically, $l=2,1,0$ correspond to columns $p\times 3$0, respectively. The recollection rule is fixed: the indicated column is placed in the middle of the stack.

Because there are 7 cards in each column, and because the selected card sits in row $p\times 3$1, the paper defines the map

$p\times 3$2

for $p\times 3$3. This function models one complete round of the classical trick. Conceptually, the rule depends only on coarse row information after the redeal; the exact initial position is progressively discarded under iteration.

2. The classical $p\times 3$4-card case and its fixed point

In the $p\times 3$5 case, the fixed point is the middle position of the 21-card pile, namely $p\times 3$6 (Champanerkar et al., 2013). The verification is direct: $p\times 3$7 so $p\times 3$8, and therefore

$p\times 3$9

This midpoint interpretation is structurally important. The selected card is not merely forced into a narrow interval; it is driven to a single invariant position under the recollection rule. The paper includes a table showing the evolution of every starting position. For example, positions $p\times q$0 all move to $p\times q$1 after one round, then to $p\times q$2, then to $p\times q$3; positions $p\times q$4 already move to $p\times q$5 after the first round and stay there. The accompanying figure visualizes this convergence.

This fixed-point description gives a precise explanation of the standard performance rule that the procedure is repeated three times total. The familiar outcome—revelation of the 11th card—is therefore the observable consequence of an invariant of the induced position map, not an isolated combinatorial coincidence.

3. Stability and finite-time convergence

The stability argument in the paper is purely discrete. Starting from any initial position $p\times q$6, one has

$p\times q$7

which forces

$p\times q$8

Writing $p\times q$9 then implies

$p,q\ge 3$0

Applying $p,q\ge 3$1 again gives

$p,q\ge 3$2

so

$p,q\ge 3$3

hence $p,q\ge 3$4. One more iteration yields

$p,q\ge 3$5

Therefore,

$p,q\ge 3$6

The paper describes $p,q\ge 3$7 as a unique attracting fixed point, and the fixed point is stable because a whole neighborhood of inputs is mapped closer and closer to the point until all positions collapse to it after finitely many steps (Champanerkar et al., 2013). In this setting, stability is not established by calculus or local linearization, but by interval compression and exact iteration. This suggests a finite-state analogue of attraction: every orbit enters successively smaller blocks until only the midpoint remains.

The guarantee of the trick is thus stronger than asymptotic convergence. In the classical case, convergence occurs after exactly three iterations for every initial state.

4. Generalization to the $p,q\ge 3$8 trick

The paper next generalizes the $p,q\ge 3$9 trick to a $7\times 3$0 card trick, where $7\times 3$1 is an odd integer $7\times 3$2 (Champanerkar et al., 2013). The deck has $7\times 3$3 cards arranged in $7\times 3$4 rows and 3 columns, and again any position is written as

$7\times 3$5

After one round, the selected card lands at

$7\times 3$6

This is the corresponding discrete map for the $7\times 3$7 trick. Its fixed point is the middle position of the $7\times 3$8-card pile: $7\times 3$9 The paper verifies this by writing

$n$0

so the row index is $n$1, and substituting into $n$2 gives

$n$3

The stability proof again proceeds by tracking row indices through repeated iterations. The first iterate satisfies

$n$4

which implies

$n$5

The paper continues this bounding argument and proves by induction that if $n$6, then after at most $n$7 iterations the card reaches the fixed point $n$8. In particular, for $n$9 one gets $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$0, recovering the 21-card trick.

The first few cases are summarized explicitly: $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$1 has fixed point $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$2, $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$3 has fixed point $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$4, and $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$5 has fixed point $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$6. The general mechanism is unchanged: repeated recollection with the chosen column in the middle forces the selected card toward the midpoint row and then to the midpoint position.

5. Generalization to the $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$7 trick

The full generalization considers a $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$8 card trick, where both $n=3k-l,\qquad 1\le k\le 7,\quad 0\le l\le 2,$9 and $k$0 are odd integers at least 3 (Champanerkar et al., 2013). The deck now has $k$1 cards arranged in $k$2 rows and $k$3 columns, and the position is written as

$k$4

The same recollection rule is used: the chosen column is placed in the middle of the stack, with $k$5 columns above it and $k$6 columns below it.

The resulting map is

$k$7

The fixed point is the midpoint of the full deck: $k$8 The paper verifies this directly by observing that

$k$9

so the middle row index is $l$0. Substituting gives

$l$1

For stability, the paper uses a discrete-block argument. First, if the selected card is already in the middle row, then one more iteration sends it directly to the fixed point: $l$2 Second, if the selected card lies anywhere in the block of $l$3 rows centered at the middle row, then after one application of $l$4 it lands in the middle row, and then a second application reaches the fixed point: $l$5

This already proves stability when $l$6, because then the entire deck lies inside that middle block of $l$7 rows. When $l$8, the deck is partitioned into blocks of $l$9 rows above and below the middle block. The key inductive claim is that if the selected card starts in a block at distance $l=2,1,0$0 from the middle, then one application of $l=2,1,0$1 moves it to a block one step closer to the middle. Repeating this finitely many times eventually places it in the middle block, after which two more steps reach the fixed point. The paper therefore proves that $l=2,1,0$2 is a stable fixed point for all odd $l=2,1,0$3.

6. Conceptual significance and relation to later generalizations

The paper’s contribution is to recast the card trick as a discrete dynamical system with an attracting fixed point (Champanerkar et al., 2013). The dealing and reordering procedure induces a function that systematically discards information about the exact starting position and preserves only coarse row information, and each iteration pushes the card closer to the middle until the entire uncertainty collapses to the unique stable midpoint. The 21-card trick is the smallest nontrivial instance of this mechanism, and the general $l=2,1,0$4 version shows that the same stability phenomenon governs a broad class of card tricks.

A related later treatment, "The 21 Card Trick and its Generalization" (Deb, 2018), reformulates the process with the notation $l=2,1,0$5, where $l=2,1,0$6 is the total number of cards, $l=2,1,0$7 the number of stacks, $l=2,1,0$8 the number of stacks placed above the chosen stack, and $l=2,1,0$9 the number of iterations. That paper gives a complete solvability classification for arbitrary deck sizes under the divisibility assumption $p\times 3$00, including the statements that $p\times 3$01 is not solvable for $p\times 3$02, that top placement yields final position $p\times 3$03, that bottom placement yields final position $p\times 3$04, and that for interior placements $p\times 3$05 solvability holds if and only if $p\times 3$06 with $p\times 3$07 (Deb, 2018). This later framework is broader in arithmetic scope, whereas the 2013 paper isolates the fixed-point mechanism itself.

A plausible implication is that the 2013 analysis supplies the dynamical-systems core of the phenomenon, while the 2018 classification reorganizes the same kind of position update into a more general ceiling-function and solvability framework. In both approaches, the key mathematical content is identical in spirit: the trick succeeds because repeated structured redealing compresses the set of possible positions until only one remains.