Henry Ernest Dudeney/Modern Puzzles/Unicursal and Route Problems

Henry Ernest Dudeney: Modern Puzzles: Unicursal and Route Problems

$154$ - The Way to Tipperary

The popular bard assures us that "it's a long way to Tipperary."

Look at the accompanying chart and see if you can discover the best way from London to "the sweetest girl I know."

The lines represent stages from town to town, and it is necessary to get from London to Tipperary in an even number of stages.

You will find no difficulty getting there in $3$, $5$, $7$, $9$ or $11$ stages,

but these are odd numbers and will not do.

The reason that they are odd is that they all omit the sea passage, a very necessary stage.

If you get to your destination in an even number of stages, it will be because you have crossed the Irish Sea.

Which stage is the Irish Sea?

$155$ - Marking a Tennis Court

The lines of our tennis court are faint and want re-marking.

My marker is of such a kind that, though I can start anywhere and finish anywhere,

it cannot be lifted off the line when working without making a mess.

I therefore have to go over some of the lines twice.

Where should I start and what route should I take, without lifting the marker,

to mark the court completely and yet go over the minimum distance twice?

I give the correct proportions of a tennis court in feet.

What is the best route?

$156$ - Water, Gas and Electricity

It is required to lay on water, gas and electricity from $W$, $G$ and $E$ to each of the three houses $A$, $B$ and $C$, without any pipe crossing another.

Take your pencil and draw lines showing how this should be done.

You will soon find yourself landed in difficulties.

$157$ - Crossing the Lines

You are asked to draw the diagram of Figure $1$ (exclusive of the little crosses) with three continuous strokes of the pencil,

without removing the pencil from the paper during a stroke, or going over a line twice.

As generally understood, it is quite impossible.

Wherever I have placed a cross there is an "odd node", and the law for all such cases is that half as many lines will be necessary as there are odd nodes --

that is, points from which you can depart in an odd number of ways.

Here we have, as indicated, $8$ odd nodes, from each of which you can proceed in three directions (an odd number),

and therefore, four lines will be required.

But, as I have shown in my book of Amusements in Mathematics, it may be solved by a trick, overriding the conditions as understood.

You first fold the paper, and with a thick lead-pencil draw $CD$ and $EF$, in Figure $2$, with a single stroke.

Then draw the line from $A$ to $B$ as the second stroke, and $GH$ as the third!

During the last few years this puzzle has taken a new form.

You are given the same diagram and asked to start where you like and try to pass through every short line comprising the figure,

once and once only, without crossing your own path.

Figure $3$ will make quite clear what is meant.

It is an attempted solution, but it fails because the line from $K$ to $L$ has not been crossed.

We might have crossed it instead of $KM$, but that would be no better.

Is it possible?

Many who write to me about the puzzle say that though they have satisfied themselves as a "pious opinion", that it cannot be done,

yet they see no way whatever of proving the impossibility, which is quite another matter.

I will show my way of settling the question.

$158$ - The Nine Bridges

The diagram represents the map of a district with a peculiar system of irrigation.

The lines are waterways enclosing the four islands $A$, $B$, $C$, and $D$, each with its house,

and it will be seen that there are nine bridges available.

Whenever Tompkins leaves his house to visit his friend Johnson, who lives in one of the others,

he always carries out the eccentric rule of crossing every one of the bridges once, and once only,

before arriving at his destination.

How many different routes has he to select from?

You may choose any house you like as the residence of Tompkins.

$159$ - The Five Regiments

The diagram represents a map of a certain district.

The dots and circles are towns and the lines are roads.

During a war five regiments marched to new positions in the night.

The body stationed at the upper $A$ marched to the lower $A$,

that at the upper $B$ to the lower $B$,

that at the upper $C$ to the lower $C$,

that at the upper $D$ to the lower $D$,

and the regiment at the left-hand $E$ marched to the right-hand $E$.

Yet no regiment saw anything of any other regiment.

Can you mark out the route taken by each so that no two regiments ever go along the same road anywhere?

$160$ - Going to Church

A man living in the house shown as $H$ in the diagram wants to know what is the greatest number of different routes by which he can go to the church at $C$.

The possible roads are indicated by the lines, and he always walks either due $N$, due $E$, or $N.E.$;

that is, he goes so that every step brings him nearer to the church.

Can you count the total number of different routes from which he may select?

$161$ - A Motor-Car Puzzle

A traveller starts in his car from the point $A$ and wishes to go as far as possible while making only $15$ turnings, and never going along the same road twice.

The dots represent towns and are one mile apart.

Supposing, for example, that he went straight to $B$, then straight to $C$, then to $D$, $E$, $F$, and $G$,

then you will find that he has gone $37$ miles in five turnings.

How far can he go in $15$ turnings?

$162$ - The Fly and the Honey

I have a cylindrical cup four inches high and six inches in circumference.

On the inside of the vessel, one inch from the top, is a drop of honey,

and on the opposite side of the vessel, one inch from the bottom on the outside, is a fly.

Can you tell exactly how far the fly must walk to reach the honey?

$163$ - The Russian Motor-Cyclists

Two Russian Army motor-cyclists, on the road $A$, wish to go to $B$.

Now Pyotr said: "I shall go to $D$, which is $6$ miles, and then take the straight road to $B$, another $15$ miles.

But Sergei thought he would try the upper road by way of $C$.

Curiously enough, they found on reference to their odometers that the distance either way was exactly the same.

This being so, they ought to have been able easily to answer the General's simple question,

"How far is it from $A$ to $C$?"

it can be done in the head in a few moments, if you only know how.

Can the reader state correctly the distance?

$164$ - Those Russian Cyclists Again

In the section from a map given in the diagram we are shown three long straight roads, forming a right-angled triangle.

The General asked the two men how far it was from $A$ to $B$.

Pyotr replied that all he knew was that riding round the triangle, from $A$ to $B$,

from there to $C$ and home to $A$, his odometer showed exactly $60$ miles,

while Sergei could only say that he happened to know that $C$ was exactly $12$ miles from the road $A$ to $B$ --

that is, to the point $D$, as shown by the dotted line.

Whereupon the General made a very simple calculation in his head and declared that the distance from $A$ to $B$ must be ...

Can the reader discover so easily how far it was?

$165$ - The Despatch-Rider in Flanders

A despatch-rider on horseback, somewhere in Flanders, had to ride with all possible speed from $A$ to $B$.

The distances are marked on the map.

Now, he can ride just twice as as fast over the soft turf (the shaded bit) as he can ride over the loose sand.

Can you show what is the quickest possible route for him to take?