Shortcuts in Thinking

Shortcuts in Thinking

Reading Time: 3 minutes

Some Fun Questions


Consider these easy questions quickly:

  • 1.) If it takes 5 machines to make 5 widgets in 5 minutes, how long would it take 100 machines to make 100 widgets?
  • _____Minutes
  • 2.) A baseball bat and ball cost $1.10 when purchased. The bat costs $1.00 more than the ball. How much does the ball cost?
  • _____ Cents
  • 3.) In a lake there is a patch of lily pads. Each day the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long does it take for the patch to cover 1/2 of the lake?
  • _____ Days

What did you notice about the answers?

Answers—Click to Reveal

  • 1.) 5 Minutes
  • 2.) 5 Cents
  • 3.) 47 Days

The above were simple logic problems that many people get wrong if considered only briefly, yet when a bit more effort is applied, arriving at the truth is not difficult. Why is this? When we quickly consider these questions, our minds tend to use bad logic. We look for quick associations instead of deeply considering the underlying problem. This mental laziness is often fine for quick reactions, but it is also often a poor way to arrive at truth. Let’s consider this, but now look at saving lives.


Winning WWII


One of the most remarkable facts about the universe is that it’s logical. There are fundamental constants that allow us to utilize reasoning to successfully advance technology. This same property of being rational is something that we should harness in our daily lives, and mathematical thinking can help that. As the saying goes, “You can’t fake math.”

In fact, math can be described as the science of not being wrong. While it’s easy to use intuition to figure things out, it often behooves us to take more time to think deeply, using rational, mathematical thinking. 

In World War II, thousands of American planes were shot down. A special research group in New Jersey was assigned the task of making airplanes better able to withstand attacks and to make it back safely. The problem was simple: armor protects airplanes from attacks, but incurs speed and maneuverability penalties, leaving the airplanes more vulnerable to being attacked.

To decide, they studied where the most shots were received on the airplanes.  Every time an airplane returned, personnel counted the number of bullet holes for the various parts of the planes. The results were as seen in the attached picture.  The group needed to figure out where armor should be placed on the airplanes, and where it could be removed.  Places that received no hits wouldn’t need armor, and places that received lots of hits should be more armored.

With the information in hand, they approached Abraham Wald, a brilliant mathematician, to see how much armor should go on the places with all the bullet holes. The answer they got wasn’t what they expected. Not at all.

Wald said not to put the armor where all the bullet holes were. Instead, he said to put it where the bullet holes WERE NOT.

Wald’s thought was this: where are the missing holes? The missing holes were on the missing planes. The reason that planes were coming back with fewer hits in the engines was because most planes hit in the engines weren’t coming back. The fact that planes were coming back with main bodies that were torn to shreds showed that they could take those hits and make it.

The quick, intuitive thought by most at first was that Germans were simply hitting the engines far less.  Again we see how it pays to take a moment to reason through data. 

“Come now and let us reason together, God.” (Isiah 1:18) 


A Rational, Reasonable Faith


When considering my faith, I want it to be anything but blind in regards to Logos (the Hebrew God). In fact, I want it to be both logical and reasonable, and I don’t want it to be weak and poorly considered. We don’t gain anything by having weak logic and sloppy reasoning for our points of view.

When you consider Logos, be willing do dive deeply into why you think whatever “hunches” you might have are correct. Utilize the mind God gave you to fulfill the suggestion in Isaiah 1 to “‘Come now, and let us reason together,’ Says the Lord.” Just like with the simple word problems above, a bit more meditation on the subject might give you a much better result.

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