Working Backwards

I came across an interesting problem in the book How to Solve It by G.Polya.

How can you bring up from the river exactly six quarts of water when you have only two containers, a four quart pail and a nine quart pail, to measure with?

Given below are the containers. Notice that they are not marked with any scales. All you can measure is either 9 quarts or 4 quarts and with that you need to measure 6 quarts. Let us assume that the river has plenty of water.

Most of us will try to solve the problem by working forward. We start with the two empty containers, we try this and that, we empty and fill, and when we do not succeed, we start again, trying something else. We might succeed, after several attempts, accidentally. Even though this approach might give us a solution we will not learn anything from it. Why? Real learning happens when we extract the pattern used for solving the problem and store it in our repertoire so that it can be used for solving similar problems.

There is another way to solve the same problem.

But exceptionally able people, or people who had the chance to learn in their mathematics classes something more than mere routine operations, do not spend too much time in such trials but turn around, and start working backwards.

In this method you start with the final solution. The image given below shows that.

From what situation could we obtain the final solution. The image given below will lead you to the final solution. All you need to do is to pour 3 quarts of water from 9-quart to 4-quart container and empty the 4-quart container.

From what situation could we go the pervious state. The image given below will lead you to that state. All you need to do is to pour 1 quart of water from 9-quarts container to 4-quarts container and fill the 9-quarts container.

From what situation could we go to the previous state. Fill the 9-quarts containers and empty it twice on the 4-quarts container. That’s it we solved the problem. Working backwards is a common sense procedure within the reach of everybody. This technique is not limited to mathematics but can be applied in several domains; including our life.

During early 90′s Jeff Bezos had an idea of starting an online bookstore. At that time he was working at D.E.Shaw as a vice president. To pursue his idea he had to leave his high paying job. At that time not many people knew about the Internet. Status Quo would have prevented most of us to not leave the high paying job. Jeff being a smart guy worked backwards. From the book The Everything Store: Jeff Bezos and the Age of Amazon

When you are in the thick of things, you can get confused by small stuff”, Bezos said a few years later. “I knew when I was eighty that I would never, for example, think about why I walked away from my 1994 Wall Street bonus right in the middle of the year at the worst possible time. That kind of thing just isn’t something you worry about when you’re eighty years old. At the same time, I knew that I might sincerely regret not having participated in this thing called the Internet that I thought was going to be a revolutionizing event. When I thought about it that way … it was incredibly easy to make the decision.

Maurice Ashley is a chess grandmaster. He uses working backwards technique to solve chess problems. Watch this excellent video in which he explains the benefits of working backwards.

In the book A Study in Scarlet; Sherlock Holmes talks about working backwards

Most people, if you describe a train of events to them, will tell you what the result would be. They can put those together in their minds, and argue from them that something will come to pass. There are few people, however, who, if you told them a result, would be able to evolve from their own inner consciousness what the steps were which led up to that result. This power is what I mean when I talk of reasoning backward, or analytically.

In the USC Law School Commencement speech; Charlie Munger tells about the importance of working backwards

In life, unless you’re more gifted than Einstein, inversion will help you solve problems. Let me use a little inversion now. What will really fail in life? What do you want to avoid? Such an easy answer: sloth and unreliability. If you’re unreliable it doesn’t matter what your virtues are. Doing what you have faithfully engaged to do should be an automatic part of your conduct. You want to avoid sloth and unreliability.

Smart people like George Polya, Jeff Bezos, Maurice Ashley, Sherlock Holmes (fictional), and Charlie Munger uses this technique for solving problems. This means that we should also use it for solving problems.