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July 10, 2015 / or4green

Gene Woolsey

I was saddened to learn in the June issue of ORMS today that Gene Woolsey passed away.  There is a poignant “In Memoriam” article about him in that issue with remembrances from former students and colleagues.  Gysbert Wessels, a former student of Woolsey’s describes the “OR/MS Guild” in place under Woolsey at the Colorado School of Mines:

Gene’s approach to teaching O.R. was unique.  All students sat with him, an open plan without any partitions.  As the number of students increased, he had a classroom converted into his office.  He taught us not only in classes and by sharing real-life incidents, but just sitting in his office with him exposed us to us dealing with clients, university politics, editing journals, TIMS and ORSA politics, etc.

Wessels closes the article with this statement:

Gene was a teacher and mentor.  He thought for himself and taught his students to think.  This meant questioning, consulting the original sources, considering the facts and drawing your own conclusions.  He made a huge contribution to operations research and management science.  He will be missed by his family, friends and students.

Although I never had the good fortune to meet Gene Woolsey, I was influenced by his writings.  In 2008, I stumbled upon a copy of the book The Woolsey Papers in our Coast Guard Academy Library.  woolsey papers coverThe lessons in it, particularly the emphasis on practicality, resonated with me.  I learned a lot from it that I have tried to apply since then in teaching undergrad O.R. courses, advising capstone projects, and in consulting.   See the greenOR posts:

Perhaps my favorite quote, which gets at Gene Woolsey’s dedication to service is this one:

In short, I worked for free, so I could work for money, with some hope of gain, so I could afford to choose which pro bono project would be the most fun to do next… This is the tip of the iceberg of what we have done for ourselves and for our state and community. What are you doing for yours?

November 30, 2014 / or4green

Bits and Pieces

Here are a few interesting bits and pieces I have come across recently:

  • Gestión de OperacionesCheck out the “Gestión de Operaciones” website.  It is a blog on operations  management and operations research in Spanish, written by Francisco Yuraszeck, professor in Operational Research at the Universidad Santa Maria in Viña del Mar, Chile.  The purpose of the site is to expose students from Spanish speaking countries to the basics and most important topics in these fields.
  • A while back I posted a bit about “The Toaster Project” in which Thomas Thwaites attempted to build a toaster from scratch … literally.  He went around England’s abandoned mines to extract the metal ore, and so on.  The project is now completed and Thwaites has a book about it.  See the website.  He was on the Colbert Report to talk about it.  Thwaites explains how a conventional toaster consists of  “400 bits”.  The exchange with Colbert was very funny.   Colbert:  “He even hunted his own bread.” and “Two questions:  Why?  … and … Why?”
  • On this blog I have often harped about the importance of repair of products in sustainability.  I have been doing a lot more repair myself recently and have found the resources on the web are very helpful.  YouTube has a huge set of videos on repairing all kinds of items.  And there is the ifixit site, a big proponent of repair.  Their “Repair Jobs Revolution” page is right on the mark:  “It’s Time for a Repair Jobs Revolution: Fostering repair will give people access to affordable products, make a huge dent in the e-waste problem, and create jobs.”  The .com side of ifixit has repair guides, especially for computers, phones and tablets, and parts and tools for sale.
  • The journal OMEGA has a Special Issue on “New Research Frontiers in Sustainability”.  The dealine for submission is December 30, 2014.   Omega has come up a lot on this blog before.  Here is an excerpt from the call for papers:

    The aim of this special issue is to publish state-of-the-art research papers which address sustainability problems and challenges on the interface between the three TBL dimensions (profit, people, and planet). Analytical models, empirical studies, and case-based studies are all welcomed as long as an article provides new insights and implications to the practice of management science concerning sustainability.


October 21, 2014 / or4green

Upcoming Talk on Carbon Emissions Game

The Carbon Game

The Carbon Game

On October 30, I will be giving a talk on a carbon emissions game I developed with Bob Day from the University of Connecticut.  I have run the game in numerous classes at the Coast Guard Academy and it has also been run by a professor at West Point.  Bob and I are working on a paper about it and as that wraps up, I hope to post more about it here over time including the rules and materials needed to run the game.

