nyuuzyou/wikis huggingface dataset:
Henry Ernest Dudeney/Puzzles and Curious Problems/260 - Stepping Stones

== by : $260$ ==
;Stepping Stones
:The diagram represents eight stepping-stones across a stream.
:
:The puzzle is to start from the lower bank and land twice on the upper bank (stopping there),
::having returned once to the lower bank.
:But you must be careful to use each stepping-stone the same number of times.
:In how few steps can you make the crossing?
:Make the steps with two fingers in the diagram, and you will see what a very simple matter it is.
:Yet it is more than likely that you will at first take a great many more steps than are necessary.
=== Click here for solution ===
== Sources ==
* : Moving Counter Problems: $260$. -- Stepping Stones
* : Unclassified Puzzles: $522$. Stepping Stones
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/260 - Stepping Stones/Solution

== by : $260$ ==
;Stepping Stones
== Solution ==
Number the stones as follows:
:
Then go:
:$1$, $(\text {bank})$, $1$, $2$, $3$, $(2)$, $3$, $4$, $5$, $(4)$, $5$, $6$, $7$, $(6)$, $7$, $8$, $\text {bank}$, $(8)$, $\text {bank}$
which takes $19$ steps.
Those in brackets are backward steps.
== Sources ==
* : Solutions: $260$. -- Stepping Stones
* : Answers: $522$. Stepping Stones
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/261 - The Twenty-Two Bridges

== by : $261$ ==
;The Twenty-Two Bridges
:We have a rough map of a district with an elaborate system of irrigation,
::as the various waterways and numerous bridges will show.
:
:A man set out from one of the lettered departments to pay a visit to a friend living in a different department.
:For the purpose of pedestrian exercise he crossed every one of the bridges once, and once only.
:The puzzle is to show in which two departments their houses are situated.
:It is exceedingly easy if you give it a few moments' thought.
:You must not go outside the borders of the diagram.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $261$. -- The Twenty-Two Bridges
* : Combinatorial & Topological Problems: Route & Network Puzzles: $419$. The Twenty-Two Bridges
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/261 - The Twenty-Two Bridges/Solution

== by : $261$ ==
;The Twenty-Two Bridges
== Solution ==
$C$ and $L$.
== Proof ==
Each department has an even number of bridges leading from it, except $C$ and $L$, which have $3$.
The underlying graph of this map is therefore traversable.
From Characteristics of Traversable Graph it follows that $C$ and $L$ are the start and end points.
Hence, for example:
:$C, G, F, C, B, A, D, H, E, I, H, J, K, L, M, G, I, F, B, E, F, I, L$
== Sources ==
* : Solution: $261$. -- The Twenty-Two Bridges
* : Answers: $419$. The Twenty-Two Bridges
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/262 - A Monmouth Tombstone

== by : $262$ ==
;A Monmouth Tombstone
:In the burial ground attached to , is this arrangement of letters on one of the tombstones.
$\qquad \begin{array} \\
E & I & N & E & R & N & H & O & J & S & J & O & H & N & R & E & N & I & E \\
I & N & E & R & N & H & O & J & S & E & S & J & O & H & N & R & E & N & I \\
N & E & R & N & H & O & J & S & E & I & E & S & J & O & H & N & R & E & N \\
E & R & N & H & O & J & S & E & I & L & I & E & S & J & O & H & N & R & E \\
R & N & H & O & J & S & E & I & L & E & L & I & E & S & J & O & H & N & R \\
N & H & O & J & S & E & I & L & E & R & E & L & I & E & S & J & O & H & N \\
H & O & J & S & E & I & L & E & R & E & R & E & L & I & E & S & J & O & H \\
O & J & S & E & I & L & E & R & E & \mathbf H & E & R & E & L & I & E & S & J & O \\
H & O & J & S & E & I & L & E & R & E & R & E & L & I & E & S & J & O & H \\
N & H & O & J & S & E & I & L & E & R & E & L & I & E & S & J & O & H & N \\
R & N & H & O & J & S & E & I & L & E & L & I & E & S & J & O & H & N & R \\
E & R & N & H & O & J & S & E & I & L & I & E & S & J & O & H & N & R & E \\
N & E & R & N & H & O & J & S & E & I & E & S & J & O & H & N & R & E & N \\
I & N & E & R & N & H & O & J & S & E & S & J & O & H & N & R & E & N & I \\
E & I & N & E & R & N & H & O & J & S & J & O & H & N & R & E & N & I & E \\
\end{array}$
:In how many different ways can these words "$\text{HERE LIES JOHN RENIE}$" be read,
:starting at the central $H$ and always passing from one letter to another that is contiguous?
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $262$. -- A Monmouth Tombstone
* : Combinatorial & Topological Problems: Route & Network Puzzles: $421$. A Monmouth Tombstone
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/262 - A Monmouth Tombstone/Solution

