Thursday, August 10, 2017

Power Up a Tower: Part 4 (aka The Magic Number 11)

I’ve climbed a lot of stairwells during the past 6 years and my two favorite stairwells are:
  • One City Place - Hartford, CT 
  • US Bank Tower - Los Angeles, CA
Although I’ve only climbed these two building a handful of times*, I can tell you with certainty that these are two of the fastest, most efficient stairwells on the circuit.

*One City Place: 3 times racing and 4 times for fun. US Bank: 2 times racing, 2 times for fun.


They both share the magic number eleven. Eleven steps per flight, to be specific.

This edition of Power up a Tower will explain why 11 steps per flight is so efficient and while covering a few related topics such as:
  • The Physics of Climbing & Turning
  • Stairwell Configurations & Efficient Footfall patterns
Part 1: The Physics of Climbing and Turning

In this section I will share several principles that are 2nd nature to most competitive climbers and provide evidence justifying why each principle is true.

Unlike my other Power up a Tower articles, there won’t be any fancy calculations. I just rely on basic physics principles and geometry, although I will recommend one or two scientific papers along the way.

Taking two steps at a time is more efficient than taking one step at a time.

Most competitive stair climbers will nod their heads in agreement, but if you want scientific evidence, I recommend reading a paper by Halsey et. al. entitled “The Energy Expenditure of Stair Climbing One Step and Two Steps at a Time: Estimations from Measures of Heart Rate”.

If you don’t have access to the paper or don’t feel like reading, here is the gist of it: The authors hooked a bunch of people on heart rate monitors and had them climb a stairwell multiple times alternating between single stepping (i.e. taking one step at a time) and double stepping (i.e. taking two steps at a time). It turns out that single steppers had a higher heart rate at the top of the stairwell which indicates that single stepping requires more energy overall than double stepping.

The paper suggests that it has something to do with bio-mechanics of single stepping vs. double stepping. Single stepping has a much higher (roughly double) turnover rate it means you end up swinging your legs (and possibly arms if you aren’t using the rails) twice as much as double stepping. This additional motion costs additional energy.

If two steps at a time is better than one step, then how about three or even four steps at a time? That is a question I’m not entirely prepared to answer. It likely has something to do with bio-mechanics as well. My guess is it takes too much power (i.e. leg strength) to climb three at a time - kind of like trying to bike up a hill using a gear that is slightly too high. Furthermore (at least from my experience) taking three steps at a time puts the body an awkward position* that isn’t suited for sustained output.

*Usually I try to lean a little bit forward when climbing and it is harder to lean forward when taking three steps at a time because I'm raising my legs up so high.

Perhaps there are certain tall individuals who might be more efficient when taking three steps at a time or certain buildings with extremely short steps (e.g. 5 inches tall) where taking three steps at a time makes sense, but in general, these cases would be the exceptions rather than the norm.

Tight inside turns using the rails is faster than other turning methods.

This should come as no surprise. Race cars will take the inside line because it has the shortest distance. The same holds true inside the stairwell. In addition, using the inside rails (using upper body strength) can help you pivot and accelerate around the turn.

I’ve been tracking my own performance inside my practice stairwell for years, so I have a lot of data to back this up. For example, in my last training session, I sprinted up a total of 12 times, with 4 – 5 minutes of rest between each climb. Four of the climbs were done using the inside rails. Four of the climbs were done using only the outside rails. Four of the climbs were done using no rails whatsoever (i.e. legs only). The ascents were climbed sequentially using the following pattern: Outside, Inside, No Rail, Outside, Inside, No Rail, etc.

Here were my average climb times (in seconds):
• Inside: 40.08
• Outside: 43.30
• No Rails: 43.80

Using the inside rails is faster than the other methods and a brief analysis of my previous training data* (spanning thousands of climbs) corroborates.

*You’ll have to take my word since I’m not ready to share all my training data. Perhaps in a later post…

Geometry of the stairwell has a role to play, especially when you consider the climb times using the outside lane. In a narrower staircase, you’d expect the time gaps to narrow. Likewise, climbing technique matters. Over the years, I’ve learned a few tricks to eke out a tenth of a second or so of speed when turning inside my practice stairwell. Every climber has their own unique climbing style, so slightly different results would be expected between climbers*.

*If you are a statistician, you might point out that data from a single climber doesn’t pass muster. Point taken. I’ll be sure to update this post once I have an adequate sample size, smartass.

