A different justification for the Schulze Method

This is a different way to think about the Schulze Method, which might convince people who don't like the "beatpath" construction. 

I wrote about the Schulze method in the previous post, giving a justification that is based on the standard presentation of the method using beatpaths.

This is a different look at the same method, implemented a different way.
Suppose you first looked for a Condorcet winner: so you look at all the pairwise matchups, and find out who is preferred to who.  You make a matrix of pairwise preferences, and maybe for convenience subtract the number of voters who preferred Y to X from the number who preferred X to Y to get the margin of pairwise preferences.  (These matrices are in the example in the Wikipedia link to the Schulze method above -- if I were more detail-oriented, I'd put an example here.) 

Any row that's all positive means that candidate beats all others head-to-head: that's a Condorcet winner!  

But suppose there isn't a row that's all positive.  What do we do?  Well, first, we can eliminate any Condorcet loser: a candidate that is not preferred in any head-to-head matchup. (We'll spot a Condorcet loser in the matrix because it's got a row of margins that's all negative, or equivalently a column that's all positive.)[see footnote]

Then, since we want the closest thing to a Condorcet winner, let's "grade on a curve", and do one of the following (these are equivalent):

• add something to all the margins: add 5 (for example) to all the margins of pairwise preferences  (add more if you haven't changed anything from negative to zero/positive, add less if you've changed too many elements) OR
• find the negative margin that's closest to zero (-3 is closer than -10) and set it to zero.

Is there a row with no negatives now?  If so, that's our winner.  If not, our curve might've produced a definite loser (with a column that's all positive or zero), so eliminate that candidate [see footnote again] and curve some more, either adding more or zeroing out another negative margin, until you've got a row with no negatives.  It isn't a Condorcet winner, but it's the next best thing -- the candidate that started with the smallest pairwise "unpreferences".

[ Here's the footnote: ] technically, you want to eliminate every candidate not in the "Schwarz set", which is not just a Condorcet loser, but any group of candidates that loses to everyone outside the set.  (In other words, it doesn't matter if candidate D beats candidate E, if D and E both lose to A, B, and C, eliminate both of them - clearly the winner should be A, B, or C.)   But in terms of justification, this is a technical point; the idea is that you're looking for the "closest to Condorcet" winner.

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Voting Methods: Making the case for the Schulze Method

If you're curious about voting methods, and/or want to make a case for a Condorcet method to your friends and co-voters, this post is for you!  

First, I think the Condorcet criterion is pretty compelling on its own.  (The Condorcet criterion is: if there's one candidate that is preferred in head-to-head matchups to every other candidate, that candidate wins.)  It's easy to come up with examples where the plurality method (this is the method that gets applied most often - whoever gets the most votes wins, even if that isn't more than half the votes -- boo plurality!) doesn't give you what seems the right solution.  If there is a Condorcet winner (there isn't always), it's the one that seems like the right answer.

So what if there isn't a Condorcet winner?  What voting method do you pick?  This is a case for the Schulze method: if there isn't a Condorcet winner, that's because there's a "rock-paper-scissors" cycle where candidate A is preferred (head-to-head) to candidate B, B is preferred to C, ... , Y is preferred to X, and X is preferred to A.  (The cycle can be size 3, like rock-paper-scissors, or can be longer - the point is it circles back on itself.)

At first glance, a cycle like that seems intractable - how do we rank any of the candidates in the cycle higher than the other?  But this is ignoring the strength of the preference:

Suppose 90% of voters prefer Rock to Scissors, 85% prefer Scissors to Paper, and 51% prefer Paper to Rock.  The cycle is there, but clearly the Paper > Rock preference is the weakest link.  The preference path Rock > Scissors > Paper has a "strength" of 85%, much higher than the 51% for Paper > Rock.  Based on path "strength", we can rank these Rock>Scissors>Paper.

Is Rock a Condorcet winner?  No.  But it's the closest thing to a Condorcet winner, in the sense that its pairwise loss is the weakest.

That's the Schulze method.  It's written in terms of "beat paths", but those are just breaking a cycle into two parts, like we did splitting "Rock>Scissors>Paper>Rock" into "Rock>Scissors>Paper" and "Paper>Rock" and comparing them.  There's some added detail to deal with multiple cycles between the same candidates, but that's really all there is to the idea of the method.

Next I hope to show another case to be made for the Schulze method, which might be more intuitive.