== by : $262$ ==
;A Monmouth Tombstone
== Solution ==
:$45 \, 760$, unless you are allowed to go diagonally, in which case it is $91 \, 520$.
== Proof ==
Let us count the routes on one quadrant of this tombstone.
The total number of routes is then found by multiplying this by $4$.
Thus:
$\qquad \begin{array} \\
\mathbf H & E & R & E & L & I & E & S & J & O \\
E & R & E & L & I & E & S & J & O & H \\
R & E & L & I & E & S & J & O & H & N \\
E & L & I & E & S & J & O & H & N & R \\
L & I & E & S & J & O & H & N & R & E \\
I & E & S & J & O & H & N & R & E & N \\
E & S & J & O & H & N & R & E & N & I \\
S & J & O & H & N & R & E & N & I & E \\
\end{array}$
Let the number of routes from the $H$ to the $E$ at bottom right be $N$.
It is clear by inspection that, in order to spell the epitaph, you need to step one to the right or one down in order.
There is one way to travel to each letter along the top and down the left hand side.
Suppose you are at a given letter $x$.
Then you can get there from the letter to the left or from the letter above.
Hence the number of ways to get to letter $x$ is the sum of the ways from the left or above.
Hence the number of routes to a given letter are given by the entries in Pascal's triangle with the $1$ at the top left corner.
There are $16$ steps to the bottom right, which is reached by going $9$ across and $7$ down.
Hence:
Hence the total number of ways the epitaph can be spelt is $4 \times 11 \, 440 = 45 \, 760$.
Suppose you are allowed to go diagonally.
Then when you get to the last $I$, you have a choice as to go either to the final $E$ in the corner, or backwards diagonally to the $E$ behind you.
Hence for every fully orthogonal route there is another using that diagonal.
Hence there are twice as many paths than orthogonal-only routes.
That is:
:$2 \times 45 \, 760 = 91 \, 520$
== Sources ==
* : Solution: $262$. -- A Monmouth Tombstone
* : Answers: $421$. A Monmouth Tombstone
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/263 - Footprints in the Snow

== by : $263$ ==
;Footprints in the Snow
:Four schoolboys, living respectively in the houses $A$, $B$, $C$, and $D$, attended different schools.
:After a snowstorm one morning their footprints were examined, and it was found that no boy had ever crossed the track of another boy,
::or gone outside the square boundary.
:
:Take your pencil and continue their tracks, so that the boy $A$ goes to the school $A$, the boy $B$ to the school $B$, and so on,
::without any line crossing another line.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $263$. -- Footprints in the Snow
* : Combinatorial & Topological Problems: Route & Network Puzzles: $420$. Footprints in the Snow
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/263 - Footprints in the Snow/Solution

== by : $263$ ==
;Footprints in the Snow
== Solution ==
:
== Sources ==
* : Solution: $263$. -- Footprints in the Snow
* : Answers: $420$. Footprints in the Snow
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/264 - The Fly's Tour

== by : $264$ ==
;The Fly's Tour
:A fly pitched on the square in the top left-hand corner of a chessboard,
::and then proceeded to visit every white square.
:
:He did this without ever entering a black square or ever passing through the same corner more than once.
:Can you show his route?
:It can be done in seventeen continuous straight courses.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $264$. -- The Fly's Tour
* : Combinatorial & Topological Problems: Route & Network Puzzles: $422$. The Fly's Tour
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/265 - Inspecting the Roads