It is interesting to note that the slowest climbs were done without using the rails. I don’t have a clear-cut answer why this was the case, but I can come up with a few reasons:

  • Although I tried to stay in the inside lane, I often steered toward the center as I was pumping my arms rather than hugging the rail.
  • My turns were slower because I could accelerate my body around the turn using the handrail; I could only use the power of my legs.
  • The “no-rail” assents seem to be the most physically demanding so fatigue near the top of the stairwell likely contributed to slower times.
Turning on the landing is easiest when pivoting on the inside foot.

Most stairwells will have either a couple large balusters (i.e. vertical members to support the handrail) or a wall which physically separates the individual flights of steps from one another. To turn efficiently – no matter which foot you pivot on – your foot needs to be as close to the baluster/wall as possible. Pivoting on the outside foot is difficult* because the barrier interferes with your inside foot as you bring it around to the next step. Obviously, the closer your outside foot is to the baluster/wall, the less clearance you have between the barrier and your inside foot, which makes the turn even more difficult**.

*unless you are as flexible as David Hanley.
**unless you are as flexible as David Hanley.

Pivoting on the inside foot is much easier because your outside foot will not interfere with the baluster/wall barrier.

You may argue that some stairwell configurations make it impossible to keep just a single foot on the landing during the turn. This is certainly true if there is a significant horizontal gap between adjacent flights or some other barrier. However, even if you end up taking two or even three steps on the landing, your final pivot around the baluster/wall is still easiest when pivoting on your inside foot.

Turning requires extra energy.

Simple physics will tell you this is true. After all, when turning in a typical stairwell, you basically come to a stop and then turn on the landing – essentially making a 180 degree turn. Then, once you are facing the other direction, you quickly accelerate back up to full speed. All this acceleration (including the turning) requires force and energy*.

*If you don’t think starting and stopping requires extra energy, try a few shuttle sprints. Though to be fair, it is easy to de-accelerate when approaching the turn thanks to gravity. I find that most of my energy is spent turning and then getting back up to speed.

Calculating precisely how much energy is required to complete a turn is a complicated matter and would require a decent scientific paper to cover this topic adequately*. For example, figuring out how much energy is required to run at a given speed in a straight line requires a good understanding of the bio-mechanics of the human body as well as a foundation in kinematics to create an accurate model. Modeling a pivot while using the rails would an even harder problem to tackle, as the movement is complicated (e.g. each limb has an independent motion) and the entire action happens over a brief period of time.

*There is a paper by Misetti et. al. “Skyscraper running: physiological and bio-mechanical profile of a novel sport activity” which broaches this very topic, but I don’t agree with their model. Otherwise this is a highly-recommended paper.

Part 2: Basic Stairwell Configurations

Most stairwells inside tall buildings will have two or more flights of steps per floor with landings* separating each flight. Flights usually alternate 180 degrees from one other at each landing. Most floors are of a uniform height so you will see the same (repeating) flight pattern on multiple consecutive floors.

* I consider the floor to be a kind of a landing, too.

Throughout this article, I will use the following notation to represent the layout of a repeating stairwell: X/Y*

  • X represents the number of steps in the 1st flight.
  • Y represents the number of steps in the 2nd flight
  • “/” represents the mid-flight landing.
For example 10/10 represents two flights per floor with 10 steps per flight.

*I also use the same notation for certain turns, where the numbers just mean the number of steps per footfall. 

This article only handles typical stairwells with two flights per floor, but this notation can be expanded to include stairwells with additional flights. For example, The “Lucky Sevens” 7/7/7 has three flights per floor with 7 steps apiece*.

*A separate article isn’t going to happen for three reasons. First of all, 3+ flights/floor is relatively rare. Second of all, there are too many combinations to document (for example 3 flights/floor has 40 different configurations). Finally, a good chunk of the time these 3+ flights/floor configurations mimic their two flights/floor brethren. In the case of the Lucky Sevens (7/7/7), it is really nothing more than a 7/7 (which is a shorter version of the Magic 11s (11/11) configuration which you’ll read about shortly). Climbing up two floors of the Lucky Sevens is virtually identical to climbing three floors of the 7/7.

There are numerous exceptions, of course. For example, the Bennington Battle Monument’s stairs wrap around the inside of the building using a series of 90 degree turns. In fact, the Empire State Building (arguably the most famous skyscraper in the world) primarily has just a single flight of stairs per floor followed by a short run to get to the next flight*.

