Category Archives: Math


Here’s a whimsical taIe I wrote some time ago to teach my grandchildren that the real value in society comes not from money but from people helping people.  They were a little young at the time so I set it aside for a while.  Now they are teenagers wondering how the coronavirus will impact their future.  This is a story of the resilience of people facing an economic firestorm.  By learning how to pull the right economic levers, the people of Tangia recovered the power that lies within each of us, the power of people helping people.  By sharing this story with your children, my hope is that it will inspire them to use that power and to believe we can make the new normal better than before.

It was an impossible fantasy because a financial loss hit every citizen at the same time.  When I wrote this, I never dreamed these conditions would ever be so closely replicated in the real world.   But we now see it with Coronavirus.

The Wisdom of Sunch

Once on a planet far, far away and long, long ago, there was a civilization much like our own.  A great nation emerged on planet’s lone continent.  This all ended when a sudden cataclysmic event struck in the center of this giant land mass, causing it to sink beneath the sea.   All that was left were two small islands, thousands of miles apart.  Survivors on the larger Island known as Porgia, recovered to populate the island with some 5,000 inhabitants; but our story begins several thousands of years after the cataclysm on the smaller Tangia, home to a mere 2,000 souls.

Cranic is a shoemaker who is teaching his son Sunch his trade as his father had done for him.  His wife and daughter prepare the meals and tend to household matters – much as it would come to be in the far distant future on a planet called Earth during its Dark Ages.  There were no schools. There was nothing more to learn than how to live and get by the way their ancestors had always done or so they thought.  There was no need to study history because as far as anyone could remember it had always been the same.  They knew nothing of the great continent that once was – nothing of the island of Porgia and distant cousins living there.  Only traces of the technology from the distant past remained.

Sunch’s ancestral line had preserved the skill of turning leather into footware just as the blacksmith across the way had learned how to form the nails Cranic used to bind the soles.  The occasional plague kept the population in check.  It never varied by more than 100.  The significance of this is an economy that never changed.  Every citizen carried a leather pouch filled with gold coins.  No one knew where they came from or as Cranic explained it to Sunch, “They must have always been here.” 

The coins had a faint silhouette of a human head on one side, but Sunch couldn’t have known the detail worn away with the years.  It was a tradition among Tangians to make sure at the end of every day that each pouch contained a full quota of ten gold coins.  There were always just enough to go around.  It’s a good thing too because the arts of mining gold and minting coins had been lost for millennia.

Every day Sunch would go to the butcher to buy meat for the table and hides for the shop. He would stop by the market for fresh vegetables, the chemist for tanning oil and the blacksmith for tools and nails.  He would return home with an empty pouch which Cranic would fill with the day’s receipts for shoes.  Somehow it always came out even.  With an unchanging economy, prices had long ago settled into a delicate balance that held steady ever since.  Accounting was not an issue.  There wasn’t even a word for it because everybody knew it would work out just right at the end of the day.  There was no greed in Tangia.  Why would there be?

They had no Congress because there were no big decisions to be made. They had no military because they had no enemies.  They had no police because there had never been a theft.  After all, having more than ten coins would be just extra weight to carry around.  In Tangia, everyone had just what they needed with no extra to go to waste.

One day Sunch went down to the fish market on the beach to find everyone looking out to sea.  They were watching what appeared to be a giant white bed sheet rising slowly out of the water.  The crowd grew as a huge boat began to appear beneath the sheets.  This boat was nothing like the hollowed log canoes or rafts used by the local fishermen.  And as it neared, the immensity of the boat became clearer and it appeared to come from the “deep.”  The deep was a zone beyond even the view from atop Mount Rumble.  No Tangian dared to go there for fear of falling off the edge.  Rumor has it that some fishermen had ventured into the deep never to return.

The boat stopped in the harbor, the sheets curled up and a man threw something into the sea with a splash.  Soon a few of the strangers paddled ashore in a small dinghy where a brave few of the Tangians greeted them.  They came bearing gifts, colored stones and objects such as no Tangian had ever seen before.  They asked the crowd to go home and gather everyone together in the morning. 

