## Two Immortals Shuffle a Deck of Cards

Posted: May 19, 2023

The following is an unedited conversation I recently had with ChatGPT.

User: I have 100 unique playing cards.

Two people are randomly shuffling these cards until they both have the exact same card order.

Given how many combinations are possible from 100 unique cards, how many times would the two people have to randomly shuffle until the both had matching sets of cards?

ChatGPT: The total number of combinations that can be made from 100 unique cards is given by the factorial of 100, represented as 100!. This number is incredibly large, approximately `9.3 x 10^157`.

The concept you’re asking about is similar to the “birthday paradox,” but instead of 365 days, we have 100! possible combinations. However, with 100! combinations, even if the two people shuffled the deck a billion (`10^9`) times a second for the entire age of the universe (roughly 13.8 billion years or approximately `4.3 x 10^17` seconds), they would still be incredibly unlikely to hit the same combination.

The exact number of shuffles required for them to have a reasonable chance (say, 50%) of getting the same order is so large that it’s practically incomprehensible. The universe would likely end due to heat death or the Big Rip long before they had a reasonable chance of getting the same card order.

User: Won’t the universe exist for trillions of years even after most of the stars explode? I believe the universe will exist in a state of darkness where only black holes exist.

Given this, would the two people be able to get the same order if they somehow found a way to survive amongst black holes in a complete dark, but still existing universe?

ChatGPT: The current understanding is that the universe will continue to exist for an incredibly long time, far beyond when the last stars have faded out. However, it’s important to understand just how big 100! is. It’s around `9.3 x 10^157`, which is a number so large that there aren’t enough particles in the known universe to even begin to represent it.

Even if our two people shuffled a billion (`10^9`) times a second for a trillion (`10^12`) years, that’s only about `3.15 x 10^31` shuffles - an astronomically tiny fraction of 100!.

Even extending this to a googol (`10^100`) years, which is an incredibly long time and vastly exceeds the currently predicted lifespan of the universe, it’s still not enough. It would be `3.15 x 10^119` shuffles, still significantly smaller than 100!.

So while it’s theoretically possible that they could both shuffle into the same order, the odds are so incredibly small that it’s virtually impossible. It’s a bit like asking if a monkey randomly typing on a typewriter could reproduce the complete works of Shakespeare. Theoretically, yes, but practically, no.

As a result, even if they were somehow able to survive in a universe composed entirely of black holes and darkness, the odds of them shuffling into the same order of cards are so low as to be virtually non-existent.

User: What are the odds they hit the exact order of each other on the first time they shuffle?

ChatGPT: The odds of two people shuffling 100 unique cards and both ending up with the exact same order on the first try is incredibly low. With 100 unique cards, there are 100! (or factorial of 100) possible orders, which is approximately `9.33 x 10^157`.

So, the odds of both shuffles landing on the same one of these orders on the first try is 1 in 100!, or approximately 1 in `9.33 x 10^157`.

To give you an idea of just how large this number is, it’s much, much greater than the estimated number of atoms in the observable universe, which is about `10^80`.

So the probability is effectively zero. It’s so unlikely to happen by chance that it’s virtually impossible.

User: Would a quantum computer be able to simulate such a scenario?

ChatGPT: Quantum computers offer a new kind of computation and can perform certain calculations much faster than traditional computers. However, even the most powerful quantum computers that we can conceive of would struggle to fully simulate this scenario.

The number of possible orders of 100 unique cards is 100!, which is approximately `9.33 x 10^157`. This number is incredibly large. For comparison, the number of atoms in the observable universe is estimated to be around `10^80`, which is far less.

While quantum computers can theoretically represent and manipulate an exponentially large number of states thanks to the principle of superposition, we’re currently nowhere near the level of having a quantum computer capable of simulating that many states.

As of my knowledge cut-off in September 2021, the largest quantum computers have on the order of 50-100 qubits. Every additional qubit doubles the potential number of states the computer can simultaneously represent, so a quantum computer with n qubits can represent `2^n` states. To represent 100! states would therefore need a quantum computer with `log2(100!)`` qubits. This is far beyond our current capabilities, and likely will be for a considerable time.

Furthermore, there’s a difference between being able to represent a large number of states (which quantum computers can theoretically do) and being able to perform useful computations on those states, which would be required to simulate the scenario you’re asking about. The issue of “quantum decoherence” poses significant challenges to maintaining and manipulating a large number of quantum states.