*which is why I frickin' hate that stairwell.

There are also numerous exceptions within a building's stairwell as there are usually maintenance floors, fire doors, and other irregularities which will mess up the repeating pattern.

However, in general, the flight pattern is repeatable and throughout the rest of this article we’re going to focus on this repeating pattern.

Common Time (e.g. 12/12)

So far we’ve learned that best method to climb a stairwell is to (1) take two steps at a time, (2) hug the inside rail on the turns, and (3) use your inside leg to pivot on the landing.

Wouldn’t it be just peachy if there was a stairwell configuration that satisfied all three conditions?

It turns out that a (4m)/(4n) flight pattern (where m & n are positive integers) satisfies this condition*.

*Set m=n=1 and if you read music, you’ll understand why this stairwell is nicknamed Common Time.

This occurs because you end up with an even number of footfalls per flight when taking two steps at a time*.

*With an even number of footfalls, you’ll end up in the same position you started with. Conversely with an odd number of footfalls, you’ll end up on the opposite foot.

Figure 1 shows an illustrative example for a 12/12 flight pattern (where n=m=3).
Figure 1: 12/12 Footfall Pattern
The figure represents two flights with 12 steps each. The red dots represent footfalls and the shaded area represents the landing. The turning arrows represent your pivot foot (which makes a 180 degree turn).

The Common Time stairwell has a solid footfall pattern, but it has a couple of limitations.

First, this step pattern only works if the horizontal gap between flight is tight and there aren’t any major barriers (such as a wall or balustrade) on the landing which hinders your inner foot placement on the landing. Otherwise, keeping a single step on the landings may be difficult or even impossible.

Secondly, turning requires additional energy (more about that in the next section) which makes pacing a concern:
  • If you try to keep a constant vertical pace (e.g. by using a metronome*) your level of exertion will no longer be constant, and the turns will sap your energy.
  • If you try to keep a constant level of exertion, your turns will need to be slower, which makes using a pacing device (like a metronome*) difficult.
*Of course if you don't use a metronome, these concerns are less important... but seriously, why would you not use a metronome?!? 

How important are these concerns? In my experience, they play a minor role in a short race, but become more important in the longer races when you are climbing for an extended period at just slightly above the red line (i.e. lactate threshold or whatever biological term you want to use). In the latter case, even minor fluctuations of energy expenditure could be enough to push you well past the redline - causing you to bonk prematurely during a race.

The Magic 11s (e.g. 11/11)

Is there a stairwell configuration which overcomes the limitations of a (4m)/(4n) flight pattern?*

*spoiler alert: Yes there is.

Yes there is!

I present to you the most efficient stairwell configuration known to mankind:
(4m-1)/(4n-1) where m and n are positive integers*.

*From now on just assume that m and n are positive integers so I don’t have to keep typing it out.

For example, when m=n=3, you have an 11/11 stairwell, just like at City Place in Hartford an US Bank Tower in Los Angeles. See figure 2 for the foot pattern.

Figure 2: 11/11 Footfall Pattern
Basically, this pattern is just like the 12/12 step pattern (i.e. it has an even number of footfalls) except you only take a single step when approaching the landing.

This has two benefits:

First, you can reach a little bit further onto the landing with your inside pivot foot since you are covering only one step, not two. This makes turning a little bit easier and efficient since allows you to plant your pivot foot closer to the optimum spot.

Second, taking a single step at a time requires less energy than taking two steps at time*.

*It is more accurate to say that two single steps require slightly more energy than one double step because of wasted motion.

Keeping this latter piece of information in mind, I propose the following equation:

Energy used taking two steps (two steps at a time) ≈ 
Energy used taking one step (one step at time) + 
Energy used taking executing a turn

I use the ≈ rather than =, < or > because I don’t have the tools available (experimental or analytical) to determine exactly how much energy is required to turn.

However, I think we can safely bound this so called “turning energy” by the following:

Zero energy < Energy required to turn < Energy required to climb two steps

Clearly, the turning energy isn’t negligible and if turning costs more energy than climbing two steps than I should have died from a heart attack long ago.

Personally, I’m estimating the turning energy is about equal to that of climbing a single step… and if my upper bound is true, then at worst, my estimate is off by less than a single step’s worth of energy.

