A recent puzzler I heard on Car Talk (NPR) prompted me to consider puzzles and decisions that are based on negative, or missing evidence and information present, but not always obvious. Negative evidence is often overlooked as an important clue. For that reason, many puzzles have been written based on this blindness. The puzzler I heard goes something like this: A warden lines up three prisoners all facing one way. They see a table with two black hats and three white hats on it. The warden tells them he will blindfold them and place a hat on each prisoner. Then without allowing them to turn and look at the others, he will ask if anyone knows what color hat he has on. A correct answer will result in immediate release and a false response will result in solitary confinement for a long time. No response is okay and has no consequences.

The hats are placed on the prisoners and blindfolds removed. The warden asks the last prisoner in line (the one who can see the back of the other two prisoner’s heads) if he knows what color hat he has. He responds, “No, I do not know.” Then the warden asks the nest prisoner (who can only see the back of the head of the first prisoner, but who heard the previous response) the same question. He also responds, “No, I do not know.” The warden then asks the last prisoner, the one who is staring at a wall and cannot see either of the others even though he can hear them, and he responds, “Yes, I know the color of my hat.”

What is it and how did he know?

Forgetting the unfortunate premise of this puzzle (I kept the format as I heard it while driving), this is a variation of an old puzzle with many siblings. In this form, it is particularly simple to solve. I did it without pencil and paper while driving. But the way the winning prisoner was able to use the lack of conclusion by the other two is part of what I mean by negative evidence.

Sometimes the puzzle takes a different form. Many years ago I had a plasma measuring instrument on a spacecraft and needed to select many different operating modes to take advantage of changing ambient conditions. However, the spacecraft was only able to provide me with two impulse commands for control. I could easily use 1000. The value of my data would be compromised if we could not utilize the flexibility built into the instrument. Our solution was to realize that while each command could only mean one thing, receiving a command of either type was a totally different type of information. So we designated one command as “zero” and the other as “one”. Then we built a shift register to store commands as they came in. The register would shift one space each time it received a command of either type, simultaneously it entered the value of the incoming command. In this way we could build up any binary number we wanted. There were a few complications like resetting the register after some time so new commands could be entered, but it worked fine. Of course that was in the bad old days when even a lunar module had less computing power than a smartphone.

But a more direct use of negative evidence is how I was able to win at Clue.

More than fifty years ago, my friends and family liked to play Clue. They would stumble about to various rooms and make the usual suggestions: Colonel Mustard did it in the kitchen with a rope. Then proof that supposition was inaccurate would be shown by some other player. For a long time, it appeared that I was the only one who realized that failure to disprove a hypothesis by another player gave me valuable information about what that person was holding by eliminating cards from the possibilities. Similarly if another player disproved something, then I knew they held at least one of the three relevant cards. Combining these sources of information often gave me an advantage and led to victory, and a switch to Monopoly as the game of choice by my family and friends.

Back to the prisoner. He was wearing white. Did you get the answer?