Here is more information about the talk:

Title:  On Decision Making in a Carbon Emissions Game


This talk describes a carbon emissions game we developed as an educational tool to be played in a classroom setting.  The game is modeled on real-world electric power markets with a CO2 emissions cap, often referred to as a “cap-and-trade” system.  Players take on the roles of competing utilities, each with a fossil fuel plant and a renewable energy plant. Any electricity generated by fossil fuel requires offsetting carbon credits, which are available via auction. Depending on the auction price, it can either be cheaper or more expensive to generate electricity with fossil fuel versus renewables.  The goals of the game are: for students to become acquainted with cap-and-trade markets; to understand how regulatory policy applied to a market can induce environmental benefits; to discover how to derive strategies mathematically; to get a taste of basic game theory, auctions, and the newsvendor problem as they apply to the game; and to meet these objectives in a fun setting.  While fairly straightforward and simplified to provide symmetry among student competitors, the game can lead to surprisingly interesting results.    This talk describes the game and how to facilitate it, presents strategies, analyzes games that were played, introducing novel quantitative measures to evaluate the quality of game play, and offers a number of suggestions for extending the game. Our hope is that other teachers and researchers will implement the game themselves, taking it in new and exciting directions.

Talk Info:

  • What:  US Coast Guard Academy Department of Mathematics Colloquium
  • When:  Thursday, October 30th 2014, 1:15 – 2: 15 PM.
  • Where:  Room S135, Satterlee Hall, US Coast Guard Academy, New London, CT.  Note you need a picture ID (e.g., driver’s license) to enter the campus.
  • Extra fun:  As part of the talk, I plan to run a small version of the game so attendees can play in order to better experience what the game is all about.
September 28, 2014 / or4green

Battery Recycling

I recycled this image from an earlier greenOR post.

I recycled this image from an earlier greenOR post.

It is nice to see a battery recycling program in the works in my state of Connecticut. I learned of this from a recent article in the Danbury (CT) Republican American by Paul Singley:

Connecticut environmental officials are working with battery manufacturers to come up with a bill that will be introduced in the 2015 legislative session. Connecticut and the battery industry hope it will be a model for the rest of the country.

The law would cover single-use alkaline batteries as well as rechargeable ones.

“This increases recycling, it removes items from the waste stream and it creates jobs around recovery of the material,” said Tom Metzner, environmental analyst for the state Department of Energy and Environmental Protection

The bill was developed by the CT Dept. of Energy and Environmental Protection (DEEP), and the Product Stewardship Institute (PSI) in conjunction with battery companies such as Energizer, Duracell, and Panasonic

Scott Cassel, chief executive officer of the Product Stewardship Institute had this to say in the article:

“The manufacturers understand that they have a responsibility to step up to the plate and not only sell the product, but they also have a responsibility to manage it properly.”

Somehow I missed this group when I was putting together my stroller recycling post but they are on my radar now.

Battery recycling has come up in the O.R. world before. In one of the first posts on this blog, I summarized a 2003 Interfaces paper on battery recycling.  Here are some excerpts of that:

Paper Summary: Closed-Loop Supply Chains for Spent Batteries by F. Schultmann, B. Engels, and O. Renz, Interfaces, November-December 2003

Germany has a “battery decree” that puts the responsibility on manufacturers and importers of collecting, sorting, and, if possible, reprocessing spent batteries. This paper focuses on portable batteries, as opposed to car batteries, industrial batteries, etc. The paper consists of two parts:
1. a flow-sheeting process model to study the impact of using spent batteries in the production of steel
2. a two-stage facility-location problem to optimize the reverse logistics for the spent batteries

The reverse logistics work may be useful for the new CT recovery law.  In all it sounds like a development in the right direction.

August 28, 2014 / or4green

How to Recycle a Stroller?