What all this means is that (despite the uncertainty) taking a single step while turning requires about the same energy as taking two steps at a time on a straight flight. 

If you use a metronome to keep pace (as you should) your energy expenditure will remain fairly constant…even while turning (and that is a good thing).

Seriously, this configuration is the bomb. Once you start pivoting on your inside leg, you don’t even have to think about it. You automatically end on your inside leg. Every. Single. Turn.

Interlude: Common Turning Patterns

Quick turns are predicated on having your feet in the right spot as you enter a turn. The 12/12 and 11/11 stairwells automatically place your feet near the sweet spot so that is why I hold them in such high regard.

Unfortunately, not all configurations are so convenient*.

*All together, there are ten basic stairwell configurations* and we’ve already covered the Common Time and Magic 11s stairwells. Several of the remaining eight are more complicated.

*Proving there are exactly 10  two-flight stairwell configurations is a neat math problem. If you have a few minutes (and are so inclined) I urge you take a short break from reading and solve this problem before continuing.

We can’t change the number of steps per flight in a stairwell, but we can change how we take the turns. Before we get into more complicated stairwell configurations, let’s discuss turning.

Turning has two primary goals. The first (and obvious) goal is to get around the current landing quickly and efficiently. The second (and less obvious) goal is setting up your footfalls to make the next turn is as quick and efficient as possible.

Here is an example to illustrate the 2nd goal.

Let’s say you are in a 12/12 stairwell and just started a pivot turn on the inside foot. Should your next footfall take one step or two steps at a time?

Back in figure 2 we showed that taking one step after the turn is optimal from an energy expenditure standpoint. But if we took only one step, we’d find ourselves on the outside (wrong) foot when we reached the top of the next flight! Clearly taking two steps at a time is the right choice*.

*Unless you are a contortionist.

The key to figuring out the best footfall pattern in a stairwell is choosing the best type of turn that doesn’t screw up your next turn*.

*Note: I used bold typeface because this sentence is really important.

Before moving to the remaining stairwell configurations, let’s study the type of turns we have at our disposal.
A: 2/2 Inside Pivot
B: 2/1 or 1/2 Inside Pivot
C: 2/2 Outside Pivot
D: 2/1 or 1/2 Outside Pivot
E: 1/1 Inside Pivot
F: 1/1 Outside Pivot
G: Double footfalls on the Landing (Inside Pivot)
H: Skip Landing (Inside Pivot)

The 2/2 inside pivot is the same turn used in a Common Time (e.g. 12/12) stairwell. See figure A. The turn spans two footfalls and each footfall covers two steps, so it is a fast and energy intensive turn. This turn may be difficult to perform if the landing has any obstacles, so this choice may not always be available. On a letter grading scale, I’d give it a solid B+.
Figure A: 2/2 Inside Pivot Turn
The 2/1 & 1/2 inside pivot are the same turns used in a Magic 11s stairwell. See figure B. The 2/1 & 1/2 are interchangeable. For example, in an 11/11 stairwell you could construct a footfall pattern using either type of turn*. The turn spans two footfalls and on average, each footfall covers 1.5 steps (2 then 1 or 1 then 2 steps). Therefore, the turn itself is moderately fast, but energy balanced (i.e. the energy it takes to get around the turn per footfall is approximately equal to the energy it takes climbing up two steps at a time on a straight staircase). Since one of the footfalls only covers one step, it is easier to perform than the 2/2 inside pivot. On a letter grading scale, I’d give it an A+.

*To prove this to yourself, go back to figure 2 and change the direction of the arrows to widdershins.
Figure B: 2/1 & 1/2 Inside Pivot Turns
The 2/2 outside pivot turn is shown in figure C. Like the 2/2 inside pivot turn, it spans two footfalls and each footfall covers 2 steps. It is fast and even more energy intensive than the 2/2 inside pivot because an outside pivot is relatively awkward and inefficient. Because this turn is so difficult, it can only be performed in a stairwell with little or no obstacles. On a letter grading scale, I’d give it an C+.
Figure C: 2/2 Outside Pivot Turn
The 2/1 & 1/2 outside pivot turns are shown in figure D. Like the 2/1 & 1/2 inside pivots, these two turns are interchangeable. Each turn spans two footfalls and each footfall covers 1.5 steps on average. It is moderately fast and slightly more energy intensive than the 2/1 inside pivot turn because an outside pivot turn is relatively awkward and inefficient. This turn can be performed in most stairwells; it is easier to perform the 2/2 outer pivot, but harder than the 2/1 inside pivot. On a letter grading scale, I’d give it an B.
Figure D: 2/1 & 1/2 Outside Pivot Turns
The 1/1 inside pivot is shown in figure E. This turn spans two footfalls and each footfall covers 1 step. It is a slow but easy to perform. As such, it can be performed in most stairwells. On a letter grading scale, I’d give it a C+, but I’d rank it higher if the landing had obstacles.
Figure E: 1/1 Inside Pivot Turn
The 1/1 outside pivot is shown in figure F. This turn spans two footfalls and each footfall covers 1 step, which makes the turn slow. The outside pivot is awkward, but easy to perform because of the single steps. As such, it can be performed in most stairwells. On a letter grading scale, I’d give it a C, but I’d rank it higher if the landing had obstacles.

