70 Rules, The Unfinished Game

70 Rules, The Unfinished Game

The Unfinished Game

I have just finished reading a wonderful book, The Unfinished Game, by Stanford University Mathematician Keith Devlin (http://www.amazon.com/The-Unfinished-Game-Pascal-Seventeenth-Century/dp/0465018963). The book takes a penetrating look at the correspondence that occurred between 17th century mathematicians Blaise Pascal and Pierre de Fermat (yes the last theorem guy). The two great minds had taken up communication in an effort to solve a mathematical puzzle that had been around for at least a couple of centuries prior to their efforts. The Problem of Points, also known as the problem of the Division of Stakes, is a mathematical problem that drew Pascal’s attention in the 1650s. Briefly stated the problem seeks to determine the fair distribution of the wagers in a game of chance that has been interrupted before its conclusion.

The collaboration of Blaise Pascal and Pierre de Fermat over the Problem of Points is widely credited as being the beginning of Probability Theory.   The ability to use mathematics to make intelligent assessments about future events changed the world. Probability became an indispensable tool in a wide array of human endeavors but especially in science and business. Whole new disciplines in sciences and whole new industries in the world of commerce flowed from the letters the two men wrote to each other 450 years ago.

My introduction to Professor Devlin was his publishing of “A Mathematician’s Lament” by Paul Lockhart, on his Math Web Site Devlin’s Angle. Lockhart’s Lament is a passionate plea for the teaching of math as a beautiful and interesting subject. Lockhart believes that the proper approach to Mathematics is as an art and should be approached in a playful and joyous manner. Mathematics should be its own reward. This has been a goal of mine as well. Lockhart now teaches Mathematics in Elementary School.   If this has piqued your interest I also recommend Lockhart’s beautiful book Measurement, which demonstrates what he means (http://www.hup.harvard.edu/catalog.php?isbn=9780674057555).

My journey down this path began in 1984. I read a book called Mathematics and the Imagination (http://www.amazon.com/Mathematics-Imagination-Dover-Books/dp/0486417034) by Edward Kasner and James R. Newman. I have been entranced by mathematics ever since. I became an Elementary School teacher as well and tried my best to make math available to my students in this manner.

I had used the Pascal-Fermat connection as the intro in an essay on my other principle interest, the nature and structure of money. So when I saw that Professor Devlin had tackled The Problem of Points I was eager to see what he had to say. The Unfinished Game has not disappointed. He makes Pascal and Fermat come to life. He explores the Problem of Points from both a historical and theoretical point of view. I was especially happy to see my favorite mathematician Jakob Bernoulli in a starring role. Galileo has a cameo.

Things also took a dramatic turn for me when Professor Devlin concluded his book with a glowing description of a Nobel Prize in Economics won by Robert Merton and Myron Scholes. Professor Devlin uses their accomplishment as an example of how the work of Pascal and Fermat continued to open new vistas in human activity. The two mathematicians received the prize in the 90s for their model predicting the “Value of Derivatives”. I have my own take on the two and their accomplishments and to say that it is the diametric opposite of Professor Devlin’s puts it mildly. I’ll get back to that.

A very special surprise for me was the appearance of Luca Pacioli, the third time this year that he has figured prominently in my studies. He was the first person to publish anything about the Problem of Points. He worked out his own solution which did not gain much acceptance in the mathematics community, but he wrote up and published it 150 years before Pascal and Fermat took up the question. If you haven’t heard of Luca Pacioli you’re not alone. He is odds on favorite (that’s a probability cliché) to win the contest for highest ratio of Importance/Obscurity in the history of Western Civilization.

It’s not that he is unknown. In some circles he is well known. I was using Leonardo da Vinci’s pencil sketch of a Luca Pacioli wood carving of the icosidodecahedron in my own book without crediting Luca. The reason was that I did not know about him. I don’t think I am alone. I had seen several da Vinci pencil sketches of the Platonic and Archimedean Solids. What I eventually discovered was that they were sketches for Luca Pacioli’s book La Divina Proportiona. This book dealt extensively with the Platonic Solids and the Golden Ratio, (the Divine Proportion, the mean and extreme ration, the Golden Section). These are two of the main topics of my book, Mathematical Measures, Mathematical Pleasures, Mathematical Treasures. (http://www.amazon.com/Mathematical-Measures-Pleasures-Treasures/dp/0615743757, free download at http://mathfortheages.com/wp-content/uploads/2012/06/book-final-edit-11-122.pdf)

I am a firm believer in paying attention to any person or piece of information that shows up in two completely different contexts in which I am interested. Pacioli had just hit the jackpot. If the Problem of Points was the third time that Luca Pacioli had been on my radar, and Leonardo da Vinci’s pencil sketch of the icosidodecahedron the first, then there must have been a second.