How do you recycle an entire stroller?  Here I describe how I went about it, inspired by concepts in The Upcycle and Cradle to Cradle.  We had been storing some old strollers and car seats we no longer used in a shed in our yard.  Some were slightly broken.  The shed roof had been leaking and so some of the items were further damaged.  In the process of fixing the shed, the time came to get rid of the strollers.  The water damage had resulted in mold on the stroller fabric.  So the strollers could not be readily reused by another family.

stroller frames

I looked into recycling programs in my city and state, but strollers were not on the list.  I contacted the manufacturers of the strollers (and car seats) – Graco, Kohlcraft, MacLaren, Safety 1st, and Babies R Us.  I let them know we had gotten good use out of the product but that in its current state it was not fit for immediate reuse. With the fabric removed, the frames themselves were still in fairly good condition and pretty clean.  I asked for recommendations for disposal, and in particular asked, “Do you have a product take-back program whereby you can reuse the stroller’s materials (see for example” (More about that link later.) I added that I wanted to avoid disposing of these products in the trash since they would then be incinerated releasing harmful emissions. And that would also be a waste of the materials (such as metal) that required large amounts of energy to extract and fashion into parts for the stroller.

Here are the responses I received:

  • Company #1:  “While we certainly appreciate your concern for the environment, we’re sorry, as a manufacturer, we do not have a take back program.
    A few suggestions would be:
    1. Check your phone directory to see if there are any recycle companies in your area
    2. Check with local retailers (such as Babies R Us), some of the retailers do a “trade in” type program where you can take used baby/children’s products. The store recycles the product and gives you a discount on a new product.”
  • Company #2:  “Ian- thanks for being so environmentally conscience![sic] However we don’t have a program like that. I would suggest that you gut the white law tags and disable the stroller- take wheels off- and then recycle the frame”
  • Company #3:  ” At this time, we cannot provide any suggestions on how to dispose of the stroller. You may want to contact local charities in your area or your city government about options you may have to dispose of the stroller.”
  • Company #4:  While we really appreciate your concern for proper recycling of  your product, we do not have any recommendations for the strollers. We know that our car seats are #5 plastic, but, unfortunately, we do not have any information regarding our strollers.  However, you may check with your local waste management facility to see what they would advise.”  

I looked into the retailer trade-in program Company #1 suggested.  Babies R Us held one a while ago, but nothing at this time.  Company #2’s suggestion was closest to what I ended up doing.  The responses were prompt and courteous, and really not surprising.  Still, I was disappointed not one of these five companies had a more substantive recommendation.

screws, springs, etc. salvaged from strollers

screws, springs, etc. salvaged from strollers

After some more digging, the only promising lead I turned up was that strollers were included on a list of items accepted at a nearby scrap metal yard.  With the Cradle to Cradle concept of breaking products up into their component technical and biological nutrients at the end of life in mind, I began to disassemble the strollers.  They are basically made of four materials:  fabric, plastic, metal and rubber.  The fabric was easy enough to remove by unsnapping buttons and removing screws.  While it would have been nice to clean off the mold and other debris, given the effort required for such a small amount of material, I decided to trash it.  I did hang on to some of the straps, thinking they might be useful for something else later on.

old stroller wheels

old stroller wheels


Next the wheels came off through some unscrewing, prying and/or hammering.  They consist of plastic and/or rubber.  For now they are boxed up, waiting to be used for some future project … not sure what that will be, but seems like they could come in handy.  Again, the alternative is incineration.

This resulted in wheel-less frames of metal and plastic.  My plan was to separate these two materials, try to recycle what plastic I could, and bring the metal to a scrap yard.  This step turned out to be the most difficult.  The plastic and metal are attached to each other by rivets.  Rivets are used to connect materials you do not want to become separated.  Think airplane fuselage.    This makes a lot of sense for a baby stroller – you don’t want it coming apart while walking junior to the park.  One way to remove a rivet is by drilling through its center.  Here was my reference video –  While this did work it took a while and I went through drill bits quickly.  I got a lot of the parts separated, but in the end left some of the rivets in.  I’m pretty sure a professional could get them out quickly.  From the standpoint of disassembly into components, a stroller does not rate too poorly in my opinion.  It is fairly straightforward to break it up through a sequence of simple mechanical operations.

metal stroller frames

metal stroller frames

Ideally, the companies would have taken back the stroller frames, checked them out, reused them if possible, and broken them apart if not, reusing what components could be reused.  I’m guessing it’s cheaper for them to just manufacture new ones overseas than employ people domestically to work on take-backs.  That’s unfortunate.