(Crap... I forgot to make Figure F. I'l have to do it later)

The double footfalls (on the landing) turn usually uses an inside pivot turn*. See figure F. This turn is slow because it wastes an entire footfall on the landing which makes the turn relatively slow and inefficient. The main benefit of this turn is that it can be performed in most stairwells, as two single step footfalls gives you a lot of extra room to turn. On a letter grading scale, I’d give it a C, but if the landing had obstacles, it would be ranked higher.

*I’d be hard pressed to find a situation where using an outside pivot turn would be preferred.
Figure G: Double Footfalls on the Landing Turn
The skip landing turn shown in figure H is probably the least well known turn. It is executed with an inside pivot on the step below the landing with the next footfall landing on the step above the landing – such that you completely avoid touching the landing. It is a fast turn on par with the 2/2 inside pivot turn, but it is slightly more difficult to execute. This turn requires a very good stairwell with convenient handrails and very little gap between the flights. On a letter grading scale, I’d give it a B.
Figure H: Skip Landing Turn
Now that we know all about turns, let's get back to business...

Part 2: Basic Stairwell Configurations - Continued

We’ve covered (4m)/(4n) and (4m-1)/(4n-1) stairwell configurations which both have an even number of footfalls per flight.

The discussion would not be complete without considering stairwell configurations which have an odd number of footfalls per flight: (4n-2)/(4m-2) and (4n-3)/(4m-3)

If you insist on taking 2 steps at a time, you’ll find yourself out of position every other flight. This happens because you have an odd number of footfalls (e.g. for a ten-step flight, you have 5 footfalls when taking the steps two at a time).

The Oddball Ten (e.g. 10/10)

Let’s consider a 10/10 stairwell (i.e. (4n-2)/(4m-2) where n=m=3). 

In this configuration, you can do one of three things:

A: Alternate between 2/2 inside and 2/2 outside pivot turns (figure 3-A).

B: Uses two footfalls on the landing (figure 3-B). This pattern turns the total number of footfalls from odd to even.

C: Uses a 1/1 inside pivot turn (figure 3-C). This pattern turns the total number of footfalls from odd to even.
Figures 3-A, 3-B, & 3-C: 10/10 Footfall Patterns
Pattern C is often overlooked because you must consciously decide to single step before and after the landing. Unless you’ve practiced this pattern, it won’t be reflexive.

Personally, I like to use pattern A as it keeps my vertical pace constant. But in practice, I usually end up switching to pattern B as fatigue sets in.

I don’t often use pattern C since the 1/1 inside pivot end up being too slow for my tastes. That said, I keep the technique in my back pocket for special cases. Certain types of landings make it very difficult to keep one foot on the landings (e.g. wide gaps between flights, rails/balustrades that stick out, etc.). In those cases, this step pattern might give you enough room to keep just a single footfall on the landings.

Double Odd 9s (e.g. 9/9)

Next, let’s consider a 9/9 stairwell (i.e. (4n-3)/(4m-3) where n=m=3)*. 

*Colloquially called the Double Odd because each flight has an odd number of steps and an odd number of footfalls.

There is only one obvious climbing choice as shown in figure 4.
Figure 4: 9/9 Footfall Pattern
This pattern alternates between 1/2 inside pivot turns and 1/2 outside pivot turns.

Turning on the inside foot is a piece of cake, but turning on the outside foot is troublesome. Fortunately, the single step ameliorates some of the awkwardness of doing an outside pivot because it allows you to plant your foot step further into the landing (and the added room makes the turn less awkward).