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Learning Outcomes for Outlast

Here is some feedback from the originator of the game idea "teach environmental sustainability through a game with zombies":

What I was thinking about were both reasoning/critical thinking skills and environmental sustainability.  What do you do when the first plan doesn't work? What is your new goal? What information do you need? What is your strategy? I thought the zombies would make it cool and relevant and more exciting, but we could also have used a colony on the moon. Ok, they would have needed a lot more background knowledge! 

Game design: Game Dojo stream pre-report

Hello!  There's a Game Dojo stream tonight where I'll talk with Sen about Outlast.

Here's a brief follow-up of what has happened from the earlier Game Dojo conversation.  The items to consider in the short term were:

• consider permanent resources (that are location features: shelter, arable land, water, e.g.) as well as temporary resources: I haven't really done this.  
• consider the minimal best 9 locations to have. What if these were all there are? (letting a location get overrun should be a gut-wrenching decision) This I have done, and played the game with only 9 locations (when a location is overrun, it disappears and nothing replaces it.)  This game is more challenging, of course, but definitely achieves the goal of treating locations as precious.  
• What's a playable "scenario" that can be "solved" (won) (A deterministic end goal, that is definitely achievable).  There should be a strategy that works regardless of unfortunate die rolls: I tried out a few straightforward scenarios: the farm (which makes it easier to produce food) next to the sustainable storage (which has a win condition if you have enough food), and the laboratory (win condition: medicine and gas) near the hospital and gas station.  I realized that a well-coordinated team can definitely win in either case, quickly.  (more on this)
• write a bio, then publicize it and this blog so people (you guys!) can follow the story of the game design.  Have not been good about writing a bio. :(

Here are some thoughts to hold off on for the moment, but should be considered down the road:

• Consider expanding the map to 25 "little" locations rather than 9 "big" ones. Not looked at yet.
• This game sounds like it wants to be a co-op area control game: This is sort of true, in the sense that a powerful tactic is to move as a pack, with one outrunner on a different location.  This looks a bit like area control: having 3 people in one place is better than 2 here and 1 somewhere else. (If there's a chance for food, for example, 3 searchers on the same location are liable to hit it and can all use it.)
• Pivot: what makes the players realize they need to switch strategies (from exploit to sustain, explore to hunker down, e.g.), and what makes the timing of the pivot crucial (waiting too long or going too soon has to have a cost): Running through the "scenarios", I am a bit concerned that there isn't a pivot - or at least, focusing solely on the goal from the start was a successful strategy.  This is one of the "counterintuitive" parts of the game -- players naturally start out trying to subsist, but choosing to starve from the start in pursuit of a win condition is a quicker path to victory.
• building a defensible position; using fortification or something to protect/claim some region: Hypothesized a bit about what "securing locations" might look like as a new action players could take.  I think there's definitely a plausible opportunity here.
• zombies chasing people around, rather than just showing up? Generally de-emphasizing the zombie attacks in favor of overruns? Not looked at yet.

Game design is happening!

So, I've been designing a game. The working title is Outlast, and it's a zombie game with a subtext of environmental sustainability. More to follow, including the pitch for the game, but if you're here to hear my tale of game design, welcome! The label "games" will show relevant material for you; you're welcome to look at everything, but if you _just_ want stuff about the game, click on that label. Without further ado, here's the hook:

When the zombies overrun society, every resource is precious. In Outlast, you and your friends must grab some resources (like food and guns) and maintain others (like a safe hideout.) As resources dwindle, you must strike out to discover new locations, while continuing to occupy the best sites you’ve already discovered. Locations you don’t use become inaccessible, so players have to develop an evolving “base of operations” consisting of several locations that work well together. If you succeed, you may be able to eliminate the zombies and create a safe haven for humanity. Good luck - we’re counting on you!

I've got a wonderful opportunity to work on this design in the Game Dojo, a mentorship group that Sen-Foong Lim of Meeple Syrup is leading. I had a great conversation with him, with a summary of how the game plays, available here.

 Here are some of my "to-dos" from that conversation:

• consider permanent resources (that are location features: shelter, arable land, water, e.g.) as well as temporary resources
• consider the minimal best 9 locations to have. What if these were all there are?
• What's a playable "scenario" that can be "solved" (won) (A deterministic end goal, that is definitely achievable).  There should be a strategy that works regardless of unfortunate die rolls.
• (letting a location get overrun should be a gut-wrenching decision)
• write a bio, then publicize it and this blog so people (you guys!) can follow the story of the game design.