== by : $265$ ==
;Inspecting the Roads
:A man starting from the town $A$, has to inspect throughout all the roads shown from town to town.
:Their respective lengths, $13$, $12$, and $5$ miles, are all shown.
:
:What is the shortest route he can adopt, ending his journey wherever he likes?
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $265$. -- Inspecting the Roads
* : Combinatorial & Topological Problems: Route & Network Puzzles: $423$. Inspecting the Roads
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/265 - Inspecting the Roads/Solution

== by : $265$ ==
;Inspecting the Roads
== Solution ==
The shortest route is $\text {ABCHCDEIEFGBHDIHGIFAG}$.
This traverses all roads once, except $CH$ and $EI$, which are traversed twice.
== Sources ==
* : Solution: $265$. -- Inspecting the Roads
* : Answers: $423$. Inspecting the Roads
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/266 - Railway Routes

== by : $266$ ==
;Railway Routes
:The diagram below represents a simplified railway system,
::and we want to know how many different ways there are of going from $A$ to $E$, if we never go twice along the same line in any journey.
:
:This is a very simple proposition, but practically impossible of solution until you have hit on some method of recording the routes.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $266$. -- Railway Routes
* : Combinatorial & Topological Problems: Route & Network Puzzles: $424$. Railway Routes
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/266 - Railway Routes/Solution

== by : $266$ ==
;Railway Routes
== Solution ==
There are $2501$ ways of going.
== Proof ==
We need consider only the routes from $B$ to $D$.
:$\begin{array} {rrrr}
\text{$1$-station journey} & \text {$1$ route} & \text {$2$ variations} & 2 \\
\text{$2$-station journey} & \text {$1$ route} & \text {$9$ variations} & 9 \\
\text{$3$-station journey} & \text {$2$ routes} & \text {$12$ variations} & 24 \\
\text{$4$-station journey} & \text {$5$ routes} & \text {$18$ variations} & 90 \\
\text{$5$-station journey} & \text {$4$ routes} & \text {$72$ variations} & 288 \\
\text{$6$-station journey} & \text {$14$ routes} & \text {$36$ variations} & 504 \\
\text{$8$-station journey} & \text {$22$ routes} & \text {$72$ variations} & 1584 \\
\end{array}$
The $1$-station route is direct to $D$.
The $2$-station route is $BCD$.
The $3$-station routes are $BCBD$ and $BDCD$.
The five $4$-station routes are $BDBCD$, $BDCBD$, $BCBCD$, $BCDCD$ and $BCDBD$.
Each is subject to a number of variations of the actual lines used.
For a journey of a given number of stations there is always the same number of variations, despite the route taken.
A $7$-station route is not possible.
== Sources ==
* : Solution: $266$. -- Railway Routes
* : Answers: $424$. Railway Routes
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/267 - A Motor-Car Tour

== by : $267$ ==
;A Motor-Car Tour
:A man started in a motor-car from town $A$, and wished to make a complete tour of these roads,''
::going along every one of them once, and once only.
:
:How many different routes are there from which he can select?
:Every route must end at the town $A$, from which you start,
::and you must go straight from town to town -- never turning off at crossroads.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $267$. -- A Motor-Car Tour
* : Combinatorial & Topological Problems: Route & Network Puzzles: $425$. A Car Tour
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/267 - A Motor-Car Tour/Solution

== by : $267$ ==
;A Motor-Car Tour
== Solution ==
:$264$.
== Proof ==
== Sources ==
* : Solution: $267$. -- A Motor-Car Tour
* : Answers: $425$. A Car Tour
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/268 - Mrs. Simper's Holiday Tour

== by : $268$ ==
;Mrs. Simper's Holiday Tour
:The diagram shows a plan, very much simplified, of a tour that Mrs. Simper proposes to take next autumn.
:It will be seen that there are $20$ towns, all connected by railway lines.
:Mrs. Simper lives at $H$, and wants to visit every other town once and once only, ending her tour at home.
:
:There are in fact $60$ possible routes she can select from, counting the reverse of a route as different.
:There is a tunnel between $N$ and $O$, and one between $R$ and $S$, but Mrs. Simper does not want to go through these.
:She also wants to delay her visit to $D$ as long as possible so as to meet a friend who lives there.
:The puzzle is to show Mrs. Simper the best route under these circumstances.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $268$. -- Mrs. Simper's Holiday Tour
* : Combinatorial & Topological Problems: Route & Network Puzzles: $426$. Mrs. Simper's Holiday Tour
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/268 - Mrs. Simper's Holiday Tour/Solution

