y = y x
(and other puzzles)
1. The locus of
points satisfying the relation x
y = y x (x, y > 0)
two components. One is the straight line y = x. At
point does the other
component cross the line y = x
- I was told this puzzle about 15 years ago, and have not found a
reference to it on Google. It goes something like this...
upon a time the potentate
of a faraway land issued
a decree compelling subjects to pay an annual tax of
grain. The measure of
grain required from each subject was
a number in weight units
equal to their age in years.
At the time of weighing, the tax
assessor (a cousin of the ruler) selected one to three stones from a
special set of seven
sacred stones--never more
than three, because “three”
was also a sacred number. The subject's grain was placed on one side of
an equal-arm balance and
up to three stones of different whole number weights were
the opposite side or else were distributed between the two sides of the
balance. Thus, each subject’s
obligation could be measured in a single weighing.
Brutal though it may seem, subjects were
taxed from the tender age of one! To measure out one unit, the grain
would be placed on
one side of the balance; then a stone weighing one unit--whatever it
was--could be placed on the
other side. To measure
two units by weight of grain, the tax collector might place a
stone weighing one unit on the same side as the grain, and a stone
weighing three units on the other side, 3 − 1 = 2. --These are just
seven stones were very special indeed. Intent upon exacting
payment from the eldest subject, as well as from infants in arms, the
mathematically inclined ruler had
worked out seven specific weights to ensure the
stones could be used for as many years of life
as possible, with no gaps.
What was the maximum age at which a grain
obligation could be weighed and what weights were the seven
stones? --The solution consists of 8 numbers, one age and seven weight
values. (my answer)
The first puzzle should not require a calculator or
computer. The second is
essentially a programming
challenge. I do not know how it could be solved without
aid of a computer, but perhaps there is a way.
3. This puzzle
by Albert A. Mullin appeared in the February 1974 issue of The American Mathematical Monthly.
I am changing the wording a little. “How many perfect 1-ohm resistors
are needed to construct a series-parallel resistor network
of π ohms
to six decimal places accuracy?”
4. One of the earliest puzzles I
remember being told is the following. (The
content somewhat dates it.) “Mary, Jane, and Alice were blond,
and red-haired, but not necessarily in that order. Of these three
statements only one is true: Mary was blond. Jane was not blond. Alice
was not red-haired. What color is each girl’s
5. This is problem number 1 in
The USSR Olympiad Problem Book (W. H. Freeman, 1962). “Every living
person has shaken hands with a certain number of other persons. Prove
that a count of the number of people who have shaken hands an odd
number of times must yield an even number.”
illustration on left. This puzzle is easy to make using a
3-hole punch ruler or a stick of similar size, as shown.
The object is to move the washer on the right across to the
so that both washers are on the same loop (illustration on right). Or the
reverse, move the washer on the left loop over to the right-hand loop—this
is equally easy or not!
Or, start with both washers on the same side (the goal) and
move one of them to the other loop, solving the puzzle in reverse.
Scissors and glue are prohibited, similarly, pocket knives,
razor blades or anything else that could be used to cut the string.
It is not okay to saw through the ruler (or stick, as the case may be).
Finally, both ends of the string are knotted in the back and must remain
Hint: The washers do not fit through the hole in the middle of the stick.
The above is a
start. I hope to add a few interesting puzzles along and along.