A group of scientists were doing an investigation into problem-solving techniques, and constructed an experiment involving a physicist, an engineer, and a mathematician. The experimental apparatus consisted of a water spigot and two identical pails, one of which was fastened to the ground ten feet from the spigot.

Each of the subjects was given the second pail, empty, and told to fill the pail on the ground.

The engineer was the first subject: he carried his pail to the spigot, filled it there, carried it full of water to the pail on the ground, and poured the water into it. Standing back, he declared, “There: I have solved the problem.”

The physicist and the mathematician each approached the problem similarly. Upon finishing, the physicist noted that the solution was exact, since the volumes of the pails were equal. The mathematician merely noted that he had proven that a solution exists.

Now, the experimenters altered the parameters of the task a bit: the pail on the ground was still empty, but the subjects were presented with a pail that was already half-filled with water.

The engineer immediately carried his pail over to the one on the ground, emptied the water into it, went back to the spigot, filled the pail, and finally emptied the entire contents into the pail on the ground, overflowing it and spilling some of the water. Upon finishing, he commented that the problem should have been better stated.

The physicist, in turn, thought for some time before going into action. He then took his half-filled pail to the spigot, filled it to the brim, and filled the pail on the ground from it. He noted that the problem had an exact solution, which he had found.

The mathematician thought for a long time before stirring. At last he stood up, emptied his pail onto the ground, and declared, “I have reduced it to the previous problem.”