== by : $268$ ==
;Mrs. Simper's Holiday Tour
:The diagram shows a plan, very much simplified, of a tour that Mrs. Simper proposes to take next autumn.
:It will be seen that there are $20$ towns, all connected by railway lines.
:Mrs. Simper lives at $H$, and wants to visit every other town once and once only, ending her tour at home.
:
:There are in fact $60$ possible routes she can select from, counting the reverse of a route as different.
:There is a tunnel between $N$ and $O$, and one between $R$ and $S$, but Mrs. Simper does not want to go through these.
:She also wants to delay her visit to $D$ as long as possible so as to meet a friend who lives there.
:The puzzle is to show Mrs. Simper the best route under these circumstances.
== Solution ==
Although there are $60$ routes altogether, only $8$ of them avoid both tunnels.
The route that complies with the conditions, avoids both tunnels, and delays the visit to $D$ till as late as possible, is:
:$HISTLKBCMNUQRGFPODEAH$
:
== Sources ==
* : Solution: $268$. -- Mrs. Simper's Holiday Tour
* : Answers: $426$. Mrs. Simper's Holiday Tour
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs

== by : $269$ ==
;Sixteen Straight Runs
:A commercial traveller started in his car from the point $A$ shown,
::and wished to go $76$ miles in $16$ straight runs, never going along the same road twice.
:The dots represent the towns and villages, and these are one mile apart.
:The lines show the route he selected.
:
:It will be seen that he carried out his plan correctly, but $6$ towns or villages were unvisited.
:Can you show a better route by which he could have gone $76$ miles in $16$ straight runs, and left only $3$ towns unvisited?
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $269$. -- Sixteen Straight Runs
* : Combinatorial & Topological Problems: Route & Network Puzzles: $427$. Sixteen Straight Runs
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs
Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs/Historical Note

== Historical Note on by : $269$ - Sixteen Straight Runs ==
The solution leaving $3$ towns unvisited was the one provided by .
The improved solution leaving just $1$ town unvisited was discovered by .
 presented it in his Mathematical Games column in , later republished in his $1975$ collection .
== Sources ==
* : Answers: $427$. Sixteen Straight Runs
* : $17$. Cooks and Quibble-Cooks
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs
Category:Historical Notes
Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs/Solution

== by : $269$ ==
;Sixteen Straight Runs
== Solution ==
=== Solution $1$ ===
=== Solution $2$ ===
== Also see ==
* $161$ - A Motor-Car Puzzle from his collection, which is a similar problem.
== Historical Note ==
== Sources ==
* : Solution: $269$. -- Sixteen Straight Runs
* : Answers: $427$. Sixteen Straight Runs
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs
Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs/Solution 1

== by : $269$ ==
;Sixteen Straight Runs
== Solution ==
:
== Historical Note ==
== Sources ==
* : Solution: $269$. -- Sixteen Straight Runs
* : Answers: $427$. Sixteen Straight Runs
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs
Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs/Solution 2

== by : $269$ ==
;Sixteen Straight Runs
== Solution ==
Here is a solution which leaves just one town unvisited:
:
== Historical Note ==
== Sources ==
* : Answers: $427$. Sixteen Straight Runs
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/269 - Sixteen Straight Runs
Henry Ernest Dudeney/Puzzles and Curious Problems/26 - Cross and Coins

