Tick marks like these are the same technique used to recount elections.

On Aug. 7, 2012, ballots were cast in primary elections – to finalize the ballot choices for voters in the Nov. 6 general election.

And  last week, on Sept. 4, 2012, the Washtenaw County board of canvassers conducted a recount of some of those ballots. Several different races were recounted, including some from Augusta Township, Sylvan Township, the city of Ypsilanti, and the city of Ann Arbor.

In the city of Ann Arbor, it was the Ward 4 city council contest that was recounted. That race had offered a choice for voters between incumbent Margie Teall and Jack Eaton.

The initial count of ballots across Ward 4 showed Teall with a total of 866 votes, compared to 848 votes for Eaton. That’s a difference of 18 votes. Another way of putting that: There was an average difference of exactly two votes per precinct in Ward 4. [Warm-up puzzle: How many precincts are in Ward 4?]

In the recounted totals, each candidate lost a vote in Precinct 9. In Precinct 6, Teall picked up one vote and Eaton lost one. That left Eaton and Teall with 846 and 866 votes, respectively. So the hand-counting of the paper ballots essentially confirmed the result of the optical scanners used on election day.

I’ve now covered four recounts for The Chronicle in the last five election cycles. At a recount event, as many as four separate tables might be set up in the room. Of course, the candidates in the races being recounted and their supporters are interested in watching the recounting of the ballots – to make sure everything is done properly. So it’s typical that four or five people stand around each of the tables watching the recounting as it takes place.

The actual recounting of the ballots for a given precinct is done by three people seated at the table. One person examines each paper ballot and calls out the name of the candidate who received a vote. The two other people each record a tally mark on a grid. At the end of the recounting, the hand-recorded totals on the two grids must match each other. If they don’t, everything must be re-recounted.

So the recounting procedure depends on the ability of the talliers to hear the person who is calling out the candidate name for each ballot. Because of that, everyone in the room always observes strict silence, without even being told by members of the board of canvassers that they must be quiet.

I’m kidding. It’s always necessary for a member of the board to shush everyone – more than once. That’s because we all fall prey to the belief that we can have our own side conversations that are quiet enough not to disrupt the counting – unlike those other loudmouths.

One reason those side conversations take place is that people need a way to pass the time. That’s because watching a recount is just plain boring. (That’s how you know it’s important.) So as you’re standing there watching, you start to wonder: How long is this going to take?

And as you look at the number of people assembled in the room, some of whom are being paid $12 an hour to do the recounting, you also wonder: How much is this going to cost?

Cost of a Recount

The city of Ann Arbor sends a newsletter to its election workers called Pollwatcher. The Summer 2012 edition of  Pollwatcher included an editorial comment about the Ward 4 recount. It claimed that the recount was unnecessary and wasteful:  ”[T]axpayers’ dollars will be wasted on this needless recount.”

That’s the kind of opinion that people are free to express as individuals, but shouldn’t be free to express on behalf of the city of Ann Arbor. And the city later removed the document from its website. For my part, I think that any recount serves the useful purpose of validating the accuracy of the optical scanners.

At any rate, it’s still a fair question to ask: How much did the Ward 4 recount cost?

That all depends on what you count as a “cost” of the recount. For example, city clerk Jackie Beaudry and deputy city clerk Jennifer Alexa attended the recount – but they are not paid for that work beyond their regular city salary. You can imagine an argument that their time should be factored into the cost. But for the purposes of this puzzle, we’ll focus just on the additional cash that the city had to pay directly in connection with the recount.

The cash totals below were provided to The Chronicle by city clerk Jackie Beaudry and by Washtenaw County chief deputy clerk Ed Golembiewski.

Puzzle One:  The total cost to Washtenaw County to recount the Aug. 7 election – for Ann Arbor, Ypsilanti, Sylvan Township and Augusta Township – was $426.35. Washtenaw County divided the cost among the four governments based on the number of precincts that were recounted for each government. That is, each local government was charged its proportionate share, based on the number of precincts that had to be recounted. The total number of precincts recounted was 22. Of the 22 precincts to be recounted, 9 were for Ann Arbor’s Ward 4. How much was Ann Arbor’s share of the $426.35?