So, while quantum computers hold a lot of promise, they’re not a magic bullet that can easily simulate scenarios involving incredibly large numbers of possibilities.

User: Alright, so how can we solve the problem of the two people shuffling 100 cards and getting the same order? Can we look into 4 dimensional folding?

ChatGPT: The problem at hand, two people shuffling 100 unique cards and ending up with the same order, is primarily a statistical one, and concepts like four-dimensional folding won’t help solve it.

Four-dimensional space or folding refers to a geometrical concept, primarily used in theoretical physics, which includes an additional spatial dimension beyond the normal three dimensions (length, width, and height). It is used in theories like string theory and is a key element in ideas like wormholes. However, this doesn’t have a direct application on the statistical problem of shuffling cards.

The statistical problem here is the sheer enormity of possible outcomes when shuffling 100 unique cards, which is 100 factorial (roughly `9.33 x 10^157`). Even with incredibly fast shuffling and a practically infinite amount of time, the odds of two independent shuffles producing the exact same order is extremely small.

There’s not really a “solution” to this problem in the sense of finding a way to make it likely that they’ll shuffle into the same order. It’s simply a case where the number of possibilities is so large that, for all practical purposes, it’s impossible for two people to independently shuffle into the same order of cards.

User: If that’s the case, then how does the concept of multiverse theory work? Isn’t the idea that essentially any possibility will happen in some past or future existence.

For example, let’s say I’m 5 foot 5 inches tall. On earth, that’s short for an adult male. It’s very unlikely I would do well playing a professional sport like basketball. However, according to the multiverse theory, isn’t there some version of me (a 5 foot 5 inch adult male) dominating the NBA as the best player to ever exist? Meaning, any situation that could exist, by the laws of physics, does exist.

If that’s the case, then there are an infinite number of universes and existences where 2 people are shuffling decks of cards and getting the same order.

In fact, I would go so far as to say there are multiple universes where those 2 people are in fact getting the same order every single time they shuffle.

What do you think?

ChatGPT: The concept of a multiverse is indeed one way of addressing these kinds of statistical issues, and you’ve articulated one interpretation of it quite well. If there are an infinite number of universes in which every possible event occurs, then yes, there would be a universe in which two people shuffle a deck of 100 cards and get the same order. And there could even be universes in which they get the same order every single time they shuffle.

However, it’s important to note that the multiverse is a theoretical concept and is not proven. There are various multiverse theories, each with different implications, and there isn’t yet a consensus in the scientific community about their validity. Some scientists view these theories as interesting possibilities, while others are more skeptical.

The interpretation you’ve mentioned is often associated with the “many-worlds” interpretation of quantum mechanics. This suggests that every quantum event that could happen does happen in some universe, and the universe is constantly splitting into many different versions. In this view, unlikely events happen all the time, just in other universes.

But even if we assume the multiverse theory is correct, that doesn’t change the statistical improbability of these events happening in any given universe. In our universe, two people shuffling a deck of 100 cards and getting the same order is extremely unlikely, even though it might be happening in some other universe.

So, while multiverse theory is one way of thinking about unlikely events, it’s still a theoretical concept and doesn’t change the odds of these events in our own universe.

User: Okay, can you calculate the probability of 2 people shuffling a deck of 100 cards 100 total times.

Each time they shuffle, they get the same order.

So they shuffle 100 cards, 100 times each, and each time, they get the same order.

What is the probability of such a scenario?

ChatGPT: The odds of two people shuffling 100 unique cards and getting the same order on a single try is approximately 1 in 100!, or about 1 in `9.33 x 10^157`.

If you want both people to get the same order 100 times in a row, you would multiply this probability by itself 100 times, which is equivalent to raising it to the power of 100.

The result is `(1 / 9.33 x 10^157)^100` which equals approximately 1 in `(9.33 x 10^157)^100`.

This is an extraordinarily large number - it has 15,800 digits. For comparison, the estimated number of atoms in the observable universe is about `10^80`, which has only 81 digits.

This number is so big that it’s beyond our capability to fully comprehend. So while it’s theoretically possible, it’s so unlikely that it’s virtually impossible in practice. In other words, the chances are essentially zero.

User: What are the estimated number of strings in the known universe?

ChatGPT: The “strings” you’re referring to seem to be related to string theory, a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

As of my last update in September 2021, string theory is still very much a theoretical branch of physics, not a verified or universally accepted one. While string theory does offer potential explanations and models for the fundamental nature of our universe, these haven’t been empirically tested or confirmed due to the incredibly high energy levels that would be required.