70 Rules

We, my classmates and I at King City Joint Union High School, were saying it a lot in June of 1970. 70 Rules. It rolled off our tongues. High school graduates are full of themselves. We almost believed it. So for a few weeks before and after graduation you could hear 70 Rules coming from cars dragging main or shooting the loop. We didn’t make the phrase up. When we were freshmen we had heard 67 Rules. When we were juniors it was 69 Rules. (You’ll notice I skipped a year. Not an accident). By the time 70 Rules died down I was left to start considering what to do with the rest of my life. Plan A had been dead for months now and Plan B was nowhere in sight.

One year earlier my Plan A had been to matriculate at Stanford University in the fall of 1970. For this plan to have worked a whole series of events would have to have taken place. These events did not take place. I was not headed for Stanford. It had always been a stretch anyway. As with most of my fantasies of adolescence there was a young lady involved. Alas.

As the summer of 1970 moved along I began to miss my classmates. They are the only group of people I have ever loved. My identity was with them and my identity had disappeared.

By the fall of 1971 I managed to end up at Rocky Mountain College in Billings, Montana. Rocky is the only institution I have ever loved. It saved my life. From 1971 to 1975 it was my Alma Mater, my Soul Mother. It took me in and sustained and nurtured the talents and skills that I had brought with me. I left it a better person than I entered, any step up from useless is not a long journey. But what it also did was to help me regain a spark of curiosity that sustained me for a decade until something else took its place.

Rocky Mountain had a legacy. It was originally Billings Polytechnic. It was started by the Eaton Brothers as a Religiously affiliated college that would accept any and all in the State of Montana and its neighboring states that wanted an education. It was called the College of the Open Door because you could arrive without a penny in your pocket and leave with a college education if you were willing to work. Billings Polytechnic had a working farm in which poor students could earn their tuition with the sweat of their brows. That legacy remained and I thank God that it was still there for me.

My tenure at Rocky Mountain College helped me gain a position in the Master’s Program in the Department of Agriculture and Resource Economics at Oregon State University in Corvallis. Oregon State was in the athletic conference Pac 8 at the time and I got to see the teams from the “Farm” come and play. Neither school was much of a threat to anyone at the time, but the convergence allowed me to wax nostalgic about my Plan A. I hadn’t made it to Stanford, but I had made it to a school that played them in sports on a regular basis. Maybe a Phd. would be in my future.

I loved the study of Economics, but I could not see myself as an Economist. My approach to Economics was always very similar to Lockhart’s approach to mathematics. For me Economics was and still is an art. I went outside the purview of my department for a thesis topic. “The Workforce Incentives of the Aid to Families with Dependent Children”. My thesis remained unfinished for almost 40 years. The subject just kept getting broader. I finally finished it to my satisfaction. It is titled “The 1% Solution” and can be read at my website worldmeetworld.wordpress.com. It was The 1% Solution that brought me to such a completely different reading than Keith Devlin has on the work of Merton and Scholes. Or maybe I’m still just trying to get to Stanford.

The Rule of 70

Luca Pacioli was what is known as a Polymath. He made major contributions in a wide range of areas. I’ve mentioned Probability and Geometry, he was also a Monk, Philosopher, and a Physician, but perhaps his most important field of endeavor at the time was his impact on business practices and accounting. He is considered the Father of Modern Accounting, and Double Entry Bookkeeping. He also created many tables for businessmen that dealt with calculation of interest on loans.

One useful tool that Luca Pacioli made available was a simple method to determine the length of time it would take for a loan to double. It was called the Rule of 70, also the Rule of 72 and the Rule of 69. These “Rules” all dealt with calculations of compound interest. They are actually all the same Rule, the multiple versions and names have to do with the ease of use. 72 is more compatible with the 12 month calendar, but the actual number of the rule is between 70 and 69, to be more exact, approximately 69.4. I like that phrase, “to be more exact, approximately.”