That’s not the whole story. Candidates in an election can’t just demand a recount willy-nilly. They need to file an application and pay $10 for each precinct that they wished to have recounted. In the case of Ann Arbor’s Ward 4, Jack Eaton had to write a check to Washtenaw County to cover that cost. And that amount was subtracted from the cost that Washtenaw County charged to the city of Ann Arbor.

But the city of Ann Arbor had additional cash costs, not involving Washtenaw County. The ballots had to be retrieved from a storage warehouse – by workers who are paid as needed on an hourly basis. That cost was $145.

Puzzle Two: Using your answer from Puzzle One, and factoring in the $90 check written by Jack Eaton and the $145 cost for the hourly workers, what was the cost per ballot to the city of Ann Arbor for the recount? (Use the recounted ballot totals for your calculation: 866 for Teall and 846 for Eaton).

How Long Does a Recount Take?

One way I pass the time at a recount is to use a stopwatch to measure the rate of recounting at different tables.

On Sept. 4, one table recounted at a rate of 10 ballots per minute. If that sounds slow to you, bear in mind that the paper ballots had two sides. Both sides of the ballot had to be checked, even though the Ward 4 city council race was just on one side. That’s because sometimes people try to vote in both the Republican and Democratic primary – and it’s not hard to make that mistake, because both sets of candidates are printed on the same sheet of paper. If someone “cross votes,” that ballot is not counted.

Puzzle Three: Assume a counting rate of 10 ballots per minute. How long, measured in hours, minutes, and seconds, would it take to count 1,712 ballots?

Solutions to puzzles are welcome in the comments section.

Post Script: Ballot Length

The physical layout of the ballots for the upcoming Nov. 6, 2012 general election has been finalized for all the governments in Washtenaw County. Chief deputy clerk Ed Golembiewski, who also serves as Washtenaw County director of elections, told The Chronicle in a phone interview that all the items finally did fit on a single sheet of paper. That’s significant, because Golembiewski said that early in the process, it looked like two sheets of paper would be required to make all the items fit.

So for this election anyway, we will not have to contemplate the added complications and delays that a two-sheet ballot would cause on election day. And if there’s a recount, only one sheet of paper will need to be recounted.

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Figure 1. How tall is the schoolhouse? Note that the drawing is intentionally not to scale. Also note that the definition of “height” in Ann Arbor’s zoning code requires not just performing a sum of two numbers, but also a division.

So I’m taking advantage of the occasion to launch an occasional series that is meant to present math puzzles I’ve stumbled over “in the wild,” in the course of covering local government. It will appear only as time allows, so this could very well be the only installment of the series.

The puzzles are meant to be accessible to kids in high school, junior high, or elementary school – so for many Chronicle readers, they will be trivial.

But these puzzles might offer readers’ children a chance to apply what they’ve learned in math class to an actual, authentic real-life example – drawn from the municipal workings of the city in which they live.

Today’s puzzle has a geometric flavor. The basic question: How tall is the schoolhouse in Figure 1?

First, let’s please agree not to argue about the quality of the drawing. I admit that it may look more like a church than a schoolhouse. I took as my starting point a photograph included in a recent piece by local history columnist Laura Bien.

The drawing is not Laura’s fault, of course. The drawing differs from that photo in many ways. For example, the drawing lacks a belfry and an American flag on the roof. I left them out, because they make the math puzzle more complicated than necessary.

Another reason I left them out: The real-world example – on which the puzzle is based – was not a schoolhouse. I chose a schoolhouse for the drawing just to honor today as the first day of school. The real world-example is a two-building apartment complex called City Place, located on South Fifth Avenue, just south of William Street. 

Puzzle One: How Tall?

Puzzle One: How tall is the schoolhouse in Figure 1?

A good response to this puzzle is: What do you mean by tall?

We could argue for years about what the definition of “tall” should be. Here’s what the definition of “tall” is, according to the official rules used by the city of Ann Arbor:

Building height: The vertical distance of a building measured from the average elevation of the finished grade within 20 feet of the building to the highest point of the roof for a flat roof, to the deck line of a mansard roof, or to the midpoint elevation between eaves and ridge for a gable, hip or gambrel roof of a building.