Here are some thoughts to hold off on for the moment, but should be considered down the road:

• Consider expanding the map to 25 "little" locations rather than 9 "big" ones.
• This game sounds like it wants to be a co-op area control game
• Pivot: what makes the players realize they need to switch strategies (from exploit to sustain, explore to hunker down, e.g.), and what makes the timing of the pivot crucial (waiting too long or going to soon has to have a cost)
• building a defensible position; using fortification or something to protect/claim some region
• zombies chasing people around, rather than just showing up? Generally de-emphasizing the zombie attacks in favor of overruns?

Math as Literary Theory (or not at all)

The other day Sarah was explaining multiplication to Zoe, especially about why any number times 0 is 0, and a number times 1 is that number.  I realized in that moment that I had never really heard (or what is called "hearing") the quite logical explanation that Sarah was giving Zoe.  Somehow, when I was Zoe's age (or older?) I had instead just memorized a rule that I either created myself or someone gave me.  The rule is this:

• The zero is an infection:  it infects anything that "times" it (or something like that).  Whenever a number faces the dreaded zero, that number is completely annihilated by the black hole of the zero's (non-)power.  The zero, in other words, is a principle of contagion.

• The 1 is a mirror that reflects back whatever "times" it.  If a number confronts the 1, all the number sees is itself.  Mere reflection.

This is of course a horrible way to understand multiplication.  But it's a great entryway into literary theory, especially theories of representation.  Are we dealing with a principle of reflection or mere representation (the 1)?  Or is this a case of reflection as distortion, or rather, not reflection at all but contagion: the one representing ends up infecting what is supposed to be represented? 

It's also a good primer on ethics, or what Levinas called "the ethics of ethics."  When I face the other, am I a "one" or a "zero"?  What would be the ethical integer?  For Levinas, ethics does not take place when I assimilate the other to me (when I infect the other, when I overwhelm the other with my own qualities).  Ethics does not begin with the zero.  Rather, ethics begins when I take up the position of the "one" (1):  my own self is annihilated in my encounter with the other.  Or rather, as a 1, I have no self, and therefore am able to allow the other to be present as such.  A certain reading of Levinas would therefore say that the ethical integer is always the 1.

Of course, literary theory also likes to confuse the difference between the zero and the one: no longer simply contagion or reflection, the "mirror" becomes passageway:

I don't know what "math" would say about that one.  Probably a lot, since Looking-Glass was, for Carroll, a math problem, or at least a chess problem.

The moral of the story:  In effect back in 6th I mean 1st grade, when I was learning multiplication, I really wasn't learning anything about math, but rather ended up assimilating math to my "self," but a self that would not actually be constituted until much later ("math" reflected a self that was not yet).  Or maybe that's what literary theory is, for me anyway:  my own non-encounter with math, my own private zero.  This is why "math" always returns to me as trauma or neurosis. 

What RA Feels Like

I just made a little animated video about how my rheumatoid arthritis makes me feel. Please watch. Then, please donate.

Siblings Size-up Margins of Safety

Bob and Sarah worked together again (see the announcement of our first collaboration here) on a post over at Medical Law Perspectives: Uncleanable ERCP Duodenoscopes: Manufacturer, Hospital, and Physician Liability?  This time, Bob clarified the comparison of two margins of safety. 

In an editorial published in JAMA last October, William A. Rutala, PhD, MPH, Professor of Medicine, University of North Carolina School of Medicine, Chapel Hill; Director, North Carolina Program for Infection Control and Epidemiology; Director, Hospital Epidemiology, Occupational Health and Safety Program Hospital Epidemiology, University of North Carolina Health Care; and David J. Weber, MD, MPH, Professor of Medicine, Division of Infectious Diseases, University of North Carolina School of Medicine, Chapel Hill, stated that the margin of safety associated with cleaning and high-level disinfection of gastrointestinal endoscopes is 0-2 log10. Rutala WA, Weber DJ. Gastrointestinal endoscopes: a need to shift from disinfection to sterilization? JAMA. 2014 Oct 8;312(14):1405-6. PMID: 25291575. Rutala and Weber compared this low margin of safety to the 17 log10 margin of safety associated with cleaning and sterilization of surgical instruments. 