If a loan was let at 5% simple interest it would take it 20 years to double. Five percent per year for 20 years yields 100%, 0.05 x 20. The Rule of 70 tells us that if it was compounded yearly it would take only about 14 years (.70 x 20) to double. You can do the math one year at a time. At the end of the first year you’d have 1.05. Paying 5% on 1.05 yields 1.1025. Paying 5% on 1.1025 will yield 1.1576 at the end of the third year. Etc. Or you can just take (1 + .05) to the 14th power to get 1.97. It doubles in slightly over 14 years. Compounding yearly for 20 years yields 2.65. Compounding every six months yields 2.685. Compounding monthly yields 2.7126… The more often you compound the closer you get to e, 2.71828…

Increasing the frequency of compounding increases the return on the loan but the rate of the increase slows each time. These two opposite tendencies run up against a limit. It runs up against one of the most famous and well used limits in the world of mathematics. (1 + 1/x) to the x, as x approaches infinity. This is the formula for the transcendental number e, 2.71828…..   e is irrational, which means it cannot be stated as the ratio of two whole numbers. e is the base of the natural logarithms, I prefer the spelling logarhythms.

Logarhythms were invented by Scottish Mathematician and Theologian John Napier, Baron of Merchiston. He did this about 40 years before Pascal and Fermat began their correspondence. The table of logarhythms did not figure in their discussion but they did in those of the man who took their work and combined it with the work of Christian Huygens and made Probability Theory a Discipline in Mathematics. This man was named Jakob Bernoulli.

Bernoulli figures heavily in Professor Devlin’s book and is given the credit he so richly deserves. Bernoulli was a member of Mathematics’ most important family dynasty. No less than 7 Bernoullis made major contributions in mathematics in the face of 4 generations. Arguably the most important was Jakob.

Jakob Bernoulli was the first to recognize the connection between the transcendental number e and compound interest. The formula for e is (1 + 1/x) taken to the x power as x approaches infinity. This is 100% interest compounded continuously.

is called the exponential function. e is nature’s growth constant. It is also the constant of radioactive decay. When graphed it yields the logarhythmic spiral. The shape of spiral galaxies and hurricanes. Jakob Bernoulli so loved the logarhythmic spiral that he left instructions to have one carved on his tombstone only to have an Archimedean Spiral carved instead.

More than a 100 years before Napier or Bernoulli or Euler became aware of e Luca Pacioli had published on the Rule of 70. The Rule of 70 comes within a breath of e.   The time for e to manifest itself is compound interest is called the e-folding time. The time for a loan to double is called the doubling time and it is 0.693 times the e-folding time.

Frederick Soddy

The mathematical process of doubling and halving became known in science as the “Half Life.” The “Half Life Period” was a term coined by Earnest Rutherford in his work on radioactive decay with Frederick Soddy in the early 20th century. They were both Nobel Prize winners in the physical sciences.

At the height of his promising career in the physical sciences Soddy left to begin a crusade to challenge the structure of the world’s monetary system. He was solving what he considered the most pressing problem that humanity faced, Money. “In four books written from 1921 to 1934, Soddy carried on a “campaign for a radical restructuring of global monetary relationships”,ring a perspective on economics rooted in physics.” Wikipedia

When the trained scientist Frederick Soddy began to understand how money worked in his world he became angry. He not only wanted banking reform he believed the bankers should be put in jail.

Soddy took Probability Theory and the principles of compound interest and revolutionized chemistry. He then took his knowledge of the Physical world and brought it back to the world of money from whence it came.   He brought it back as a revolutionary who has largely been ignored.

The Continuous Game

What Blaise Pascal and Pierre de Fermat explored was a gentlemanly and fair distribution of wagers based on some ability to gauge one gentleman’s position vis a vis another gentleman’s position in an uncertain situation, a game of chance. The uncertainty derived from the fact that the rules of the game being played had not taken the situation of the interruption of the game into account. The rules very well could have. A rule could have been: if the game is interrupted then whoever is ahead gets the whole pot. Or, if the game is interrupted then the pot is divided on a percentage basis of the points already scored. Both of these solutions would be correct and fair as long as the rules are known ahead of time and participation is voluntary.

The imposition of fairness is a value judgment. The proper place of value judgments is in game design, the setting of the rules. There is no mathematically correct choice between either of these other choices and the solution that Pascal and Fermat proposed.  What Pascal and Fermat determined was fair was what they considered the rules should be in an interrupted game. They believed that in interrupted games each player should be compensated for their relative earning potential. What they really determined is the expected value of each player’s position in a game that is not interrupted. That they did it mathematically is a testament to the power of the tools they brought to bear. The interruption is simply an arbitrary place in time but it is a place in time that turned out to have profound implications. What Pascal and Fermat invented in addition to Probability Theory was Meta-Game Theory, the mathematical analysis and synthesis of chance of human intention. The significance and value of the latter has barely been touched.