When they returned in the morning one of the strangers, the one in the three-cornered hat who appeared to be in charge began to speak.  He announced they had come to collect taxes, one gold coin from every citizen.  The crowd murmured while some vocally protested and declared they wouldn’t pay.  The leader turned and spoke to one of his men.  The man then raised the smooth stick he had been holding.  Sunch recalled that a farmer had found a similar stick buried in his field, but it wasn’t shiny like this one.  The leader raised his hand and the man then pointed the stick at a seagull flying overhear and there was a loud bang such as the Tangians had never before heard.  The seagull plummeted to the ground.  The leader explained that’s what’s in store for anyone who doesn’t pay taxes.

The Tangians complied and watched until the sheets disappeared into the deep.  The next day Sunch started on his daily rounds, but his pouch was empty when he reached the blacksmith shop.  When he told the blacksmith about the problem, the blacksmith suggested a revolutionary idea, “Raise your prices so you will have enough when you come back tomorrow.”  But we need the nails so we may have shoes to sell.” The blacksmith said, “Everybody needs to meet in my shop tonight.” 

So all the merchants gathered at the blacksmith shop that evening.  With no schooling in math, they all thought the blacksmith had a good idea and unanimously agreed to raise their prices.  Of course, the next day things didn’t go so well.  Sunch’s pouch was empty before he got to the chemist’s shop.  Despondent, he returned home with food, but no supplies for making shoes.  Cranic consoled his son saying, “That’s okay because few of our regular customers showed up to buy shoes today.”  That meant he had enough shoes left over for tomorrow.  The whole family was supportive, but they all knew they were in trouble. 

Sunch went into the woods to think.  He thought and thought.  He thought raising the prices only made things worse.  Leaving prices where they were didn’t work either because no one had enough gold coins to buy what they needed every day.  If raising prices made things worse, maybe lowering prices would make things better.  So Sunch called for another meeting at the blacksmith shop.

“Everybody needs to agree,” Sunch explained. “Otherwise this plan will never work.”  The blacksmith, the leather smith, the clothier and the chemist quickly agreed because they too had emptied their pouches.  But the farmers, the fishermen and the butcher refused to go along.  They never had it so good – their pouches were overflowing with gold coins.  People still needed food so they were willing to pay the higher prices.

A dejected Sunch retreated again into the woods.  How could he convince the farmers, the fishermen and the butcher to give up this windfall?  What would become of the tradesmen who would soon be unable to buy even food?  What good is it for some people to collect all that gold if there’s no place to spend it?  “That’s it!” he thought, and called for another meeting.

This time he explained how the farmers, the fishermen and the butcher would soon have all the gold coins, while the tradesmen would have none to feed their families or to make the goods needed.  No blacksmith to make the plows for the farmers, the hooks for the fishermen or the knives for the butcher.  There would be no more saddles for the farmers, no more canoes or oars for the fishermen and no more clothing or shoes for growing children.  Gold coins would be worthless when there was nothing to spend them on.

Sunch’s argument must have been a good one because everyone agreed to go with the lower prices.  In fact, they went a step further and redistributed all the gold coins so that everyone had exactly nine coins in their pouches.  With that their community returned to normal and every citizen realized that nothing of value had been taken from them after all.  In fact, they were better off because they were all just a little bit smarter.  And so they all lived happily ever after.

That seems a good place to end a story, but I have to tell you the rest because it didn’t work out so well for the island of Porgia.  The Porgian pirates returned home with 2,000 gold coins.  Of course, they were just fishermen when they started their journey, but they became pirates when they saw how easy it would be to take the gold from the Tangians.  They divided the booty between the crew who all went on a spending spree the moment they returned home.  The Porgians were a more advanced society, and their economy was more complex.  The increased spending sent prices through the roof, sending their economy into a tailspin from which it never recovered.  The Porgians never figured it out.

The moral of this story is all value in any society comes from its people, like potential energy stored in each one of us begging to be released.  Economics has no meaning without people or more aptly, without our labor.  The point is when we come out on the other side of this coronavirus crisis, we will still have the same value locked up inside of every one of us begging to be released.  Call it labor, call it GDP, or call it money in motion, we have to find a way to get it moving again lest it grind to a halt.

What Sunch taught us is that money or gold is just a place marker, a universal IOU that keeps track of our good deeds.  It’s mobile in both time and space; but perhaps from time to time, it needs to be recalibrated.  When it gets recalibrated, all our metrics get shifted. 