But I had no idea how 2 log10 compared to 17log10. Bob explained that, at best, the margin of safety of endoscope reprocessing is 15 to 17 orders of magnitude less than the margin of safety for reprocessing of surgical instruments. Not 15 to 17 times less. 15 to 17 orders of magnitude less. That is literally over a quadrillion times less. So, the margin of safety associated with the cleaning protocol for duodenoscopes recommended in instructions provided by the manufacturer (Olympus Corporation) and the FDA is over a quadrillion times less than the margin of safety for reprocessing surgical instruments.

Chicken Hunters

I mean, the chickens are hunters. I tried to get some videos of them chasing other birds off their lawn, but it was difficult to catch them in the act.  They mostly acted innocent.

A Hollywood Tail

Back on November 1st, we discovered two kittens abandoned in our yard. After a daring rescue, they were cold and flea - infested, but adorable.

Unfortunately, David is allergic to cats. So we had to find them a forever home. We emailed; tweeted; and put flyers up in the Department of English, Comparative Literature, and Linguistics, a known cat person hang out. But there were no takers.
Then our friend Bridget, a recovering cat breeder, stepped in. She fostered the remaining kitten (yes, :( DO NOT TELL OUR CHILD) and nursed it back to health and flea-freedom. Then she found it a forever home with a mom and two sons who just lost their elder cat ... in San Jose, which is over a six hour drive without traffic. In trying to find a ride for the kitten, by then named, Oliver, she contacted a friend who is an animal wrangler in Hollywood. Not only did the animal wrangler volunteer to drive Oliver to San Jose, but she lined up some work for him. 

As it is not kitten season, Oliver was able to command a four digit payday for two days of work. He was the model in a photo shoot for Shutterfly.

 

Now, Oliver has earned his initial veterinarian visit and neutering fees for his forever family and gas money for his ride up the coast. 

 

While Oliver's story has a happy ending, please remember that if you find young kittens in a relatively safe area, such as in the grass or on a sidewalk, it’s best to leave them alone. Mom is probably moving her babies or out looking for food. Check out this post from the ASPCA for more information. 

Piesgiving

So I invented a new pie this Thanksgiving. 

It uses the Salty Pluff Mud Pie filling from the New York Times wonderful dining section containing a recipe from all fifty states for Thanksgiving.  But instead of a regular pie crust, which I always find at odds with Mud Pie, I made the crust out of NYTCCC dough without the chocolate chips.  I buttered the pie plate, then covered the bottom and sides with about 1/3 inch of dough.  I popped it in the fridge overnight.  Then I baked it in a 350 degree oven for 20 minutes.

At this point, it looked like there would be no room for filling, but I popped it back into the freezer to chill until the filling was ready and it settled down a bit.  Then I followed the Salty Pluff Mud Pie recipe.

It came out of the pie plate easily and held its shape well. It was AMAZING!  The bottom perimeter of the pie, where the proportion of crust sometimes fails and ruins the whole thing, turned into this caramelized, chocolately, yum.  It was alchemy.

Before serving, I dusted the top with extra fine sea salt, which truly does bring out the flavor of the chocolate, and topped it with a dollop of vanilla ice cream.

Siblings Study Statistics

It's been pretty quiet here at Gnomicon.  But if you're wondering if we're still up to our usual shenanigans, the answer is yes.  For example, Bob and Sarah just collaborated on a post over at Medical Law Perspectives:
CDC Study Shows Skyrocketing Skin Cancer Costs; What Lawyers Can Learn By Examination and Persuasive Presentation of Statistics.
We even made illustrative bar graphs based on hypothetical data sets.  Yup, the family that critically reviews use of statistics in the public media together stays together.  Or some other catchy adage referring to the Common Core standards for high school math.

I Might Have Invented a New Cookie

What happens when you volunteer to make cookies for a fundraiser at your kid's daycare and you discover at 10 p.m. that you're 1/2 stick of butter left?  You discover a whole new cookie: peanut butter bacon cookies with banana buttercream frosting.  That title is a mouthful.  How about Elvis Cookies?  Or if the Elvis Presley estate is concerned about that, how about King Cookies?  Salty, fruity, peanuty, bacony goodness.  You don't have to like the sandwich to love this cookie.

