The Game
Have you ever played FreeCell?
I play it during my idle moments because it’s fun, the rules are simple, and there is a chance for me to beat my score – measured in both time and number of moves. It is a version of Solitaire where a full deck of cards is spread randomly over eight columns with a goal to move all cards in suited sequence to their respective piles. There are four ‘free cells’ available to temporarily hold a card each to enable movement of other cards around the tableau. Playing the game for a few minutes sharpens my focus and provides some sign as to whether I need coffee.
There is also a tremendous mathematical lesson hidden in the game that ties back to growth and business – Degrees of Freedom. I worked through railroad problems related to degrees of freedom when managing a fleet of trains. And many of our growth-oriented clients often navigate some significant constraints (i.e.: few degrees of freedom) – which we are learning can severely suppress growth rates – and partially explain the value that venture capitalists can have on accelerating the development of promising new ventures.
In my article from November 3, 2024 about Modern SME Growth Drivers, I wrote about the importance of financial resilience as a critical strategy for navigating tough times and being able to take full advantage of emergent opportunities. Having adequate financial capacity adds degrees of freedom to a company, which can change its rate of growth.
Degrees of Freedom and the Relationship with Velocity
Degrees of Freedom
In physics, particularly in the study of motion, degrees of freedom and velocity are closely related concepts. Degrees of freedom in this context refers to the number of independent ways in which a system can move. For instance, a particle moving in three-dimensional space has three degrees of freedom: it can move along the x, y, and z axes. Each degree of freedom corresponds to an independent variable that can change without affecting the others.
Velocity
Velocity is the rate at which an object changes its position with respect to time. It is a vector quantity, meaning it has both magnitude (speed) and direction. I am a recovering railroader, and so many concepts that I apply as a management consultant were learned from studying the physical and operating principles of complex transportation networks. In railroading, we might think of network velocity as being affected by how much traffic occupies a limited amount of track, and when many parked trains fill up sidings, overall velocity goes down. Why? In part, because there are fewer degrees of freedom – fewer options available to unblock a congested network.
Relationship
The relationship between degrees of freedom and velocity lies in how motion is described. For each degree of freedom, there is a corresponding component of velocity. The degrees of freedom of a system determine the number of independent velocity components that describe the system’s motion. When there are fewer degrees of freedom – fewer options available – velocity goes down.
The Relationship between the Theory of Constraints and Degrees of Freedom
The concept of degrees of freedom and the Theory of Constraints (TOC) both deal with the limitations within a system, but they approach these limitations from different angles. In a system, degrees of freedom are essentially about the flexibility of the system to change and adapt. The Theory of Constraints is a management philosophy that focuses on finding and alleviating the bottleneck or constraint within a process. The idea is to recognize the most limiting factor and optimize it to improve the overall system performance. Both concepts involve understanding and managing constraints. Degrees of freedom are concerned with the available options within the constraints, while TOC focuses on optimizing the most critical constraint. In practical applications, knowing the degrees of freedom can help identify potential areas for improvement, which aligns with TOC’s goal of enhancing overall system performance by optimizing constraints. Our management consultants often help our clients to understand where those bottlenecks are and how to add the right capacity to improve growth rates.
Back to the Cards
In FreeCell, I’ve learned that the degrees of freedom go down as more cards fill up the free cells. In the game, when the free cells are full, my time to complete the game is always dramatically slower because the tableau is constrained, and I have fewer movement options.
Applied to Growth in Business
Working inside a business, can you think of what resources you need to accelerate your growth? You need time, funds, and skills. You might think of having lots of time as the same as having more degrees of freedom. Extra free time affords you the capacity to pursue different kinds of growth investments, take advantage of emergent opportunities, and likely more time can reduce stress and improve personal health and wellness – essential for sustaining a high productivity rate. A well funded organization can add value-generating assets and programs into the business, can hire more people who can do more work, and can avoid operational consequences by being able to pay bills on time. Having the right skills available when needed can accelerate the completion of complex work.
Now consider those businesses that are heavily constrained for time, funds, or skills – all representative of fewer degrees of freedom? A business with few options can face exponentially compounded growth problems resulting from missed opportunities, financial and operational penalties, weak productivity, and a slow completion rate.
In railroading, an eloquent strategy to improving an operating ratio is to remove traffic from the network, thereby increasing the degrees of freedom and allowing the network to speed up. In growing businesses, clear goals and priorities (which is often as much about defining what a company won’t do as what it will do), adequate funding, and the right skills and experience collectively add degrees of freedom that in turn can accelerate growth.
About
Jeff Peterson is the Founder and CEO of Blue Monarch Management and is a professional Management Consultant specializing in Strategy, Governance, and Organizational Development for companies designing and driving transformational investments. He is pursuing a Doctor of Business Administration degree at the University of Calgary, with planned research studies in adaptive change, organizational design, and entrepreneurship.