Back to my project, most of the plastic and metal was now separated.  Some of the plastic was recyclable.  Some other pieces seemed like they might come in handy for some other project down the road and so I saved them.  And I also had a pile of plastic fitting neither category that I threw out.

Next I loaded up the remaining metal, threw in some old broken beach umbrellas with the fabric removed and a few other random scraps of metal that had been lying around and headed to the scrap yard.  It was an interesting place with large piles of all different kinds of material.  I was directed to the light metals dumping area, with my car being weighed before and after dumping.  My total was around 80 pounds, for which I received $6.34.

scrap metal pile from a distance

scrap metal pile from a distance

In all, it was interesting to learn about take-back programs (and the lack thereof in this case), the construction of strollers, how to remove rivets, where the local scrap yard is and what it accepts.  That and avoiding the incineration for most (at least by weight) of the material were the benefits to me.  On the one hand, this process is not what the Cradle to Cradle camp has in mind for our future.  The stroller was not upcycled, in fact, some of the parts will necessarily be downcycled (plastic, metal).  The gap between the default end-of-life choice (“throwing it out”) and the cradle to cradle ideal is wide and we are not moving quickly enough to bridge it.  Most people are not going to spend the time to do this.  This is why it is important that take-back programs exist.  Best Buy takes back electronics every day.  Babies R Us occasionally takes back strollers and the like, but that needs to be every day as well and by any store above a certain size selling those products, and with support from the manufacturers.

Back to the website I mentioned earlier, PPRC is the Pacific Northwest Pollution Prevention Resource Center.  Their “Product Stewardship for Manufacturers” page has information on take-back programs.  They point out:  “”Most take-back programs in the U.S. are voluntary, while legislation in many European countries require manufacturers take responsibility for waste costs incurred by products and packaging.”  It lists some take-back programs including Dell computers, Xerox copiers, and Bosch power tools.  Hopefully this will spread.

Here are more photos – sorry for the low quality (used my ancient phone to take some):

About to be weighed

scrap metal pile view looking back from my car

scrap metal pile view looking back from my car

straps salvaged from strollers
straps salvaged from strollers

Update (2/8/2015) – BabiesRUs is running a take-back event ending 2/21/2015 accepting these items:

  • car seat
  • stroller
  • travel system
  • high chair
  • play yard
  • bassinet
  • infant swing
  • bouncer
  • walker
  • entertainer
  • crib
  • toddler/twin bed

See this link.


June 26, 2014 / or4green

The Upcycle

The UpcycleI recently finished reading The Upcycle by William McDonough and Michael Braungart.  It is the follow-up to their inspiring 2002 book Cradle to Cradle, which I wrote about in one of the first posts on this blog in 2007.

The Upcycle extends some of the key concepts from Cradle to Cradle such as its critical take on recycling as “downcycling”, the elegant notion of designing materials to be easily separated back into “biological nutrients” and “technical nutrients”, and more.  It takes the philosophy of the earlier book about solving sustainability challenges through design further and deeper.  And it also delves into new areas especially energy usage.  The authors reject a common tenet of many sustainability thinkers that we must consume less, focus on polluting less, and generally veer towards a more ascetic lifestyle.  McDonough and Braungart write optimistically about utilizing the plentiful energy from the sun to meet the needs as well as the nice-to-haves (the new housing complex, the long hot shower, etc.) of the current and future generations, and do so in a way that not only does not hurt the planet, but that benefits it.

Below are a number of interesting bits from the book.  I’ve roughly categorized them as “problem”, “solutions”, “philosophy”, and “follow-ups”.