King Cookies
Peanut Butter Bacon Cookies

• 1/2 C unsalted butter
• 1/2 C cold bacon grease
• 1 3/4 C crunchy peanut butter
• 1/2 C PB2
• 1 C white sugar
• 1 C packed brown sugar
• 2 eggs
• 2 3/4 cups all-purpose flour
• 1 t baking powder
• 1/2 t salt
• 1 1/2 t baking soda

Procedure

• Cream together butter, bacon grease, peanut butter, PB2, white sugar, and brown sugar for at least 10 minutes. 
• Beat in eggs.
  In a separate bowl, sift together flour, baking powder, baking soda, and salt. Add your dries to your wets and mix slowly until totally combined. 
• Cover and refrigerate overnight, or as long as you can stand to wait.
  When you're ready to bake your cookies, take the dough out of the fridge and let it reach room temperature. 
• Preheat the oven to 350.  
• Roll into 1 inch balls and set on baking sheets. Flatten each ball with the palm of your hand to about 5/8 inch thick.  You might have to push the edges back towards the center with your fingers if the edges break.  
• Bake for about 14 minutes or until cookies begin to brown. Do not over-bake. Take them out and allow them to cool on the cookie sheet.

Banana Buttercream Frosting

• 1/2 C mashed fresh bananas, not frozen (cold banana will make the butter harden)
  1/8 t citric acid (1/2 t lemon juice could be used as substitutes)
  1/2 t vanilla extract
• 1/2 C butter
• 3 1/4 C confectioners' sugar
• yellow gel paste food coloring (optional)

Procedure:

• Mash together the banana, citric acid, and vanilla.
• Cream the butter alone. Add half of the confectioners' sugar 1/2 C at a time, creaming it in completely after each addition.
• Add the banana mixture. Take some time to cream this in, scraping down the bowl completely a few times to make sure the butter isn't separating and stuck to the side of the bowl.
• Add the remaining confectioners' sugar 1/2 C at a time, creaming it in completely and scraping down the bowl after each addition.
• Add the gel paste food coloring until the frosting reaches a shade of yellow that will reassure nonbelievers that this is really banana-flavored.

Assembly:

• Check that the cookies are no warmer than room-temperature.  If they're at all warm, the frosting will melt and possibly slide off.
• Frost the the cookies thickly.  The volume of the frosting should equal the volume of the cookie to balance the salty/bacony powerhouse of the cookies with sweet/banana-y awesomeness.
• Twirl the frosting in the middle to make a little pompadour. Get it?

Grateful Guinea Pigs

My friend Tristan Dalley designed this amazing logo for our 2014 Walk to Cure Arthritis team, the Grateful Guinea Pigs. The team name comes from what medical research test subjects are nicknamed: guinea pigs. I have a kind of rheumatoid arthritis that doesn't respond to the usual medications. So I get to be a guinea pig for orphan drugs. Orphan drugs are developed through an FDA program to fund research and development of drugs and biologics intended for the safe and effective treatment, diagnosis or prevention of rare diseases that affect fewer than 200,000 people in the U.S., or that affect more than 200,000 persons but are not expected to recover the costs of developing and marketing a treatment drug. The Arthritis Foundation works with the FDA's Office of Orphan Products Development to lobby for funding and research for rare forms of arthritis. Being on the cutting edge of medicine is scary sometimes, but I am a grateful guinea pig.

If you'd like to donate to the Arthritis Foundation through our team's fundraising website, please go to http://ocwalktocurearthritis.kintera.org/gratefulguineapigs

If you'd like to purchase some Grateful Guinea Pigs merchandise, 10% of the price will go to the Arthritis Foundation.  

Zoe's European Shoes

Zoe got new sneakers (with 'Z's), for her European walkabout (3-year old version). She is currently breaking them in by holding them in the air.  Like so:

Homemade Cottage Cheese

 

After an unsuccessful attempt at making homemade cottage cheese, I achieved delicious curds. The difference? I used the suggested calcium chloride. The results was a nice break in the curd. You can see how well the curds stayed together once they were cut and being reheated. I didn't stir the curds, but I did swirl the pot to help distribute the heat. I followed the directions in Ricki's Basic Cheese Making Kit from the New England Cheese Making Supply Company. The only thing I also wasn't too concerned about was only raising the cheese 2 degrees every 5 minutes to get to the required 110 degrees.

 

After draining the whey from the curd, I had a big solid mass. Once crumbled, it looked more like cottage cheese, but didn't have the smooth curds like store bought cottage cheese. I also did mix some cream into the cottage cheese at the end to moisten it.

 

I feel ready to tackle feta next!

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Bug!

 


I saw this bug on my tire this morning. It was about the size of my thumb, so around 2 1/2 inches long by 1 inch wide. That's what I call a big bug.

