Amusements in Mathematics HTML version

Mr. Wilson put the case so very simply that the three men saw how it might happen
without any marriage within the prohibited degrees. Perhaps the reader can work it out
for himself.
"Look at the clock!"
Ingoldsby Legends.
In considering a few puzzles concerning clocks and watches, and the times recorded by
their hands under given conditions, it is well that a particular convention should always
be kept in mind. It is frequently the case that a solution requires the assumption that the
hands can actually record a time involving a minute fraction of a second. Such a time, of
course, cannot be really indicated. Is the puzzle, therefore, impossible of solution? The
conclusion deduced from a logical syllogism depends for its truth on the two premises
assumed, and it is the same in mathematics. Certain things are antecedently assumed, and
the answer depends entirely on the truth of those assumptions.
"If two horses," says Lagrange, "can pull a load of a certain weight, it is natural to
suppose that four horses could pull a load of double that weight, six horses a load of three
times that weight. Yet, strictly speaking, such is not the case. For the inference is based
on the assumption that the four horses pull alike in amount and direction, which in
practice can scarcely ever be the case. It so happens that we are frequently led in our
reckonings to results which diverge widely from reality. But the fault is not the fault of
mathematics; for mathematics always gives back to us exactly what we have put into it.
The ratio was constant according to that supposition. The result is founded upon that
supposition. If the supposition is false the result is necessarily false."
If one man can reap a field in six days, we say two men will reap it in three days, and
three men will do the work in two days. We here assume, as in the case of Lagrange's
horses, that all the men are exactly equally capable of work. But we assume even more
than this. For when three men get together they may waste time in gossip or play; or, on
the other hand, a spirit of rivalry may spur them on to greater diligence. We may assume
any conditions we like in a problem, provided they be clearly expressed and understood,
and the answer will be in accordance with those conditions.
"I say, Rackbrane, what is the time?" an acquaintance asked our friend the professor the
other day. The answer was certainly curious.
"If you add one quarter of the time from noon till now to half the time from now till noon
to-morrow, you will get the time exactly."
What was the time of day when the professor spoke?