The Tangians of course, had no concept of metrics – like gross domestic product (GDP).  To them, their economy seemed constant.  With a little help, it was the same after the pirate raid as before.  Had they measured GDP (in gold coins), they would have seen that it had gone down by 10% because they were spending 10% fewer coins per day.  This decrease in GDP may have concerned and confused them.  It may have lead them to make mistakes, in turn causing a downward spiral in the economy as happened with the Porgians who did rely on GDP metrics.  We see they were all buying the same amount of goods and services for less gold than before.  This upward revaluation of money or gold is what saved their economy.

Now I don’t imagine that we’d have much luck today getting everyone to agree to divvy up the money.  Fortunately there are other ways in a complex economy to redistribute wealth.  Our response to the coronavirus crisis is to put money in the hands of those who are likely to spend it, increasing demand.  In accounting terms, we put money on one side of the ledger without putting labor on the other side.  This would mean the ledger doesn’t balance.  The rational is that increased demand as we emerge from the crisis, will raise prices, attracting investment and creating jobs.  In other words, we expect to balance the ledger as people fill these jobs, go back to work, and spend the money. 

But where did the money come from?  Didn’t the government just print new money?  In effect, yes they did, but it didn’t come without a price tag.  So who paid the price?  To answer this, you have to think of money as shares of the economy – like pieces of a pie that add up to the whole.  When we come out of this crisis, the wealthy “investor” class will still be holding most of the shares while the much larger “consumer” class will have little or none left to spend. 

Giving everyone a fixed amount of $2,000 wouldn’t change the size of the whole pie:  it just cuts the pie into more pieces – smaller pieces.  Unlike the Tangian solution inflating the value of gold coins, gifting money to everyone simply devalues the dollar making prices got up.  So anyone holding a lot of dollars will see their share of the pie shrink.  Like the crisis in Tangia, redistributing wealth to those who are likely to spend it is the crucial factor.  Whether we inflate or deflate our currency doesn’t matter.  This massive redistribution of wealth to the consumer class is the right move.  I know you will hear otherwise because it won’t feel much like a boon to the average consumer. 

During the Great Depression of the 1930s the world was slow to find this solution.  Bread lines and soup lines served a humanitarian role, but did nothing to spur discretionary spending (money spent by consumers on things other than necessary things such as food, clothes, and fuel).  Eventually infrastructure projects like the Hoover Dam and the Tennessee Valley Authority (TVA) followed by WWII were able to accelerate spending producing a boom in the late forties and the fifties.  Interestingly neither the projects nor the war resulted in any consumer products for us to spend money on.  It was full employment and fair wages that created the demand for consumer goods. 

Eisenhower’s interstate projects in the fifties and sixties extended the boom.  Since then our economic profile has been dominated by lobbyists for big business.  The prevailing theory is that only the sale of consumer products bringing profits to investors that matters.  At the same time businesses invest in ways to replace labor forces through cheaper advance technology, as they should.  Greater efficiency benefits us all.  The problem is it starves the consumer class through low employment or low wages.  This reduces demand for the very products they want to sell.  This in turn demands more low-cost technology to replace more of the labor force.  The cycle continues. 

Individually these companies are doing everything right (microeconomics).  What they don’t do (and in fact they can’t do unless every business in the country agrees) is finance the infrastructure needed to make all business and the economy as a whole run more efficiently (macroeconomics). 

“Everybody needs to agree” Sunch explained “otherwise this plan will never work.” 

Tangia had no central government to make these types of decisions so everybody had to agree.  We not only have a central government but we have lessons from our own history to show us how.  If we do this right this crisis could lead us to the next infrastructure boom. 

Since the early 1960s big business has been cast in the role of the bad guys exploiting low-wage workers.  The poor are criticized for not working hard enough and claiming benefits they don’t deserve.  There is little justification for either criticism because the system is rigged.  And it’s rigged against both the rich and the poor. 

Imagine your company manufactures a product for retail sales.  When your warehouse is full you lay off workers until there’s room again for more product. The same thing is happening to other companies across the country.  Laid off workers stop buying product because they don’t have the money or they have lost confidence in the economy.  Companies and workers would both benefit if there were a way to put more money in the hands of workers without overflowing the warehouses.

Giving out money to consumers eventually just gets offset by higher prices.  Instead we want to give out jobs that do not produce immediate consumable goods.  That might include the military which provides long term security.  We already have that so let’s look instead to infrastructure to streamline transportation, communication and energy.  Like the Hoover Dam, TVA and interstate highways, these projects benefit everyone and we will reap the benefits for decades to come.  Let’s look to green solutions to head off global warming, the next great crisis. 

