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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