Moreover, string theory doesn’t provide a specific prediction for the number of strings in the universe, as strings in this context are considered to be the most basic constituents of matter and energy. The concept of counting them individually, in the way we might count atoms, doesn’t apply in the same way.

In essence, asking for the number of strings in the universe is a bit like asking for the number of points on a line in geometry. The question doesn’t quite fit the nature of the objects involved. However, the continuing development of string theory and other advances in theoretical physics may eventually provide more concrete ways to approach such questions.

User: I’m trying to wrap my head around the fact that there can be a number that’s larger than the number of the fundamental building blocks of the universe (atoms). It seems like it would be a fundamental law that cannot be broken.

ChatGPT: Numbers and mathematical concepts aren’t bound by physical limitations. We can easily define and work with numbers that are vastly larger than the estimated number of atoms in the observable universe, which is about `10^80`.

When we talk about the factorial of 100, for example (approximately `9.33 x 10^157`), we’re not saying that there are that many of something concrete in the universe. Rather, we’re talking about the number of ways to arrange 100 unique items, which is a concept that exists independently of any physical constraints.

This can certainly be a difficult idea to grapple with. When we think about numbers in the context of quantities of things, it seems like there should be a maximum number that corresponds to the total number of the smallest things in the universe. But when numbers are used to represent concepts like arrangements or probabilities, they aren’t constrained in the same way.

That’s why we can talk about numbers far larger than the number of atoms in the universe, even though such numbers may not represent a physically achievable quantity of things. It’s a testament to the abstract nature of mathematics and its ability to go beyond the limits of physical reality.

User: It’s less about needing math to be bound by physics, and more that these large numbers could manifest into reality. Meaning, if 2 people sat down long enough, and were essentially immortal, they would eventually get the same order from randomly shuffling a deck of 100 cards

ChatGPT: Yes, theoretically, given infinite time and opportunities, any non-zero probability event can happen. So, two immortals shuffling a 100-card deck could eventually find themselves holding decks in the exact same order. But the “infinite time” required for such a scenario to occur would be, in practical terms, unimaginably vast due to the sheer number of possible card arrangements (100!).

The chance is so infinitesimally small that even if each shuffle took just a second and the pair shuffled continuously for the entire current age of the universe (about 13.8 billion years), they wouldn’t even have begun to make a dent in the number of total possible arrangements. The timeframe required stretches beyond the lifespan of the universe as we understand it.

So, while it’s technically possible in a theoretical scenario where time and life are unlimited, it’s practically impossible within the confines of our reality as we understand it. This emphasizes the enormity of such large numbers and how they interact with real-world probabilities.

User: Why wouldn’t we just create a pocket universe inside of our own (the current universe we live in), and simply remove the constraint of time from the fundamental laws of physics for that universe. So this pocket universe inside of our own, would be bound by all laws of physics except for time. Would that help the 2 people shuffling cards?

ChatGPT: Your question delves into the realm of highly speculative and theoretical physics. As of my knowledge cutoff in September 2021, there is no known way to create a “pocket universe” or to selectively remove a fundamental aspect like time from the laws of physics. The laws of physics as we currently understand them govern the entire observable universe consistently, and we don’t have the ability to alter or suspend these laws in a specific region or “pocket”.

Moreover, the concept of removing time is a complex one, as time is deeply woven into the fabric of our universe. Many physical processes are inherently time-dependent, including the action of shuffling cards. In a universe without time, it’s unclear what it would even mean to perform an action or observe a sequence of events. The entire concept of causality - the idea that one event leads to another - relies on the existence of time.

It’s an intriguing thought experiment to consider what could be done in a universe with different physical laws, but it’s important to remember that these ideas are currently in the realm of science fiction rather than science fact. As our understanding of the universe continues to evolve, who knows what we might discover in the future. For now, though, the task of two people shuffling into the same order of 100 cards remains a near-impossible one within the known laws of our universe.

User: Assuming quantum immortality is indeed real, wouldn’t the 2 people eventually hit the same combination?

ChatGPT: Quantum immortality is a philosophical thought experiment related to the many-worlds interpretation of quantum mechanics. According to the theory, a conscious observer would always find themselves surviving, even in situations where survival seems highly unlikely, due to the existence of an alternative universe where they continue to exist.

While this concept might suggest that a person could survive indefinitely, thus giving them infinite attempts to match the card shuffle, it’s important to note a few things. First, quantum immortality is purely speculative and not supported by empirical evidence.