While we must morn the toll of coronavirus, we must also find a way forward.  Why not solve this economic crisis by solving the next one?  The extra income will raise the standard of living for the consumer class and raise profits for the investor class. 

As for the “happily ever after” part, that depends on you.


The scientists gave us the relevant numbers, but it somehow just didn’t compute with our politicians.  It’s pretty well established that lawyers and politicians can’t do math that doesn’t start with a $.  The “No Child Left Behind” Act that mandated every student rank above the 40th percentile is a case in point.  Here’s a story I read in the late 90s that demonstrates how even professionals can overlook the math.

A young woman questioned her professor about a B+ she received on a paper in 1994.  The professor agreed that it was well written and would have been an A had she not gotten her facts wrong.  He showed her where she wrote “the number of children killed on the street by gunfire has doubled every year since 1954.” She asked what was wrong with that.  He said I don’t know how many children were killed in 1954 but suppose it was just one.  He said doubling every ever would make the total number this year 240 or 1,099,511,627,776.  But she was able to show him this quote taken word for word from a reputable magazine.  It turned out they took from another periodical who made an error in copying it from a child advocacy hand out where it read “the number of children killed on the street by gunfire every year has doubled since 1954” , about the same growth as the population. 

I didn’t need a calculator the see the problem.  Perhaps dealing with large numbers majoring in physics made it easier for me because we did a lot of rounding.  I know that 210 = 1,024, approximately 1,000 and each multiplier of 1,000 adds three more zeros giving me one trillion as a low-side estimate.

If they had a category this would certainly make the Guinness world record as the largest computational error.  The young woman’s B+ was a lenient penalty for such a humongous error.  But in science the penalty is often severe. 

When faced with a small number of local cases of coronavirus, we tend to focus on that “small number” hoping it limits what any function applied to that small number, can do.  In this case it doesn’t.  The function tells us that it will double every three days.  We know this won’t reach one trillion in four months (30 days = 10 doubles = 1,000 fold), but it can be in the millions.  Of course “better late than never” mitigating factors can still make a huge difference. 


A Typical Ranked Choice Ballot

Its forthright simplicity, directness and familiarity make it much more likely to gain voter acceptance by voters than other more complex forms of ranked choice voting.  You can think of it as plurality voting on steroids.  The ballot is no different than the one we are all used to and if you choose you can simply put an “X” or the numeral “1” in the box next to your top pick.  But you may also rank your second and consecutive choices on the ballot, as few or as many as you like.  This is a “ranked choice ballot.”  Either way, if your top pick gets a majority of first place votes he or she is the winner. 

But a simple plurality doesn’t cut it. 

What’s wrong with selecting a plurality winner?  Well I’m virtually in the process of writing a book on this topic called “Finding Our True Political Center”, but I will give you one example here.  We’ve all seen polling on head-to-head matchups.  If I were to win head-to-head matchups against every one of my opponents, it makes sense that I should win the plurality vote as well.  So it may surprise you to learn that in a polarized race there is a good chance that I will lose to a polarizing opponent.  But wait for it; that same polarizing opponent may actually be the plurality winner even if he loses head-to-head matchups against every one of his opponents.   In voting theory this is known as a Condorcet loser. 

When no one gets a majority of first place votes with consensus voting we add the second place votes to the total to see if some majority of voters rank any candidates in either first or second place.  If so the candidate with the most votes wins (note as many as three candidates can meet this requirement).  If none do, we simply add additional tiers until a winner is named.               

As a moderate you have liked both Klobuchar and Buttigieg for the 2020 Democratic nominee. Splitting the moderate vote you hoped one would drop out soon so the other can pick up their supporters and gain more delegates in March.  This conundrum called the “spoiler effect” is a common problem with plurality voting.  Let’s see how consensus voting eliminates this spoiler effect.

For simplicity let’s make our hypothetical election a nation-wide consensus election for the Democratic nominee. Since we expect no single candidate to have a majority of first place votes we know a consensus winner cannot be named before the second place votes get added.  By ranking one of our favorites first and the other second you effectively cast a full vote for each.  The theory is you give them each the number of votes they would have received from you had the other not been in the race.  This argument actually bears out pretty well in a diverse field of voters but I will leave the proof to the mathematicians.

