The Markov Card Trick!

A few years back, my young son received an invitation to his friend’s birthday bash, where we relished a delightful celebration brimming with birthday melodies, delectable sweets, chocolates, and, naturally, a scrumptious cake. Just as we were preparing to bid adieu, the cousin of the birthday boy mentioned having an intriguing card trick up his sleeve.

The trick was rather straightforward. Armed with a double deck of cards, the cousin removed special cards such as Kings, Queens, Knights, and the Joker, assigning the Ace card a value of one. He then arranged the cards into a matrix comprising five columns and sixteen rows, with each card’s value prominently displayed. The rules of the trick were as follows:

1. Select one card from the initial row.
2. Determine the step size based on the number on the chosen card (where the Ace is valued at one).
3. Proceed with the step size in a lexicographical manner, moving from left to right until reaching the end of the row, then advancing one row downwards.
4. Repeat steps 2 and 3 until reaching a card where further progression is impossible.
5. The last card encountered is regarded as the Stop card.

Incredibly, when the birthday boy signaled the end of the trick, the cousin confidently predicted the card where the procession had stopped — and his guess proved to be spot on. Without skipping a beat, the cousin repeated the entire process, rearranging the cards and executing the trick once more, and then again, and again. It felt like an endless loop of wonderment.

Fascinated by this phenomenon, my son, who hadn’t been among those selected to pick a card, discreetly selected one without drawing any attention. To everyone’s astonishment, his final card matched the last card chosen by the girl who had initially selected a card in secret. This astonishing coincidence repeated itself with several other children who were also chosen.

As we journeyed home, my son turned to me with a perplexed expression and inquired about the extraordinary sequence of events that had unfolded.

I encountered a fascinating card trick that hinges on a fundamental principle known as the Markov property. This principle dictates that how you arrive at a card becomes inconsequential once it’s reached, essentially implying that the system lacks memory. In simpler terms, knowing the present renders the past irrelevant.

At the outset, selecting the first card sets in motion a pathway. Initially, we have five distinct pathways to follow. However, if two pathways happen to converge at a card, they merge until they ultimately lead to the final card.

Indeed, there’s a striking resemblance between the card trick and the birthday paradox.

The birthday paradox, despite its name, isn’t a true paradox but rather a captivating statistical phenomenon. It sheds light on the surprising probability that within a relatively small group of individuals, there’s a significant chance of at least two people sharing the same birthday. This concept often defies intuition: while it may appear unlikely, the probability escalates exponentially with the group’s size.

This phenomenon stems from the combinatorial nature of the problem, where the focus shifts from the likelihood of any two people sharing a specific birthday to the broader probability of any shared birthdays among the group. Thus, the birthday paradox serves as a compelling illustration of how statistics can challenge common perception and unveil unexpected patterns within ostensibly random occurrences.

Continuing along this trajectory, let’s explore the concept of complementary probability. What are the odds of not correctly guessing the last card? Initially, there’s a twenty percent chance that both the volunteer and the trick master select the same initial card. If this scenario doesn’t unfold, they each opt for distinct initial cards, resulting in two separate paths. What’s the probability that these paths remain separate?

Similar to the birthday paradox, this probability diminishes as the number of cards increases. Surprisingly, in over ninety percent of arrangements, all paths converge to the same final card. Furthermore, in ninety-five percent of cases, at least four cards lead to the final one. Considering these probabilities, it becomes evident that the boy performing the trick was banking on highly favorable odds.

My son astutely remarked that this isn’t merely a trick but rather a form of pattern recognition. And he is right.

To summarize, the card trick operates on the principle that if two paths converge, they will be identical from that point onward, illustrating a system devoid of memory. This characteristic is formally known as the Markov property in mathematical terms, and it serves as the foundation for Markov chains. The memoryless property is prevalent in various systems, including biological and computer systems. Leveraging this property can yield significant advantages in various applications.

I want to thank Zaryana Yabolchkin for her great help in the graphics.

The Markov Card Trick! was originally published in Coinmonks on Medium, where people are continuing the conversation by highlighting and responding to this story.