Real problems arise when the rules of the game are malleable and participation is anything but voluntary. This is the situation in which the modern world and many of its inhabitants find themselves in today. They have no choice except to participate in a game that is called the economy. The game is rigged and notions of fairness do not enter into its design. Their other choice is starvation for themselves and their families.

Robert Merton and Myron Scholes waded into the “game of economy” with an equation that was used to estimate the value of an investment instrument known as the Derivative. The first red flag should have been the pretentious allusion to the work of Newton and Leibniz in the name Derivative. Merton and Scholes and colleague Fischer Black used the most sophisticate tools that mathematics had to offer to create a predictive model, an equation that would yield information on how the market for this new method of investing would play out. It would help to establish the Value of Derivatives.

Derivatives are so known because they constitute a process by which people do not invest directly on assets, they are one step removed. People and institutions invest on some sort of grouping or packaging of an underlying asset. The packaging may or may not include bets on the future value of the underlying asset. The packaging has developed to the point that it can basically include anything.

It’s just like two guys sitting at a bar and betting on whether the next guy who walks in will be wearing a black or white coat or if a guy goes to the bathroom whether he will relieve himself in the left or right urinal. This is still just plain wagering. But if the modern world of communication gets involved and thousands of their friends begin to place side bets and then those side bets and the original bets are packaged and you can also bet on those packages, you have a basic idea of what Derivatives are. This is still a harmless use of an afternoon. It even has some vague similarity to the gentlemanly context in which Pascal and Fermat operated.

The game in which Derivatives operates, the game which Merton and Scholes and Black analyzed was not a harmless afternoon; the stakes were food, clothing, and shelter for the people of the world. And had they stopped at analysis that would have been idle academic speculation. But they did not stop at analysis, they put themselves in the game. They added their academic cover for a group known as Long Term Capital Management.   A group of very high rollers made fistfuls of dollars in the Derivative markets in the 90s until they didn’t. Then their equation proved to be a complete and utter bust for themselves and their partners. And this would have been fine as well, except for what happened next. The financial establishment bailed them out.

Their equation was not used to seek fairness in a gentlemanly contest. Their equation was used to give advantage and power to the already advantaged and powerful in a game of life and death. By placing their considerable mathematical skills in the service of derivatives and derivative traders they have sought to give unfair advantage. Derivatives are designed to obscure. They are the white collar version of the pick pocket. They help some people make a killing. They lead other people over a cliff. The difference between the game that Merton and Scholes analysed and the one that Fermat and Pascal analisied is that the main outcome being predicted by Merton and Scholes was the behavior of other people. The “predictions” they put into play quickly became a prime mover in the behavior of other people. They essentially became dice shavers for those who knew how they were being shaved. I have no indication that either knew what they were doing but they were at the very best useful idiots. This is not scholarship worthy of being associated with Fermat and Pascal.

What’s true about the modern world is that it is always an unfinished game for some people. For others the game is long since over, or never started, because a requirement of playing the game is that you have to have money. For some people the purchasing power of money is zero, because they have no money. For some people the Value of Derivatives is Infinite because they have no money. Just as Blaise Pascal and Pierre de Fermat examined the tool of their game, Dice, anyone in academia who wishes to make any predictive statements about markets should have some knowledge of the nature and use of money. Money is the dice of the market.

Creating a money that is as indifferent and disinterested in the outcome of the game as a good pair of dice is a necessary prerequisite to introducing any notion of fairness into the economy.   That is a worthy use of the legacy and tools that Luca Pacioli and Blaise Pascal and Pierre de Fermat and Jakob Bernoulli have left us. The 1% Solution is an effort to do just that, create a money that is as disinterested and indifferent to the various players in the game economy as a pair of “Fair Dice”. The whole world is invited to come play.


Keith Devlin has done a great service to the world in drawing attention to a moment in the history of mathematics that is worth exploring and savoring. The lives of these two men are a beacon to the capacity of the human mind.

If he can find the time I would hope that he could give my approach a look and a comment. And if he invited me to Stanford I’d probably accept. I’d only be 44 years late.

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