  • Burning fossil fuels moves valuable carbon from the ground into the atmosphere.  There it is effectively out of reach and it is hard to get it to cycle back to land and sea at its natural rate.  “It would be like sending gold dust into the sky, or diamonds,” they say.  p38
  • On the large and costly ($1 billion/year) regulatory structure in place for nuclear power:  “Regulation is an indicator of the need to redesign.”  p97


  • “Imagine a building that uses renewable energy for its operations and to offset the energy used to make the building, contributes extra energy to the grid!”  p69
  • A nice connection to multiple-objective optimization:  a company called Steelcase, wanted to swap out PVC (polyvinyl chloride) for something less toxic in the production of their “Think Chair”.  Thermoplastic polyurethane (TPU) is an option but has greater embodied energy.  Embodied energy measures the total lifecycle energy associated with a material from production to transportation to usage through to disposal.  Steelcase made the call that toxicity was of greater concern.  The logic there is that it is easier to make energy renewable than to detoxify what has been polluted.  (This adds an interesting perspective to the incandescent versus CFL light bulb debate, to be made moot by LEDs in the near future.)    p75
  • Thanks to large supplies of geothermal and hyrdo energy, Iceland has the potential to export renewable energy.  But transmission is a problem.  Instead the energy could be embodied in materials.  Materials function as batteries (effectively, not literally).  In other words, use that excess energy to produce materials that are then efficiently exported in a way that energy could not be.  Iceland went with this concept applied to producing aluminum.  Unfortunately there ended up being some sustainability issues in that, such as the construction of a huge dam and others. p101
  • Chapter 4 “Soil not Oil” closes with a very interesting discussion on converting wind energy to light to grow produce in greenhouses.  The greenhouses can be anywhere (on roofs, underground, etc.) since they are relying on artificial light.  The LED lights would be tailored to the light frequency requirements of the particular fruits and vegetables.  The system can take advantage of wind gusts occurring in the middle of the night, a time when there is generally not much demand for energy.  Roughly, I interpret the idea like this – in nature, if you rely on rain, it can be sporadic.  But it is normally useful when it does come.  If it is windy but irregularly so, and that wind translates to light, then the light comes and goes like the rain.  As long as there is enough light (as there also needs to be enough rain), then the plant will grow.  If not, additional energy can be added to the system.   p142
  • IBM Building atrium by Matthew Bisanz via wikipedia

    IBM Building atrium by Matthew Bisanz via wikipedia

    When I lived in Manhattan several years ago, I would sometimes spend time in public spaces of huge buildings like the Sony, IBM, and Citicorp buildings in midtown.  I enjoyed these spaces and at first assumed the companies altruistically offered them to the public.  In fact, they are usually part of a deal between the company and city, provided in exchange for certain building rights (e.g. the right to build higher).  McDonough and Braungart refer to this in Chapter 5:  “It is not uncommon for real estate developers to be asked to provide public benefits in exchange for permission to develop building projects in sought-after locations.”  The authors put a different spin on this idea.  Instead of providing public space, the builders might be asked to provide energy, in a manner of speaking.  They give an example of a developer building 400 new apartments in Beijing and having to “mine the old city for energy” in return.  So instead of a new building having to build a park, the builders would have to upgrade aging inefficient energy generation in existing buildings.  And ideally the overall outcome is that more people are accommodated in housing using less total energy – the new building is highly efficient while the old ones are made significantly more energy efficient than they were. p174

  • “The US Postal Service committed to understanding every single material in its supply chain and to phasing out, right away, the highest priority substances – namely, anything teratogenic, mutagenic, or carcinogenic… advancing through a progression of inventory, assessment, and optimization…”  P192