Homemade Sand/Water Table

To keep the Wee Bairn entertained this summer in our Wee Backyard, I cobbled together a sand and water table out of an old planter and some bins from IKEA.

Now I've got a space between the two bins that won't fit another bin, but looks a little sad.

Any thoughts on what should go there? The gap is about 14 1/2 inches deep and 10 7/8 inches wide.

Awesome Shortcake

I have no pictures to share because we ate these up so fast, but if you have strawberries, they are worth making. Recipe for Shortcake Biscuits 2 cups AP flour 2 T sugar 1 T baking powder 1/2 t salt 8 T cold unsalted butter, cut into small pieces 1 cup heavy cream Stir dry ingredients together in a large bowl. Cut butter into flour mixture until butter is pea sized. Add cream and mix until moist. Kneed gently. Roll out until about 1 inch thick. Cut into biscuits. Bake at 400 degrees for 12-15 minutes. Serve with strawberries and whipped cream. Delicious.

Challah

Back in September of 2008 I read an article in the New York Times Magazine's Recipe Redux column about challah. Four years later, I finally made it. Not the 2008 version, which, much like the article, is offensive on a lot of levels. I made the 1976 version. Well, I made my food processor version of it. In fact, I've made it about seven times in the past three months, and I think I've perfected it.

The original recipe from Sarah Schecht of Brooklyn appeared in an article by Craig Claiborne in 1976. I’ve divided it in half, given food processor directions, and braided like a sane person instead of in an eight-strand braid. However, the pictures you see in this post are actually of a double batch following the instructions below (literally, I made it once, then made it again . . . otherwise it doesn't fit in the food processor, or in any bowl I own to rise) and then following Ms. Schecht's instructions for braiding, which even I could follow.

I highly recommend using Penzeys Ceylon cinnamon, which has a heavenly, ethereal taste, unlike the heavy ground bark you get from your average grocery store cinnamon. I used Penzeys cinnamon blend this last time, and while it is leaps and bounds better than grocery store cinnamon (which is probably stale China cinnamon), it might have resulted in challah too breakfast-y to use for lunch sandwiches.

If you are planning to use your challah for sandwiches, allow the second rise (the one after you braid it) to continue a little longer. On cold days, I let it rise more than an hour. The challah pictured here rose for two hours and 15 minutes in a kitchen with an ambient temperature around 71 degrees. The reason traditional challah might fall apart when you cut it for sandwiches is because it is supposed to be easy to pull apart to share on Shabbat, like a loaf of dinner rolls.

2012: Challah

• 4 1/2 C unbleached flour, plus additional flour for kneading
  
• 1 t dry active yeast
  
• 1/4 t baking powder
  
• 1/4 t cinnamon
  
• 1 1/2 t salt
  
• 1/4 C plus 2 T plus 1/8 t sugar (save the 1/8 for the egg wash)
  
• 1/2 t vanilla extract
  
• 3 large eggs (save one for the egg wash)
  
• 1/4 C plus 2 T canola oil
  
• 3/4 C lukewarm water

Procedure:

• Place the flour, yeast, baking powder, cinnamon, salt, and sugar in the food processor bowl. Pulse to combine.
  
• Add the vanilla, 2 of the eggs, and the oil and run on bread setting for 20 seconds. Add 3/4 c lukewarm water and run again for 20 seconds. The dough is ready when it doesn’t stick to your hands. Transfer to an oiled bowl, cover the bowl with plastic wrap, then lay a clean towel over that, and let rise for at least an hour.
  
• Turn the dough onto a flat surface and knead briefly. Cut off 1/3 of the dough, knead quickly, shape into a ball, flour lightly and let rest for 15 minutes. Repeat with remaining 2 pieces. Using your hands, roll each piece into a 12-to-15-inch-long rope. Continue with remaining balls.
  
• On a baking sheet lined with a Silpat non-stick baking sheet liner (or analog), align the ropes, side by side. Gather the tops together, one at a time, pinching down to seal. Braid them. When braided, gather the bottom ends of the ropes and pinch them together.
  
• Cover the loaf with a towel and place in a warm spot until the loaf is doubled in size, about an hour. Preheat the oven to 325 degrees.
  
• Beat the remaining egg with the remaining 1/8 t sugar. Brush the loaf with the egg wash. Bake until puffed and golden, about 45 minutes.

Makes 1 loaf.

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