The term “ranked choice voting” refers to any of the many forms of voting that allow ranking of choices on a ranked choice ballot.  The most popular of these, “Instant run-off voting” is often mistakenly called ranked choice voting, seemingly to the exclusion of a host of other prime candidates.  I submit it is the most popular because it’s the first, not because it’s the best. 

Many ranked choice forms reduce the field one candidate at a time then repeat the algorithm on the remaining field in “elimination rounds.”  The “instant run-off” eliminates the candidate with the fewest first place votes.  The “Coombs method” or the “bad apple sort” eliminates the candidate with the most last place votes.  No matter how good these more complicated algorithms are, the fact that voters are confused by and skeptical of elimination rounds is problematic.  Consensus voting takes a straight-forward approach to selecting one winner rather than eliminating a series of losers.

My personal mission has been to find a more inclusive algorithm that doesn’t scare voters away.

Inclusivity is a measure of how well the voting method weighs the wishes of all factions (including such as racial, ethnic and religious groups in addition to party affiliation) in proportion to their numbers to find the right balance. 

The instant run-off is neither inclusive nor user friendly. The Coombs method is quite inclusive but perhaps even more difficult for voters to follow.  Consensus voting is inclusive, intuitive and familiar.  This leaves us a couple of questions, why do we want an inclusive method and how does consensus voting accomplish this? 

The “Why” takes us back to polarized elections and divisive campaigns. 

A polarizing candidate is one who attracts a lot of first place votes and a lot of last place vote with few in the middle.

Of course this is by design because the objective in a plurality race is to “be the first choice of a lot of voters” so it really doesn’t matter what the rest of the voters think.  His Amazon rating might be 3-stars, because that’s what he gets with a lot of ones and a lot of fives.  In other words we elect a lot of average candidates who are also divisive politicians.  Getting all 4-star ratings gets you squat in a plurality race.  A lone polarizing candidate is possible but more often they come in (polar opposite) pairs. So we need an algorithm that gives weight to those second and third choices.

The best way to understand the “How” is to see it in action especially in a polarized race.  Consider a three-way race between a liberal, a conservative and a moderate.  Both of the extremists have lots of firsts and lots of lasts and certainly one of these will win a plurality race.  The moderate dominates the center but will have the fewest first place votes making him the first-man-out in an instant run-off race.  One of the two more divisive candidates will win promising four more years of divisive and obstructionist government.

If no candidate receives a majority in a consensus race we add in the second place votes making the moderate a certain winner with nearly 100% of the (first or second place) votes.  This introduces a new problem.  What if the guy in middle is a doofus?  He wouldn’t be the first doofus we’ve elected.  But would we really elect a doofus in this situation?  The answer is yes. In a polarized race defeating the opponent is as important, sometimes even more important than electing the preferred candidate so yes, we will always place him last even if it means electing a doofus.

But this is a problem that really isn’t a problem.  Don’t get me wrong; because of this apparent problem it took me a while to find this very simple algorithm and no doubt this is the reason no one has endorsed or promoted it before.  What we need to understand is plurality voting and the instant run-off open the outer lanes for divisive candidates while consensus voting closes these lanes an opens the center lanes for moderate candidates.  With the center lanes wide open many qualified moderates would compete for these lanes and there are lots of qualified moderates.

We are not trying the change the behavior of voters because we can’t.  But we can change the behavior and the incentives for qualified candidates to become unifiers rather than dividers.

How to Steal an Election from the NRA

A call to action

My recent blog article created a stir when I said,

Do you or your group want to take out incumbents who haven’t supported responsible gun control legislation?  Register with their party and try to take them out in the primary first.” 

Just to be clear this is not about dirty tricks; it’s about using your vote where it will do the most good.  Plurality voting as used in most primary elections gives us ample opportunity to waste our votes.  If voting for your favorite candidate means wasting your vote on a certain looser, you may cast your vote instead for the lesser of evils that stand a chance of winning.  If your vote is not needed because your choice is sure to win anyway, you may not waste your time voting at all; you may have even done this in the last election.  There are voting methods that resolve these dilemmas on a simple ballot, but we have to make the most of the system we have.  Continue reading

The Sparrow Math Series

This video series is designed for students of all ages who want to improve their math skills.  Singapore Math has been incorporated in the curricula in many primary schools.  It has  proven that these skills can be mastered faster and retained better by teaching student to picture the solution.  The Sparrow Math series takes this principle above and beyond into higher levels of math.