x^{
y} = y^{ x
}

(and other puzzles)

1. The locus of points satisfying the relation x^{
y} = y^{ x} (x, y > 0)
has
two components. One is the straight line y = x. At
what
point does the other
component cross the line y = x
?

2. Seven sacred stones - I was told this puzzle about 15 years ago, and have not found a reference to it on Google. It goes something like this...

Once 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 placed on 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 examples.

The 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 the 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, brunette, 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 hair?”

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.”

6. See 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 left, 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.

Rules: 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 so.

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.

1. The locus of points satisfying the relation x

2. Seven sacred stones - I was told this puzzle about 15 years ago, and have not found a reference to it on Google. It goes something like this...

Once 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 placed on 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 examples.

The 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 the 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, brunette, 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 hair?”

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.”

6. See 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 left, 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.

Rules: 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 so.

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.