  • “Crade to Cradle designers and manufacturers know that they are engaged in what Buddhists call “right livelihood”, a way of making a living within the framework of right behavior that allows them to happily present themselves to their children.” p20.
  • There is the story of McDonough’s childhood gardener in Japan, Oji-san, who cut the grass by hand with a sickle.  Oji-san’s face fell when he was brought a push mower.  “… the silence and the careful, constant tending are part of what makes Japanese gardens authentic.  Efficient, mechanical, quick mowing would change all of that and make Oji-san less valuable and necessary.  …  He practiced what the Japanese call shokunin – social obligation, physical, mental, and spiritual – to care for this beautiful place.”  This reminded me a little of the least popular of the suggestions I entered into the GreenGov Challenge in 2009 about raking leaves instead of using leaf blowers, etc.
  • There are many implicit OR connections in a book like this.  These are inherent in sustainability because of its competing objectives.  I was surprised by how many times the word “optimize” was used.  I counted at least ten but I am sure it was many more than that.  Here are a few examples:
    • On how people set their thermostats, p64: “There is no optimization occurring in terms of fuel use or anything else.”
    •  “Design could optimize systems and then let human hands have the control.”  This is connected to material about a NASA building and climate control within, people can open windows, etc … p155.
    • “We think always of the upcycle: Optimize materials or their ingredients.  Optimize product pathways.  Optimize nutrient management.”
    • About carpet, p163:  “A more optimal solution exists.”  Oops – “more optimal” doesn’t really make sense.  Reminds me of the O.R. talk (Jeff Camm?) in which the speaker mocked the Kia Optima, a plural vehicle.
    • The book ends with 10 points to take away, #5 of which is “Optimize, Optimize, Optimize” (p214).    “It doesn’t make us happy to see your downward-sloping lines of fewer carbon emissions, fewer toxins.  We want to see your rising lines of positive aspirations and beneficial commitments.”  In a way they are not really talking about optimization here, but about optimism versus pessimism.  Rather than reduce harmful emissions, let us increase beneficial ones, they say.  But on the other hand, there is the sense that a constraint or goal (meaning a “goal programming” type goal) is negatively or at least narrowly defined while an objective can have a more positive, wide open feel.  While constraining the bad stuff to be below some level is good, it is somehow limited in imagination.  Maximizing benefits on the other hand is more open-ended, has greater possibilities, or at least can feel that way.
  • “The most effective transformational foundation of Cradle to Cradle is, to the surprise of some, not environmental.  Nor is it ethical.  It is economic.  If Cradle to Cradle fails as a business concept and innovation engine, then it fails, period.  … It succeeds when it celebrates economic growth … when it upcycles the economy, and ecological and ethical benefits accrue.”  p189

Follow-ups (just a few of the many mentioned in the book, there are four pages of end notes)

  • Albert Howard’s An Agricultural Testament (1940) … led to beginnings of organic farming, composting, …
  • Kurosowa’s Dersu Uzala (film)  I watched it as a result of reading about it in The Upccyle.  I would highly recommend it.
  • – Cradle to Cradle Products Innovation Institute.  This website and organization accompanies the book.  It has a decent lookup tool for c2c products (I found some insulation there) and discussion forums.

All told, the book is filled with ideas and examples that feel like they could be part of a sustainable future, ideas that will one day be commonplace, though they may seem far-fetched today.  Like Cradle to Cradle it is an inspiring read.  I will follow up soon with a post about some action I took as a result of reading it.

March 20, 2014 / or4green

Tournaments and Logs

With the NCAA basketball tournaments kicking off, here is a post about an exercise I put together for our pre-calculus class last semester on tournaments and logs.  I find a lot of students do not have a good intuitive grasp of the logarithm function and why it might be useful.  This exercise actually originated when my son was around 5 and began racing his toy cars.  (See the video above – it was hard to determine the winner by eye so we filmed it and ran the finish in slow motion; you can freeze-frame it at around 0:02 to see who won.)  He started collecting matchbox cars and when he got to 16 of them, I suggested he do a tournament.  This kept him busy for a while.  A few days later he came back to me, puzzled, saying “How do I do that thing with the cars if I have 23 of them?”  So now we had to enter the world of byes or play-in rounds.  This led to following “workshop” I ran for my pre-calc students.  We had just covered exponential and logarithmic functions.  I paraphrase the assignment below with some explanation.  For the actual handout I used, click here.