Personally, I dislike the outside pivot turn, but otherwise this is a solid pattern with limited downside.

Part 3: Asymmetric Configurations

Before continuing, I need to define the term symmetric and asymmetric with respect to stairwell configurations.

Symmetric means we have the same number of stairs per flight*.

Asymmetric means we have different numbers of stairs per flight*.

*Technically I need to tack on the phrase “when we set m = n” to the end of these sentences. For example, the 12/12 configuration is clearly symmetrical. It comes from the symmetric (4m)/(4n) pattern when we set m=n=3. But if we set m=2 and m=3, we get the 8/12, which is also symmetric using this expanded definition.

So far, I’ve only described the symmetric stairwell configurations, so now let’s turn our attention to their asymmetric counterparts.

Close Cousin 11/12

Remember the Common Time (e.g. 12/12) and the Magic 11s stairwells? Both feature an even number of footfalls per flight.

Their asymmetric cousin is the 11/12* pattern, formally written as (4m-1)/(4n), which also features an even number of footfalls per flight.

*By the way, the 11/12 pattern is identical to the 12/11 pattern in a repeating stairwell. To prove it to yourself, start your step count from the floor start counting from mid-floor landing. Since we’re in a repeating stairwell, it shouldn’t make any difference where you start counting from. For sake of consistency, though, I always list the smaller number first (e.g. 11/12 rather than 12/11).

This configuration (like most other asymmetric configurations as it turns out) is a bit hard to visualize in a simple figure. Rather than do a fancy bit of resizing and re-scaling (which ultimately is still visually confusing) I’ve opted to show the 11/11 and 12/12 patterns side by side to help visualize the 11/12 pattern. See figure 5. You may ignore the shaded region, but I’ve left it semi-transparent so you can still see the related 11/11 and 12/12 patterns as a reference.
Figure 5: 11/12 Footfall Pattern
As far as stairwells are concerned, this pattern is among the best. It alternates between using the 2/2 inside pivot turn (grade B+) and the 2/1 inside pivot turn (an even better A+).

As such, I like this stairwell a little bit better than the Common Time (e.g. 12/12) stairwell, but it falls slightly behind the Magic 11s.

Close Cousin 9/10

Next, we have the asymmetric cousin to the 9/9 and 10/10 patterns; the 9/10 pattern - formally written as (4m-3)/(4n-2). They are cousins in the sense that they all share an odd number of footfalls per flight.

Since there are quite a few different ways to handle the 10/10 pattern (go back and see figures 3-A, 3-B, and 3-C), there are a bunch of ways to handle the 9/10 pattern as well. Here are two of the best:

Pattern A: This footfall pattern alternates between 2/2 inside pivot turns (grade=B+) and 1/2 outside pivot turns (grade=B). The outside pivot turn is awkward, but the single step makes it palatable. See figure 6-A below.
Figure 6A: 9/10 Footfall Pattern (A)

Pattern B: This pattern is hard to visualize, but that’s the price you pay for asymmetry. Rather than repeating every floor, the pattern repeats itself every other floor. On odd floors it cycles through a couple of 1/1 outside pivot turns (grade = C). On even floors it cycles through 2/1 & 1/2 inside pivot turns (grade = A+). See figure 6-B below.
Figure 6-B: 9/10 Footfall Pattern (B)
The main difference between patterns A and B is the that A uses a total of 10 footfalls whereas  B uses 11 footfalls per floor. Pattern A is faster but requires a bit more effort on the turns. Pattern B takes the turns very efficiently at the expense of more footfalls. Which option would I choose? Probably option A since I use a metronome, but option B is a solid 2nd choice - especially if the stairwell has troublesome landings.

The Ugly Stepchildren

The final configurations, the 9/11, 9/12, 10/11, and 10/12 (I’ll write the formal terms later) have the following in common: One flight has an odd number of footfalls (9 or 10 steps) while the other flight has an even number of footfalls (11 or 12 steps).

This means that the total number of footfalls per floor is odd (since odd + even = odd) and if you take two steps at time*, you will always end up on the opposite foot when you reach the next floor.

*I should clarify to say “two steps at a time wherever possible”. Obviously, you’ll have to take at least one step at a time before or after the landing if you have an odd number of steps in a flight (like 9 or 11 steps).

