The mathematical universe and the shopkeeper's paradox

The shopkeeper's fake currency paradox

The mightiest discipline!

Today is the birthday of Miss Shakuntala Devi, also known as the human computer. She's so popular that a simple Google search would take you to the Wikipedia article on her, and Google India has even assigned her a special doodle to mark the occasion.

To mark the occasion, perhaps, the owner of India+ page and India community on Google+ and a good friend of mine, Mr. Rajesh Narayanan, posted an interesting riddle concerning fake currency and economy, both spicy topics so far as the current Indian trend is concerned. So, his original post is shown below:

Today is birth anniversary of Mathematical genius Shakuntala Devi, here is an interesting riddle.  Try to solve it :)

A lady buys goods worth 300 and pays 1000 note to the shopkeeper. The shopkeeper gets the change from the neighbouring shop, keeps 300 for the goods and returns 700 to the lady. After sometime the neighbouring shop owner returns saying that the 1000 note is fake and takes his money back. 

How much loss did the first shopkeeper suffer?

After he personally pinged me, I took notice of the matter, and kicked off my own solution. Obviously, I did not succeed at my very first attempt, and it took around 5 minutes for me to crack the problem. However, after solving it using elementary game theory and algebra, I got really fascinated to see how such a simple deal can help us conclude some deep secrets of the universe!

The solution 

As evident, the problem is not so direct as it appears first. After quite a while, only two of us could actually answer, including Mr. Sunil Yadav (a fellow moderator of India+ and another friend of mine) and me. Naturally, the first bit of fascination comes from the simple fact that such a simple-looking problem can indeed be so tricky. But here we do miss one point of human psychology: we always happen to consider the shortcut. However, after trying to deal with the problem in the straightforward arithmetical way, I instead considered giving algebra a shot.

So, here's the logical sequence: 

After I tackled the problem on my board

1. First, let A,B and C denote the lady, the first shopkeeper and the second one respectively. Since the question revolves around B, let B's total initial amount be Rs. x.

2. Now, at first A gets goods worth Rs. 300 from B. So, B's amount becomes x - 300.

3. A pays B with a fake note of Rs. 1000, having zero value. So, B's amount remains the same, i.e Rs. (x - 300).

4. Now, B goes to C and hands over the fake note to him. So, B's amount remains the same.

5. C pays B Rs. 1000 (real deal this time, really!), making his amount Rs. (x - 300) + 1000 = Rs. (x + 700).

6. Now, B pays Rs. 700 to A. So, B's amount becomes Rs. (x + 700) - 700 = Rs. x.

7. C comes to take his Rs. 1000 share back from B, making B's amount Rs. (x - 1000).

Hence: 

A. A's total loss = total initial amount - total final amount = Rs. x - Rs. (x-1000) = Rs. 1000. 

B. Had the 1000 Rupee note been real, there would have been no loss or gain on anyone's part. 

Application of Game Theory 

Neutron decay is an example of zero sum game

Elementary game theory dictates that most activities are of two kinds, zero-sum and non-zero sum. Zero sum games are those activities where if one player/party gains, there must be an equal amount of loss for another player/party. For example, if X pays Y Rs. 10, X's amount decreases by Rs. 10, while Y gets richer by Rs. 10.

On the other hand, non-zero sum games are more complex. Here, one's gain isn't accounted for by the loss of another's amount. For example, if X teaches Y how to fish, then X brushes up his lesson himself and doesn't lose anything, while Y learns something new and thus gains. Thus, Y's gain doesn't necessarily imply or require X's loss.

It is believed by many, that the entire universe is the result of a zero sum game. As per modern advancements in inflationary big bang theory (IBBT), we know that the universe might have started from an absolutely small state, so small that the laws of particle physics (as per Prof. Guth) had been applicable o the universe as a whole at the beginning. According to Prof. Stephen W. Hawking, the universe could be the ultimate free lunch, having started off actually from zero. Equal amounts of positive and negative energies could have been created at the onset of the (causeless?) quantum event which supposedly triggered the formation of the universe as we see it. So, building the universe is like making digging off a finite volume of sand from a desert, creative a hole and a hill simultaneously. This hill symbolically represents the universe. If the hill is deposited back into the hole, everything becomes plain and leveled again. Thus, the creation of the universe might actually be a zero sum game.

Evidence? 

Since we have deal with the mathematical and physical singularity while trying to figure out the early universe, which we still have no way of tackling, we have to wait for a more basal theory to emerge, which would be more versatile than our existing stock of the two big players, quantum mechanics and general relativity (mutually incompatible). This theory might actually be a candidate for the Theory of Everything (ToE), and the contrasting theories above mentioned should be derivable from the new theory.

However, we have events in nature which already are examples of zero sum events. Three of these are:

Hawking radiation explains how radiation
is emitted from the event horizon of a
black hole

1. Decay of a chargeless neutron into a negatively charged electron and an equally and oppositely (positively) charged proton. The rest portion of the neutron's mass is ejected out in the form of a antineutrino. Here, charges are created from chargelessness in a way that their algebric sum is exactly 0. This is basically due to the first law of thermodynamics.

2. The law of conservation of energy, another result derived from the first law of thermodynamics. Here, if there's a certain amount of energy apparently 'vanishing' at a point, an equal amount of energy should 'appear' at another point. No matter what we do, energy is always conserved.

3. Formation of a pair of particle and anti-particle at the event-horizon of a black hole (Hawking radiation).

Getting back to the problem once again, we notice how a similarity between the problem and something so vast and complicated as the universe itself is found. One thing we notice about the problem itself almost immediately, is that it takes places within an isolated system. There are three players, A,B, and C, and that's why since A encounters gain, there must be someone who loses an equal amount. Since C doesn't lose/gain anything, it must be B.

So, two of the final insights that we obtain from so simple a riddle are as follows:

Proud to be a part of it. :)

1. The universe is also an isolated system, quite possibly. Although there might be other universes as per the multi-verse concept, physically speaking we don't have enough evidence to claim their existence. Multi-verse is essentially a construct that is generated to account for clearing off apparent 'biased'-ness of the universe in choosing one event from a set of more than one possible outcomes, by the universe (or reality, as we see it).

2. The universe is actually mathematical. Now why is the nature mathematical? We don't exactly have an answer, but the twin anthropic principles from M-Theory give us very logical and some satisfactory statements (which aren't exactly straightforward answers). Strong Anthropic Principle (SAP) states that, the setup of the universe is tuned in a way that intelligent life, capable of asking the important question of how the universe came into being, should gradually emerge out of it (or evolve from it). Thus, using logical notation,

p => q, so, ~q => ~p 

Simple contrapositive of the statement yields us another valuable insight. If the universe had not been mathematical and tuned in the way it is, we would not be here to question why it is the way it is. :)

Lastly, it is really puzzling to see the transformational transition of human thoughts. From so simple and straightforward a riddle, emerged something so grand and divine as thus. Really, math is brilliant!

Thanks to Mr. Rajesh Narayanan for sharing his post. And if you're on Google+, are from India or just interested to know about our country, be sure to follow India+ (https://plus.google.com/u/0/104771037582332553233) or join the India community (https://plus.google.com/u/0/communities/100453920415382664577) right away!