== by : $26$ ==
;Cross and Coins
:Here is an interesting little puzzle that will probably keep the reader amused for some moments.
:Take any $11$ of the $12$ current coins of the realm,
::and using one duplicate coin, can you place the $12$ coins, one in each division of the cross,
::so that they add up to the same value in the upright and in the horizontal?
:
:To prevent any possible misunderstanding, the $12$ current coins are of the value of $\pounds 1$, $10 \shillings$, $5 \shillings$, $4 \shillings$, $2 \shillings 6 \oldpence$, $2 \shillings$, $1 \shillings$, $6 \oldpence$, $3 \oldpence$, $1 \oldpence$, $\tfrac 1 2 \oldpence$, $\tfrac 1 4 \oldpence$.
:Now remember, there must be $11$ coins all of different value and only one duplicate.
:You see, the puzzle is to determine which coin we shall duplicate,
:and then so to arrange them that both rows shall add up alike when the question arises which coin shall be placed at the crossing to be twice added?
=== Click here for solution ===
== Sources ==
* : Arithmetical and Algebraical Problems: Money Puzzles: $26$. -- Cross and Coins
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/26 - Cross and Coins/Solution

== by : $26$ ==
;Cross and Coins
== Solution ==
The solution given by is:
:
who then goes on to explain:
:The illustration shows a solution.
:Only one coin -- the farthing -- is repeated, and in both directions the sum is $\pounds 1, 2 \shillings 7 \tfrac 1 2 \oldpence$
:Readers may possibly ask how many different arrangements are possible, all correctly adding up to this amount.
:Well, the penny must, in every solution, remain in its present position, but the seven coins in the upright may be permuted in $2520$ ways.
:(the two farthings being regarded as indistinguishable),
:and the four coins in the horizontal may be permuted in $24$ ways.
:Therefore, $2520 \times 24 = 60 \, 480$ -- the number of different ways in which the coins may be arranged.
:But we can exchange the three coins $2 \shillings$, $6 \oldpence$ and $\tfrac 1 2 \oldpence$ in the horizontal with the $2 \shillings 6 \oldpence$, $\tfrac 1 4 \oldpence$ and $\tfrac 1 4 \oldpence$ in the upright.
:This new arrangement will give $5040 \times 12 = 60 \, 480$ solutions also.
:so the total number of ways is twice $60 \, 480$ or $120 \, 960$.
== Sources ==
* : Solutions: $26$. -- Cross and Coins
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions
Henry Ernest Dudeney/Puzzles and Curious Problems/270 - Planning Tours

== by : $270$ ==
;Planning Tours
:The diagram represents a map (considerably simplified for our purposes) of a certain district.
:The circles and dots are towns and villages, and the lines roads.
:
:Can you show how $5$ motor-car drivers can go from $A$ to $A$, from $B$ to $B$, from $C$ to $C$, from $D$ to $D$, from $E$ to $E$, respectively,
::without ever crossing the track or going along the same road as another car?
=== Click here for solution ===
== Also see ==
* Henry Ernest Dudeney: Modern Puzzles $159$ - The Five Regiments: exactly the same puzzle as this, couched in a different scenario.
== Sources ==
* : Unicursal and Route Problems: $270$. -- Planning Tours
* : Combinatorial & Topological Problems: Route & Network Puzzles: $428$. Planning Tours
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/271 - Avoiding the Mines

== by : $271$ ==
;Avoiding the Mines
:Here we have a portion of the North Sea thickly sown with mines by the enemy.
:A cruiser made a safe passage through them from south to north in two straight courses, without striking a single mine.
:
:Take your pencil and try to discover how it is done.
:Go from the bottom of the chart to any point you like on the chart in a straight line,
::and then from that point to the top in another straight line without touching a mine.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $271$. -- Avoiding the Mines
* : Geometrical Problems: Dividing-the-Plane Puzzles: $300$. Avoiding the Mines
Category:Henry Ernest Dudeney/Puzzles and Curious Problems
Henry Ernest Dudeney/Puzzles and Curious Problems/272 - A Madam Problem

== by : $272$ ==
;A Madam Problem
:In how many different ways is it possible to read the word $\text {MADAM}$ in the diagram?
:
:You may go as you please, upwards and downwards, forwards and backwards,
::any way possible along the open paths.
:But the letters in every case must be contiguous, and you may never pass a letter without using it.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $272$. -- A Madam Problem
* : Combinatorial & Topological Problems: Route & Network Puzzles: $429$. A Madam Problem
Category:Henry Ernest Dudeney/Puzzles and Curious Problems