by Peter Meyer and Richard Meyer

Richard Meyer is a professor emeritus at the University of Wisconsin specializing in mathematical physics. Peter Meyer, principal of the Meyer Group in Scotts Valley, California, speaks and writes internationally on management issues and on getting more results from fewer resources. He invites comments. Feel free to Email him at

Perhaps it has happened to you, perhaps it is happening now. Months of depressing thought only yield you a clear explanation of why the tools you have cannot solve the problem you face. Problem solving is often done using a scientific process to get from question to answer. Often, it works. Sometimes it falls apart. What do you do when a scientific process fails? This article uses a real case to detail one way to get to a solution quickly.

One of the authors was trying to develop a workable gas turbine, but there was not enough information to know how to proceed to do so. There was a great deal of data, but it did not answer the questions that needed answers; so he needed to find another way to solve the problem.

The first gas turbine had been built in 1944, but it was only a military toy. Stodola's analysis showed it could not compete with piston engines unless it could operate at very high gas temperatures. These are temperatures far beyond those at which any then known material can support the centrifugal stresses of the engine.

The first hopeful idea came from a German discovery in the 1920s that the process by which a fluid stream heats a solid surface is confined to a thin boundary layer of fluid adjacent to that surface. The rest of the fluid plays no significant part in the heat transfer: In principle, cooling just that thin layer ought to insulate the steel--the turbine blades would never "know" the temperature of most of the gas. If this could be done in an engine, it would revolutionize aircraft engines. But could it be achieved economically in a gas turbine?

As a British scientist late in the last Great War, one of the authors was told to figure out whether cold air injection would work here.

Unfortunately, there was a snag: a turbulent layer. Boundary layers come in two types, each similar to an extreme of flow in rivers. Consider a slow river. Where the water moves slowly, the flow may be smooth and steady. However, in white-water rapids, the river is violently turbulent. The smooth type of boundary layer had become more amenable to mathematical analysis, but the bounndary layer on turbine blades is of the turbulent type. It is full of eddies of all sizes which interact in ways that even today we are just beginning to understand.

Sometimes, useful insights can come from applying the basic principles of logic and physics, even when little or nothing is known about the nature of the phenomenon. That is how the pitifully rudimentary snippets of knowledge about turbulent boundary layers had been obtained. Long months of work, however, yielded no more than a convincing argument why this approach could not give any credible indication of the prospect for economic viability of cold-air injection to cool the turbine blades. Clearly, this was one of those projects that was producing frustration, rather than results.

So, the key question was changed to a less ambitious one: what is the minimally required information needed to get even a half-credible guess at the prospects for this cooling method?

Do the basic principles of logic and physics always result in success? Not always. To solve this one in a proper scientific manner might well take decades. Although these tools are a good start, we often need another path. By accident, the author turned to a modern management tool.

Stealing Puzzles From Traditional Management

Guessing the relevance of connections between, and the implications of, many disparate bits of information is not only the task of engineers and inventors. A business manager gets a daily flood of pieces of information calling for decisions, and his or her success depends on making decisions correctly and on time.

A jigsaw puzzle presents a problem analogous to both the turbine problem and problems regularly faced by business managers: where to start; what to do next with all those pieces? Most people start to solve a jigsaw puzzle by looking at the top of the box to get a clear picture of what they are trying to recreate. The clearer the picture, the more easily and quickly the puzzle can be completed. A model of the finished puzzle allows a person or team to decide where a particular piece belongs and which pieces to work on next. It is also easier to identify pieces that do not belong in the puzzle and can be set aside.

For any work group is faced with a puzzle, it will usually seem much easier to sort out the puzzle if they can visualize a "boxtop" to guide them. In development you have a clear view of many objectives. You can use that view much like the picture on the top of the box the puzzle came in that shows the completed puzzle. That speeds up the process of discarding irrelevant pieces of information and of locating the part of the puzzle where a relevant piece may fit.

This form of jigsaw management works for administration and for many business issues. Basic research is different.

In research and invention there is no boxtop to build from. The visualization process that works for overall management does not work in creative endeavors like these. The puzzles encountered in research and invention resemble the challenge presented by a jigsaw puzzle when the top of the box has been hidden away before the pieces are dumped on the table.

When this happens, intuition becomes the key. Intuition causes one to look at the pieces for likely corners or parts of an edge. These promise a framework and reference for other pieces. From them, you can work inwards to approach a first glimmer of the picture to be recreated. It sparks a more confident process of trial and error to accelerate the solution of a puzzle with a large number of pieces.

The Minimalist's Puzzle

The ultimate jigsaw puzzle comes in a box from which the boxtop and many pieces have been lost. Is it still possible to infer what kind of picture it once was? This gives you another way of attacking a problem that does not resolve itself neatly with the scientific method. The scientist was reminded of this when the strictly scientific approach to cooling turbine blades failed. It had failed, not from lack of patience, but from lack of information.

Such a puzzle prompts a search for pieces which look like corners of parts of edges. This will not replace the normal process to resolution for research problems, but may be useful as a step along the path to the answer. Starting with jigsaw pieces gives you another way of attacking a problem that does not resolve itself neatly with the scientific method.

For the cooling of blades, parts of two puzzle edges were known. One edge was still a question: how does the amount of cold-air injection per unit of surface area depend on the gas speed and on the temperature of the gas and steel? The opposite edge included the few facts known from basic physics.

However, arranging and rearranging the other pieces still failed to give enough of a picture. It always left too many crucial holes. The jigsaw analogy now suggested a new question: did one really need the whole puzzle? What might be all that is required to solve the problem? This point of view shifted attention to a more modest aim: tracing a bridge of information from one edge to the other.

Conclusion: Progress On the Third Try

The bad news was that all potential bridges failed to cross from edge to edge with a logical flow. These failures, however, gave a strong clue to the minimal amount of missing information. With this new aim, the scarceness of information had an advantage: it limited the options.

Could those missing pieces be guessed? Exploration revealed a crooked path across the puzzle along which widely different guesses made little difference to the answer. It showed convincingly that the required amount of cold air must be quitesmall. That was all the information really necessary. This concept of an economically viable engine is realistic. A basis now existed for the next development step of planning an experiment.

The jigsaw principle is a useful tool for focusing the searcher's view on the crucial questions. It allows him or her to think laterally without going too far astray. It accelerates the sorting of information. If a piece adds to the picture, one may see that immediately; if not, the next piece may make the fit clearer. In the gas-turbine problem, jigsaw management helped find a solution, despite a series of frustrating failures, by focusing attention on the missing and indispensable information.

There are many problems that are nicely solved by the traditional scientific method. Sometimes that method does not work, and sticking with it lengthens the time to solution. A way of limiting options is needed. Playing with puzzles is the metaphor that helped solve this quandary. As part of the overall process, it can give a clear view of the critical gaps in what you know. That can save you time and effort.

(R. E. Meyer, The Boundary-layer Cooling of a Flat Plate, Brit. Aeron. Res. Counc. Rep. & Mem. 2420 (1946), H. M. Stat. Office, London.

Copyright, 1996 by the Meyer Group, all rights resv'd. For more information, please call (408) 439-9607