Workshop:  Set up a Tournament using Logarithms

Assume a single-elimination tournament with winners advancing to another contest while losers are eliminated, repeating until only one team is left.


the number of competitors, n

Devise a procedure to indicate:

  1. the number of rounds in the tournament, r
  2. the number of byes in the first round, b

As to what meant by “procedure”, I gave some direction, but not a lot, saying it should consist of short, simple steps mostly involving mathematical operations.  Students are certainly used to following procedures, methods, algorithms, etc. but seemed a lot less comfortable coming up with one themselves.

I split my students into groups of around 3 to work on this.  It didn’t take long for them to figure out that for 16 teams there are 4 rounds, for 32 teams there are 5 rounds, for 64 teams there are 6 rounds, etc.  And so in general 2r = n as long as n is a power of 2.  If is not a power of 2, they realized that some kind of byes are necessary.  I provided an example of this with 5 teams on the handout for the exercise.  Some groups started to realize the importance of the power-of-2 numbers near the number of competitors.

Using these ideas, one group emerged with the following step in their procedure:   “Round n up to the nearest power of 2.”  That’s a great start, and isn’t hard if we know our powers of 2.  But what if we want to automate the process, i.e. use a computer?  For instance, what’s the command to “round up to the nearest power of 2” in Excel, Mathematica, c++, etc.?

With a little encouragement, some groups began to consider using the function log2x in their procedures.  It dawned on them that they could solve for the number of rounds in their formula 2r = n by taking logs:  log2n = r.   In other words, the number of rounds in this kind of tournament is equal to the log2 of the number of teams, assuming n is a power of 2.  If not, one can round up by taking the ceiling of log2n (i.e. let r = ceil(log2n)) and that gives the total number of rounds with some teams receiving first round byes.

How many byes are there?  To figure this out, note that after rounding upwards there are 2r slots for n teams.  Teams with byes effectively take up 2 slots in the first round, # of non-bye teams is n – b, so totaling the slots we have 2b + (n – b) = 2r, and solving for b yields b = 2r – n.

Putting it all together, here is one procedure that will work:

  1.  y := log2n
  2.  r := max (y, ceil(y))      ( = y if y is an integer,  = y rounded up if not )
  3.  b := 2rn

Some of the student groups made it this far in the 50 minute class period. So that was good.  Others struggled with the whole exercise.


There are a lot of variations on this kind of workshop.  Here are a few ideas:

  • Cut-throat tournaments in which 3 competitors vie in a contest, instead of 2 (e.g. “21” basketball game, cut-throat racquetball, …).
  • Simple exponential population models to illustrate inverse function relationship of the exponential function and logarithmic function.
    • For instance, a cellular organism divides into 2 every 15 minutes.  Say we start with 34.  A little later we have 272  (or maybe 261 – what other assumptions might we make there?).  How long did that take?
    • You can solve the previous problem without using logs by doubling.  But the doubling method doesn’t scale well.  Consider this variation:  Start with just 2 organisms.  Suppose a bit later on, there are 68,720,000,000.  How much time has passed?   Do you really want to double 2 up to 68 billion?  (Answers at the end of this post.)


The NCAA men’s basketball tournament now consists of 68 teams. So the calculation goes like this:

  1. y := log2(68) ~= 6.09
  2. r: = max(y,ceil(y)) = max(6.09,7) = 7    — so 7 rounds
  3. b: = 2rn = 27 – 68 = 128 – 68 = 60  — so 60 byes

The other way of looking at it is 68 – 60 = 8 teams “play in”, the so-called “First Four”.  These just happened – Cal Poly, Tennessee, Illinois, and Albany won, so now we’re at a power of 2 number of teams from here on out.

Hopefully this kind of exercise can help build intuition about log functions.  If you double something up to a higher number, log tells you how many doublings there were.  Start with one winner from a contest of two, and those two were in turn each winners from contests of two, and so. Doubling your way up gives you the number of times you’ve doubled, which is the number of rounds, and is given by the log2 of the number of teams you end up with.

The handout for this workshop is available here.  Feel free to use it and let me know if it ends up being useful.

I think the main thing my 5-year old took away from this was that there were these “special numbers” that made setting up the tournament simple.  So his plan was to work up to 64 matchbox cars, then maybe even 128.

Answers to exponential population questions:  45 minutes; 9 hours