Compare that with the symmetric configurations and their close cousins which have an even number of footfalls per floor (since even + even = even; odd + odd = even).

Naturally, the ugly stepchildren are fairly hard to map out since the map will often cover two floors (i.e. four flights) worth of steps.

The Bastard (e.g. 9/11)

The Bastard 9/11 stairwell, formally known as (4m-3)/(4n-1) is in a difficult position. If the stairwell is perfect, it has one top-notch pattern…but all bets are off if their are any obstacles.

The most efficient pattern uses a combination of 2/2 inside pivot turns (grade B+) and skip landing turns (grade B). This pattern works because it has the same number of footfalls as the symmetric 4m/4n pattern*. See figure 7-A.

*Its kinda cool when you think about it. The 9/11 has exactly 20 steps and 10 ten footfalls just like the symmetric 8/12*. Since the skip landing turn spans two separate flights (i.e. covering one step in each flight), it essentially converts the 9/11 into a symmetric stairwell. In fact, if you really wanted to, you could even use the skip landing pivot turn in a regular (4m)/(4n) stairwell. I guess you could call this pattern the legitimate “bastard” of the (4m)/(4n)**.

*Remember that 8/12 is in the (4m)/(4n) family just like the familiar 12/12 we studied back in figure 1.
**Your father is symmetric & your mother is asymmetric, but you are close enough to your father to claim an inheritance.
Figure 7-A: 9/11 Footfall Pattern (A)
The skip landing turn makes the 9/11 into a solid stairwell choice, but if there are any obstacles which prevent the skip landing turn, you are going to have to settle for another complicated two-floor pattern*.

*Which turns the 9/11 into an illegitimate bastard.

There are several variations of the two-floor pattern. I’m going to only show one of the variations to illustrate, but I’ll give you enough information to construct the others for yourself.

Check out figure 7-B. This pattern (and all variations of this pattern) uses 11 footfalls per floor which means we are taking a few single steps here and there to help us get around the turns.
Figure 7-B: 9/11 Footfall Pattern (B)
There are four basic types of turns illustrated
A: 2/2 inside pivot turn (grade B+)
B: 1/2 inside pivot turn (grade A+)
C: 1/2 outside pivot turn (grade B)
D: 1/1 outside pivot turn (grade C)

You can create other variations of this pattern by pairing up the turns as follows: A pairs with D and B pairs with C.

This give us the following variations:
A, B, C, D (as shown in figure 7-B)
A, A, D, D
B, B, C, C

Of which, I think BBCC is the clear winner since it uses 1/2 pivot turns.

You can also construct similar variations by including double footfalls on the landing. I personally don’t like these kind of patterns because if you must switch feet to get around a turn, you might as well climb a step while you are at it (and with an odd number of steps per flight, you’re going to have plenty of opportunities to do so).

The Doppelganger (e.g. 10/12)

The 10/12 stairwell, formally known as (4m-2)/(4n), can look a lot like the symmetric 10/10 stairwell. The difference is it will nominally take two floors to complete a cycle rather than just one floor*.

*This makes logical sense. 10/12 has an odd number of footfalls per floor whereas 10/10 has an even number. In order for 10/12 to get back to an even number of footfalls, you’ll need to complete two circuits (since odd + odd = even).

If you just take two step at a time (and you can because each flight has an even number of steps) the step pattern is straightforward. It is very similar to the the 10/10 pattern shown back in figure 3, but it covers two floors to complete a cycle rather than just one. This 10/12 pattern is shown below in figure 8-A.

Just like the 10/10 pattern, the 10/12 pattern will alternate between standard 2/2 inside pivot turns and awkward 2/2 outside pivot turns. The only difference is that the pivot turns alternate between floor rather than between flight (i.e. you get the same pivot turn twice in a row before it alternates).
Figure 8-A: 10/12 Footfall Pattern (A)
10/12 can mimic other patterns as well. One particularly simple configuration cycles only once per floor. See figure 8-B below.
Figure 8-B: 10/12 Footfall Pattern (B)
This 10/12 pattern is like the 10/10 pattern shown in figure 4, but it is more efficient. In 10/10 figure 4, you have a double footfall turn each landing, but in this 10/12 pattern, you have a double footfall turn every other landing*.

*You can thank the 12 step flight. Since it has an even number of footfalls, it effectively delays the double step to once every other landing rather than every landing.

