A group of scientists were doing an investigation into problem-solving techniques,

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 physicist 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 engineer and the mathematician each approached the problem similarly.
Upon finishing, the engineer 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 physicist 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 engineer, 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. Again he noted that the problem had
an exact solution, which of course 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, "The problem has
been reduced to one already solved."