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World's Simplest Impossible Problem

Don Morrison, founder of New Mexico's C.S. Department, is credited with posing the "World's Simplest Impossible Problem."

The Problem: "I'm thinking of two numbers. Their average is 3. What are the numbers?"

What are your answers?

Cleve Moler, co-founder of The MathWorks, once discussed this problem, noting that some possible answers are 3 and 3, 4 and 2, 6 and 0, 23 and -17, or 2.71828 and 3.28172.

Then, Moler set up the software program MATLAB to solve the problem, using the marix equation A*x = b where A = [1/2 1/2] and b = 3. What do you think the technology produced?

First, using MATLAB's forward backslash, he typed x = A\b. The resulting solutions were 6 and 0.

Second, using the matrix inverse approach, he typed x = pinv(A)*b. The resulting solutions were 3.000 and 3.000.

To Moler and Morrison, this problem is simple to state but is impossible because it is ill-posed and does not have a unique answer. Yet, I find it interesting that technology is still able to produce answers.

Moler closes with this thought: "Now I'm thinking of three numbers whose average is pi. What are the three numbers?"

Source: "Cleve's Corner," MathWorks Newsletter, Dec. 1990.