[Paper Review] Stable Fixed Points of Card Trick Functions

Jyoti Champanerkar, Mahendra Jani|arXiv (Cornell University)|Aug 13, 2013

Artificial Intelligence in Games6 references3 citations

TL;DR

This paper mathematically explains the 21-card trick using a linear discrete dynamical system, proving that the selected card always reaches the stable fixed point at position (pq+1)/2 after a finite number of iterations in a generalized p×q card trick, where p and q are odd integers ≥3. The stability of this fixed point ensures predictable convergence regardless of initial position.

ABSTRACT

The 21-card trick is a way of dealing cards in order to predict the card selected by a volunteer. We give a mathematical explanation of why the well-known 21-card trick works using a simple linear discrete function. The function has a stable fixed point which corresponds to the position where the selected card reaches at the end of the trick. We then generalize the 21(7 x 3)-card trick to a p x q - card trick where p and q are odd integers greater than or equal to three, determine the fixed point and prove that it is also stable.

Motivation & Objective

• To provide a mathematical explanation for why the 21-card trick reliably locates the selected card using dynamical systems theory.
• To generalize the 21-card trick to p×q card tricks where p and q are odd integers ≥3.
• To prove the existence and stability of a fixed point in the generalized card trick function.
• To determine the number of iterations required for convergence to the fixed point, depending on p and q.
• To present a pedagogically accessible proof using induction that demonstrates mathematical development.

Proposed method

• Model the card trick as a linear discrete function f(n) that maps the position of the selected card after each dealing and collecting cycle.
• Represent the card position using the form n = qk - l, where k is the row index and l determines the column (0 ≤ l ≤ q-1).
• Define the function h(n) for the p×q case, which maps the current position to the next via column stacking with the selected column in the middle.
• Use induction on the block distance from the central q-row block to prove convergence to the fixed point for all initial positions.
• Analyze the transformation of row blocks by showing that cards outside the central block are progressively mapped closer to the center.
• Prove stability by demonstrating that repeated application of the function h(n) converges to the fixed point (pq+1)/2 regardless of initial position.

Experimental results

Research questions

• RQ1 Why does the 21-card trick always locate the selected card in three iterations?
• RQ2 What is the mathematical function that models the card position transformation in the 21-card trick?
• RQ3 How can the 21-card trick be generalized to p×q cards with odd p and q?
• RQ4 What is the fixed point of the generalized card trick function, and is it stable?
• RQ5 How many iterations are required for convergence to the fixed point in the generalized p×q case?

Key findings

• The 21-card trick converges to a stable fixed point at position 11, which is (21+1)/2 = 11.
• For the generalized p×q card trick with odd p,q ≥3, the fixed point is (pq+1)/2, and it is stable under the function h(n).
• The function h(n) ensures that any selected card reaches the fixed point in a finite number of iterations, regardless of initial position.
• For p ≤ q, convergence occurs in at most two iterations.
• For p > q, the number of iterations depends on the ratio of p to q, with three iterations required for q+1 ≤ p ≤ 3q and four for 3q+1 ≤ p ≤ 5q.
• The proof uses induction on the block distance from the central q-row block, showing that cards outside the central block are mapped progressively toward the center.