The final 10/12 pattern ranks right up there with the Magic 11s because it uses both 2/1 & 1/2 inside pivot turns . See figure 8-C.
Figure 8-C: 10/12 Footfall Pattern (C)
This pattern is very interesting because it mimics aspects from both the 10/10 and 12/12 patterns:

  • The right (10 step) side of the figure is just like the right-hand side of the 10/10 pattern show in figure 3 - which has easy, but slow turns.
  • The left (12 step) side of the figure is just like the left-hand side of the Common Time 12/12 pattern shown in figure 1 - which has fast, yet energy intensive turns.
When combined, these two patterns mimic the Magic 11s*. The only difference is with this 10/12 pattern, you alternate between 2/1 and 1/2 inside pivot turns rather than just using one or the other.

*Which isn’t too surprising since they share the same number of steps per floor.

Overall, the 10/12 stairwell offers a lot of versatility. The footfall patterns shown above are all viable… but let’s face it, the final pattern gives the Magic 11s a run for the money*.

*I still give the nod to the Magic 11s stairwell because you automatically take 2/1 inside pivot turns – and that’s hard to screw up. With the 10/12 you must remember where to place your feet, so you need to be on the ball to get the correct pattern. I will concede, however, that the alternating 1/2 & 2/1 inside pivot turns are perfect balanced between legs. In either the 1/2 or 2/1 patterns, one leg will end up doing a little more work than the other (since one leg will be always be doing a single step rather than a double step around the turn).

The Politician (e.g. 10/11)

On the surface 10/11 (i.e. the (4m-2)/(4m-1)) has a lot going for it. It has the same number of footfalls as it’s big brother the Doppelganger (10/12) and the 11 step flight makes it appear similar to the Magic 11s.

But looks can be deceiving.

You can construct a lot of different patterns with the 10/11 and all of them are compromises. Look at figures 9a and 9b to see what I mean.
Figure 9-A & 9-B: 10/11 Footfall Patterns
In figure 9-A, the top turn is a 2/1 inside pivot turn (yay!) but the bottom turn features a 1/1 inside pivot turn which makes the turn easy, but slow (boo!).

In figure 9-B, the bottom turn is a 2/1 inside pivot (yay!) but the top turn features double steps on the landing, which is slow (boo!).

Next, let’s take a look at figure 9-C.
Figure 9-C: 10/11 Footfall Pattern (C)
This pattern cycles one every two floors and features the following turns:

  • A 2/2 inside pivot turn (grade B+)
  • A skip landing turn (grade B, but only in a perfect stairwell)
  • (2x) 2/1 outside pivot turns (grade B)
The bottom line is that no matter what pattern you choose, you always end up with a less-than-stellar compromise.

The Shortcut (e.g. 9/12)

The final stairwell configuration is the 9/12 (i.e. the (4m-3)/(4m)), I like to call it the Short Cut. 

The basic pattern is straightforward. See figure 10-A. It features a double footfalls turn (grade C) and a 2/1 inside pivot turn (grade A+).

(Crap.. once again I forgot to include the figure. I'll add it later. Good news is that the pattern is illustrated on the left hand side of figure 10-B. Just ignore the green arrows.)

This pattern ain't too shabby. But guess what, there is a shortcut! See figure 10-B. 
Figure 10-B: 9/12 "Shortcut" Footfall Pattern
This pattern starts off the same the basic one, with a double footfalls turn (grade C). For the next turn, rather than use a 2/1 inside pivot, instead use a 2/2 inside pivot (grade B+). This will set us up for a couple of back-to-back skip landing turns (grade B).

These two patterns are equally viable. The difference is that the basic pattern uses 12 footfalls per floor whereas the "shortcut" uses a combination of 12 and 10 footfalls on alternating floors. This means that the "shortcut" pattern is faster, yet more energy intensive on the whole. 

Final Thoughts:

This article has covered the common footfall patterns for all two flights/floor stairwells. It provides a lot of detail – admittedly too much detail for the casual reader – so I’ve made a quick summary of the different configurations in Appendix A for easy reference.

(OK - I haven't actually made up the Appendix, but I will eventually)

Many of the patterns require a near perfect stairwell with good landings and rails. But in real life you’ll find plenty of stairwells which require an extra step or two.

Therefore, I leave you with the following advice: Study your stairwell and figure out an efficient footfall pattern suitable to your turning style.

You can thank me after you shave a few seconds off your PR.