That’s a lot of words. Many of them don’t apply to our puzzle. So let’s focus on the words in bold italics:

Building height: The vertical distance of a building measured from the average elevation of the finished grade within 20 feet of the building to the highest point of the roof for a flat roof, to the deck line of a mansard roof, or to the midpoint elevation between eaves and ridge for a gable, hip or gambrel roof of a building.

The phrase “finished grade” has a special meaning in that sentence. It doesn’t mean a grade in school you completed. It basically just means the ground. And “midpoint elevation” is just a fancy way of saying “the halfway point.” So let’s summarize the parts of the definition we need:

Building height: The vertical distance measured from the ground to the half-way point between the eaves  and ridge.

In Figure 1, the height between the eaves and the ridge is given as 25.0 feet.

That’s everything you need to figure out the height of the schoolhouse. If you’re so inclined, leave your solution in the comment section. Please show your work.

City Place

The dimensions given in the puzzle are the same as the dimensions of the City Place apartment buildings. That project has a long, complicated history.

Here’s one little part of that history – even though it still glosses over many details.

As the City Place project was going through the city’s approval process, people who lived in that neighborhood disagreed with the way the city calculated the height. That’s because the apartment building isn’t as simple as the schoolhouse drawing shown in the puzzle.

The apartment building actually includes a large dormer. And according to some neighbors of the City Place project, the large dormer changed the true location of the “eave” of the building. So they said that the building was actually over 35 feet tall, according to the city’s definition.

Thirty-five feet is taller than the number you should have calculated in Puzzle One. And it’s taller than what’s allowed in that area of the city.

The city didn’t change its mind about the way the height should be calculated for the building. And the two apartment buildings were constructed this past summer.

Puzzle Two: What Shape Should the Dirt Be?

There’s now a new disagreement – between the neighbor just to the north of the project and the builder of the apartments.

Figure 2. How can you add dirt to keep the official height calculation the same as it was before?

The disagreement stems from a change in a planned height. We’ve already solved a puzzle about building height. But as the City Place project was going through the approval process, a different kind of height changed in the project’s plans. The height that changed was not the height of the building itself, but rather the height of the building above sea level.

The earlier drawings showed the north building at an altitude of 857 feet above sea level. But some later drawings showed the north building at an altitude of 858.5 feet above sea level. That’s 1.5 feet higher.

Figure 2 shows that same kind of situation for the schoolhouse in our puzzle. In Figure 2, the schoolhouse is raised 1.5 higher, compared to sea level.

That sea-level height change has an impact for the building height calculation. Remember the part in the definition of height that says you measure from the ground? If the entire building is raised by 1.5 feet relative to sea level, as shown in Figure 2, that will increase the official height of the building according to the definition … unless we change the height of the ground around the building, too.

According to the definition, it’s not all the ground everywhere that has to be changed – just the ground within 20 feet of the building. The definition states that we have to look at the “average elevation of the finished grade within 20 feet of the building.”

So one way to do that is to pile a squared-off block of dirt 20 feet wide and 1.5 feet deep all around the building. But that would leave a 1.5-foot tall miniature “wall” 20 feet away from the building. Let’s think about other ways to add ground, that don’t have this vertical wall of dirt 20 feet away from the building.

Puzzle Two: Describe an exact shape for added ground (and its dimensions) around the schoolhouse that will keep the “height” of the schoolhouse identical to the original “height.”  The shape of the added ground cannot have a vertical edge 20 feet away from the building.

If you solved Puzzle Two, then you probably came up with something similar to what the builder of the apartment complex did. Your solution likely involved a nice gentle smooth slope from the building to a point 20 feet from the building.

So what happens when rain hits a sloped surface? It runs down that surface, of course. And the neighbor to the north of City Place contends that the water is now draining onto his property – because of the added dirt. That’s now the subject of a lawsuit – because it’s not legal to cause the rainwater from your property to drain onto your neighbor’s land.

If you are so inclined (pun intended), describe your solution to Puzzle Two in the comments.

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