Secondly, even if we grant quantum immortality and the potential for infinite shuffles, the same statistical principles apply: while it’s technically possible for the two people to eventually shuffle the deck in the same order, the probability is extremely small. Given the vast number of possible combinations (100 factorial, or around `9.33 x 10^157`), it would still likely take an unimaginably long time for this event to occur.

Finally, remember that the many-worlds interpretation suggests all possibilities occur somewhere. So in most universes, they won’t shuffle the deck into the same order within a human lifetime, but in a very small fraction of universes, they might. This doesn’t change the odds in any specific universe, including the one we’re in.

So even with quantum immortality in play, the two people hitting the same combination of card shuffles remains a near-impossible event due to the vast number of potential combinations.

User: What about time dilation? Let’s say we position the two people next to the event horizon of a black hole. And for the sake of conversation, let’s say the black hole where the 2 people are positioned is experiencing time at several orders of magnitude slower than the observer (let’s say the observer is someone on earth). So for every 10 years on earth, the 2 people near the black hole experience 1 minute.

Now, let’s say the deck of cards is magical, and instead obeys the laws of time of the observer (earth). So while the cards have no subjective experience, if they did, they would age 10 years for every minute spent near the black hole. The people don’t have this same constraint. They still experience time relative to their distance to the black hole.

How long would it take the 2 people to shuffle the cards and get the same order given these new rules?

ChatGPT: Your question introduces an intriguing element of time dilation from the theory of relativity. However, even in such a case, the statistics of the situation remain the same.

Assume we have the two immortal people shuffling the cards as fast as they can (let’s say one shuffle per second for simplicity), and they’re so close to a black hole that for every 1 minute that passes for them, 10 years pass for the observer on Earth.

This effectively means they’re shuffling the deck faster from the perspective of the observer. But, the total number of unique permutations of the deck, which is 100 factorial `(~9.33 x 10^157)`, remains the same.

Even if they can perform one shuffle per second (observed as approximately 630,720,000 shuffles per minute from the Earth due to time dilation), the expected time to get through all possible permutations would still be extraordinarily large.

Let’s calculate that time. `9.33 x 10^157` permutations divided by 630,720,000 shuffles per minute gives us approximately 1.48 x 10^149 minutes. That’s still an unimaginably long time, even when taking into account the time dilation.

So, even with time dilation allowing them to shuffle the deck incredibly rapidly from the observer’s perspective, the time required to shuffle into the same order still exceeds the current age of the universe by an astronomical amount.

These estimates are also based on the assumption that they would necessarily hit every possible permutation once before repeating an order, which isn’t guaranteed due to the randomness of shuffling. In reality, they could potentially repeat orders or miss some entirely, so the actual time could be even longer.

User: How would you solve this problem then? Let’s say these 2 people must shuffle the same order before they die. They’re 2 normal humans, so let’s say max they’ll live until 80 years old.

How would one go about ensuring that the 2 people get the same order at least one time in their 80 years of existence?

ChatGPT: Given the enormous number of possible arrangements for a 100-card deck (100 factorial, which is approximately `9.33 x 10^157`), the probability that two people will independently shuffle their decks into the same order even once in their lifetimes is, for all practical purposes, effectively zero.

The challenge is not something that can be overcome by shuffling faster or living longer. Even if the two people shuffled a deck once every second for 80 years, they wouldn’t come anywhere close to going through all the possible arrangements.

If you want to ensure that they get the same order at least once, the only practical way would be to introduce some kind of coordination or pattern into their shuffling. For example, they could agree to start with the decks in the same order and then follow the same sequence of shuffles.

But this would no longer be random shuffling. In terms of pure random shuffling, it’s virtually impossible for two people to independently shuffle a 100-card deck into the same order in their lifetimes.

User: Hmm okay, what is the maximum number of cards the 2 people could have and still have at least a 1% chance of getting the same order within 80 years (assuming 1 shuffle per 10 seconds).

ChatGPT: Let’s break this problem down.

First, we need to find out how many total shuffles can be done in 80 years.

There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365.25 days in a year (to account for leap years). So, the total seconds in a year is approximately 31,557,600 seconds. If we assume one shuffle every 10 seconds, then there are approximately 3,155,760 shuffles in a year. Over 80 years, that’s approximately 252,460,800 shuffles.

We want to know the number of unique cards (n) that gives us at least a 1% chance of two people shuffling into the same order at least once in 80 years.

