Henry Ernest Dudeney/Modern Puzzles/Unclassified Problems

Henry Ernest Dudeney: Modern Puzzles: Unclassified Problems

$212$ - A Chain Puzzle

A man has $80$ links of old chain in $13$ fragments, as shown in the diagram.

It will cost him $1 \oldpence$ to open a link and $2 \oldpence$ to weld one together again.

What is the lowest price it must cost him to join all the pieces together so as to form an endless chain?

A new chain will cost him $3 \shillings$ (that is, $36 \oldpence$)

What is the cheapest method of procedure?

Remember that the small and large links must run alternately.

$213$ - The Six Pennies

Lay six pennies on the table, and arrange them as shown by the $6$ white circles in the diagram,

so that a seventh penny can be dropped into the centre and touch each of the other $6$.

It is required to get it exact, without any dependence on the eye.

In this case it is not allowed to lift any penny off the table, nor can any measuring or marking be employed.

However, you are allowed to slide the pennies after they have been placed on the table.

You require only the six pennies.

$214$ - Folding Postage Stamps

Take a $4 \times 2$ sheet of $8$ postage stamps, labelled $1$ to $8$, as shown in the diagram.

It is an interesting exercise to count how many ways they may be folded up so they will all lie under the one stamp, as shown.

There are in fact $40$ ways to do this so that No. $1$ is always on the top.

Numbers $5$, $2$, $7$ and $4$ will always be face down.

You can always arrange for any stamp except No. $6$ to lie next to $1$,

although there are only two ways each in which $7$ and $8$ can be made to lie in that position.

They can be folded in the order $1$, $5$, $6$, $4$, $8$, $7$, $3$, $2$ and also $1$, $3$, $7$, $5$, $6$, $8$, $4$, $2$, with $1$ at the top face upwards,

but it is a puzzle to work out how.

Can you fold them like that without tearing any of the perforations?

$215$ - An Ingenious Match Puzzle

Place $6$ matches as shown, and then shift just one match without touching the others so that the new arrangement shall represent a fraction equal to $1$.

The match forming the horizontal fraction bar must not be the one moved.

$216$ - Fifty-Seven to Nothing

Place $6$ matches as shown, so as to represent the number $57$ in Roman numerals.

Remove and replace any $2$ of them (without disturbing the others) to make an expression representing zero ($0$).

There are two completely different solutions.

$217$ - The Five Squares

$12$ matches are arranged to form $4$ squares.

Rearrange the same matches (so they lie flat on the table) to make $5$ squares.

All the squares must be empty.

$218$ - A Square with Four Pennies

Can you place four (old) pennies together so as to show a square?

They must all lie flat on the table.

$219$ - A Calendar Puzzle

I have stated in my book, Amusements in Mathematics, that, under our present calendar rules,

the first day of a century cannot fall on a Sunday or a Wednesday or a Friday.

I am frequently asked the reason why.

Try to explain the mystery in as simple a way as possible.

Note that $1$st January $1901$ was the first day of the $20$th century, not $1900$.

$220$ - The Fly's Tour

I had a ribbon of paper, divided into squares on each side.

I joined the ends together to make a ring, and tossed it down onto the table.

Then I watched a fly land on the ring and walk in a line over every one of those squares on both sides,

returning to the point where it started, without ever passing over the edge of the paper.

Its course passed through the centre of the squares all the time.

How was this possible?

$221$ - A Musical Enigma

Here is an old musical enigma that has been pretty well known in Germany for some years.

$222$ - A Mechanical Paradox

A remarkable mechanical paradox, invented by James Ferguson about the year $1751$, ought to be known by everyone, but, unfortunately, it is not.

It was contrived by him as a challenge to a sceptical watchmaker during a metaphysical controversy.

"Suppose," Ferguson said, "I make one wheel as thick as three others and cut teeth in them all,

and then put the three wheels all loose upon one axis and set the thick wheel to turn them,

so that its teeth may take into those of the three thin ones.

Now, if I turn the thick wheel round, how must it turn the others?"

The watchmaker replied that it was obvious that all three must be turned the contrary way.

Then Ferguson produced his simple machine, which anyone can make in a few hours,

showing that, turning the thick wheel which way you would,

one of the thin wheels revolved in the same way, the second the contrary way, and the third remained stationary.

Although the watchmaker took the machine away for careful examination, he failed to detect the cause of the strange paradox.