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McNuggets of Information

Is truth found in the outside world, or in understanding?

by James Leroy Wilson
March 26, 2008

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McNuggets of Information
In an interview at Red Ice Creations, author Jim Elvidge noted that mathematics is about proof, whereas science is about evidence. I hadn't thought of it that way before.

A mathematical proof is about logic - the logic of numbers. Science is about collecting information in the physical world. The thought reminded me of a riddle on the Car Talk radio show three years ago. The problem:

A fast-food restaurant sells chicken nuggets in boxes of three sizes: 6-piece, 9-piece, and 20-piece (at the time, this was obviously McDonalds with its McNuggets). But one customer regularly comes in asking not for these particular sizes, but for a total number of nuggets that no combination of these box sizes can fill to the exact amount. When the customer sees he will be charged for two 6-pieces when he orders ten nuggets, or one 20-piece when he orders nineteen nuggets, he gets angry and leaves.  The question: what is the largest number of nuggets the customer can order, that the restaurant can't fill to the exact amount?

The show's hosts had arrived at an answer - by collecting information. For instance: they figured out that 21 pieces requires two 6's and a 9; 22 can't be filled, 23 can't be filled, 24 would be a 6 and two 9's, 25 can't be filled, 26 is one 6 and one 20, 27 is three 9's, 28 can't be filled, etc. Working their way up,they reached a number above which it seemed that any number could be filled - at least, every number they tried.

They arrived at the correct answer, but their method only suggested their answer was probably true; it wasn't proven.They were merely collecting information, testing each number individually. But this did not sound to me like the best way to solve mathematical problems. Yet I am not a mathematician; how could I solve it?

Since they had given away the answer, I knew it would remain in the back of my mind, and my solution would be "cheating" since I knew the answer in advance. But, in any case, here is my answer.

The first thing I noticed is adding three to any number divisible by three (6,9,12,15, etc.), you will be able to fill an order with combinations of 6-piece and 9-piece boxes: 12 = 6+6, 15=6+9, 18=9+9, 21=6+6+9, etc. So, combinations of 6-boxes and 9-boxes could fill one-third of all nugget orders from 6 to infinity.

Then I realized that all numbers not in this "3+3" progression are in one of two other progressions. The first is the "(3x-1)+3" progression of 2, 5, 8, 11, 14, 17, 20, etc., since adding one to any of these numbers will result in a number divisible by three. The second progression  is (3x+1) +3" progression: 4, 7, 10, 13, 16, 19, 22, etc. Subtracting 1 from a number in this progression results in a number divisible by three.

So the question is, where would the box of twenty fall? It turns out to be in the 3x-1 progression, because 20 = 3(7) -1. So we know we can fill one-third of every order of 26 or more with one box of twenty and combinations of sixes and/or nines: 26, 29, 32, 35, 38 etc. In addition to all the orders in the 3+3 progression that we can fill, we have two-thirds of all numbers 26 and above accounted for.

That leaves one-third of all orders over 26. But it doesn't take long to see that 2 boxes of 20 makes 40, which is part of the last, "(3x +1)+3" progression, because 40=3(13)+1. Therefore with two boxes of 20 and combinations of sixes and nines, we can fill one-third of all orders 46 or over: 46, 49, 52, 55 etc. And, as we've shown, we can also fill every other order over that number in the "3x+3" progression or the "(3x-1)+3" progression.

This means that the biggest number that can't be filled is the last one before 46 in the "(3x+1)+3" progression, because all numbers in the other progressions between 26 and 46 are accounted for. The number, then, has to be 46 - 3, or 43.

Perhaps someone else can explain it more clearly (I just found out that Wikipedia has an entry on the Chicken McNuggets Problem.), but the point is that I proved the answer was 43 through reason, as opposed to assuming it was 43 based on the "evidence."

All that is required in mathematics is understanding. You don't need to collect information or perform experiments. Understanding - and mathematical tools - can help us analyze and interpret the data we collect to suggest theories, but all theories are provisional because additional information may change our theories. New information, and new theories that explain more information, will replace old theories that explain less, but they won't arrive at the whole truth.

Perhaps the only "true" or "real" things are those that do not require "evidence" at all, but are rather "self-evident" and can be found in its own nature. Mathematics is like that.

What I'm suggesting then, is that information, including statistics, can not "prove" anything, they can only add "evidence" to persuade us of a provisional assumption. Most of these assumptions can be accepted as true for most practical purposes. But the fact is, we don't know the underlying structure of the Universe in all its dimensions. We don't know what's really going on. At best, we may arrive at the "truth" the same way the Car Talk hosts solved the McNugget problem: their answer was true only insofar that no evidence had yet refuted it. It took logic to prove their answer was correct.

Everyone has an odd relationship with evidence. With data. With information. I wouldn't say "love-hate" relationship, because I don't think people are conscious of their biases. People use evidence to support what they already believe to be true, and explain away contrary evidence. It varies from person to person, but the truth is either believed by faith, or it stems from understanding.

For instance, one man serves on a jury. He votes to convict based on the weight of circumstantial evidence, persuaded that, statistically, the chances the accused is innocent are remote; he doesn't believe the circumstantial evidence could all be coincidental. Yet, when there is a great political crisis with powerful people behaving strangely, he automatically assumes coincidence and dismisses both conspiracy and synchronistic programming from a higher power. He doesn't believe the latter options can be true. He dismisses the "intelligent design" of conspirators and of God out of hand. That leaves coincidence, no matter how improbable, as the only option to embrace.

Then there is economic data. It can often show that government enacted A, and then the economy grew. Government did B, and the economy grew. Government did C, and the economy went into a recession, but then Government did D, and the economy recovered.

Does any of it prove anything? Not really, because the data can't measure what would have happened had the government not intervened. Also, correlation does not mean causation: economic growth may have occurred most where the government intervened least. More importantly, no statistic can measure quality of life. "Quality of life" can only be judged by each individual assessing his own life; national economic and vital statistics have nothing to do with it. For some, access to a swimming pool enhances their quality of life; for others, it is proximity to a theater. For some, a large bank account is what makes them sleep at night, and for still others, it is a close-knit family.

Government can steamroll the dreams of some for the benefit of others. Perhaps they can favor some industries over others, and manage to increase the Gross Domestic Product and tax base along the way. But real economics is about choices individuals make for their lives, and money is just one factor among many. Individuals don't care about their nation's GDP if they are personally starving. Economic is about human action, not government policy.

Just as one doesn't need outside information to solve mathematical answers, so it is that one doesn't need statistics to understand economics. Truth is not found in information, it is found in the mind.

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