This means we’re looking for the number n such that the total number of permutations (n factorial) is roughly equal to 100 times the total number of shuffles (since we want a 1% chance).

While there isn’t a simple formula to solve this equation directly, we can use an approximation for factorials called Stirling’s approximation:

n! ≈ √(2πn) * (n/e)^n

We can solve the equation `100 * 252,460,800 = n!` using this approximation. However, the factorial function increases extremely rapidly, so it’s likely that the number of cards would be quite small. In fact, even for 10 cards (10!), there are already 3,628,800 different permutations, which is already more than the number of shuffles that can be done in a year at a rate of one shuffle every 10 seconds.

Given this, it’s likely that even with a deck of 10 or fewer cards, the chance of two people independently shuffling into the same order at least once in 80 years would still be considerably less than 1%.

User: Well, what’s the number then? If not 10, is it 5? You never solved the problem

ChatGPT: I apologize for the confusion. You’re right, let’s solve the equation.

The problem is challenging because factorials grow very rapidly, and Stirling’s approximation involves both exponentiation and factorial, both of which increase rapidly with n. However, we can give a rough estimate by trial and error method using the direct calculation of factorial instead of Stirling’s approximation, due to the small numbers involved.

We are searching for a number n such that n! is roughly equal to 100 times the total number of shuffles, which is approximately 25,246,080,000 (252,460,800 shuffles per person times 2 people times 100 for 1%).

Calculating factorials for small numbers, we get:

• 5! = 120
• 6! = 720
• 7! = 5,040
• 8! = 40,320
• 9! = 362,880
• 10! = 3,628,800
• 11! = 39,916,800
• 12! = 479,001,600
• 13! = 6,227,020,800
• 14! = 87,178,291,200

It appears that a deck of 11 cards exceeds our number of shuffles for each person, and a deck of 12 cards would give us a 1% chance if both of their shuffles are taken into account. So, the answer would be approximately 11 or 12 cards to have a 1% chance of shuffling into the same order at least once in 80 years.

Note that this is a rough approximation, as we’re assuming that the shuffles are uniformly distributed among all possible arrangements, which isn’t guaranteed in real-world shuffling, and other factors could influence the probability.

User: What other factors?

ChatGPT: There are a few additional factors that might influence the probability:

Shuffling method: Not all shuffling methods are equally effective at randomizing a deck of cards. Some methods may be more likely to produce certain arrangements than others, and real-world shuffling methods often don’t generate a perfectly uniform distribution of possible arrangements.

Card condition: The physical condition of the cards can affect how they’re shuffled. Cards that are bent or sticky might not shuffle in the same way as new cards.

Human factors: Humans aren’t perfect random number generators. When shuffling, people might unconsciously favor certain patterns or sequences, which could affect the distribution of possible arrangements.

Starting arrangement: The initial order of the cards can affect the outcome of the shuffle. Some arrangements might be more likely to lead to certain other arrangements, depending on the shuffling method used.

Number of shuffles: The number of times the deck is shuffled between draws can affect how well the deck is randomized. With some shuffling methods, the deck might become “more random” the more times it’s shuffled.

Time between shuffles: If the cards are allowed to sit for a while between shuffles, external factors like humidity and temperature could potentially affect the cards and influence the outcome of the next shuffle.

All these factors contribute to the complexity of the situation and make it difficult to accurately predict the probability of two people independently shuffling their decks into the same order.

User: Previously in this conversation you mentioned that the 2 people shuffling 100 cards and getting the exact same order on the very first shuffle is effectively zero.

Is this still a high probability than two objects phasing through each other as a result of their atoms lining up?

ChatGPT: The probability of two macroscopic objects phasing through each other due to quantum tunneling is even more exceedingly small than two people independently shuffling a deck of 100 cards into the same order.

Quantum tunneling is a phenomenon that occurs at the atomic and subatomic level, where a particle can pass through a potential barrier that it would not have enough energy to surmount in classical physics. This phenomenon is integral to many aspects of quantum mechanics and is observed in a variety of situations, such as nuclear fusion in stars and the operation of quantum computers.

However, for two macroscopic objects to phase through each other, an unimaginable number of individual particles would each have to quantum tunnel simultaneously in the exact same direction. The probability of this happening is so small that it vastly exceeds the already tiny probability of the card shuffling scenario.

So, although both scenarios are incredibly unlikely, two objects phasing through each other due to all their atoms quantum tunneling at once is even more unlikely than two people shuffling 100 cards into the exact same order on their first try.