nyuuzyou/wikis huggingface dataset:
Henry Ernest Dudeney/Modern Puzzles/144 - The Bricklayer's Task

== by : $144$ ==
;The Bricklayer's Task
:When a man walled in his estate, one of the walls was partly level and partly over a small rise or hill,
::precisely as shown in the drawing herewith, wherein it will be observed that the distance from $A$ to $B$ is the same as from $B$ to $C$.
:
:Now, the master-builder desired and claimed that he should be paid more for the part that was on the hill than for the part that was level,
::since (at least, so he held) it demanded the use of more material.
:But the employer insisted that he should pay less for that part.
:It was a nice point, over which they nearly had recourse to the law.
:Which of them was in the right?
=== Click here for solution ===
== Sources ==
* : Geometrical Problems: Various Geometrical Puzzles: $144$. -- The Bricklayer's Task
* : Geometrical Problems: Plane Geometry Puzzles: $305$. The Bricklayer's Task
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/144 - The Bricklayer's Task/Solution

== by : $144$ ==
;The Bricklayer's Task
== Solution ==
:A glance at the illustration will show that if you could cut off the portion of wall marked $1$ and place it in the position indicated by $2$,
::you would have a piece of straight wall, $BC$, enclosed by the dotted lines, exactly the same as the wall $AB$.
:
== Sources ==
* : Solutions: $144$. -- The Bricklayer's Task
* : Answers: $305$. The Bricklayer's Task
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/145 - A Cube Paradox

== by : $145$ ==
;A Cube Paradox
:I had two solid cubes of lead, one very slightly larger than the other.
:Through one of them I cut a hole (without destroying the continuity of the four sides)
::so that the other cube could be passed right through it.
:On weighing them afterwards it was found that the larger cube was still the heavier of the two.
:How was this possible?
=== Click here for solution ===
== Sources ==
* : Geometrical Problems: Various Geometrical Puzzles: $145$. -- A Cube Paradox
* : Geometrical Problems: Solid Geometry Puzzles: $315$. A Cube Paradox
Category:Henry Ernest Dudeney/Modern Puzzles
Category:Veridical Paradoxes
Henry Ernest Dudeney/Modern Puzzles/145 - A Cube Paradox/Solution

== by : $145$ ==
;A Cube Paradox
:I had two solid cubes of lead, one very slightly larger than the other.
:Through one of them I cut a hole (without destroying the continuity of the four sides)
::so that the other cube could be passed right through it.
:On weighing them afterwards it was found that the larger cube was still the heavier of the two.
:How was this possible?
== Solution ==
The smaller cube is the one which has had the hole made in it.
== Proof ==
From Prince Rupert's Cube, the largest tunnel that can be made in a cube has a side of length $1.06+$ the length of the edge of the cube.
So the larger cube has an edge whose length greater than $1$ and less than $1.06$ times the length of the edge of the smaller cube.
Then the smaller cube has the hole made in it, through which the larger passes.
And, of course, the larger cube is still the heavier of the two.
== Also see ==
* Prince Rupert's Cube
== Sources ==
* : Solutions: $145$. -- A Cube Paradox
* : Answers: $315$. A Cube Paradox
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/146 - The Cardboard Box

== by : $146$ ==
;The Cardboard Box
:If I have a closed cubical cardboard box, by running the penknife along seven of the twelve edges (it must always be seven)
::I can lay it out in one flat piece in various shapes.
:
:Thus, in the diagram, if I pass the knife along the darkened edges and down the invisible edge indicated by the dotted line, I get the shape $A$.
:Another way of cutting produces $B$ or $C$.
:It will be seen that $D$ is simply $C$ turned over, so we will not call that a different shape.
:Now, how many shapes can be produced?
=== Click here for solution ===
== Sources ==
* : Geometrical Problems: Various Geometrical Puzzles: $146$. -- The Cardboard Box
* : Geometrical Problems: Solid Geometry Puzzles: $316$. The Cardboard Box
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/146 - The Cardboard Box/Solution

== by : $146$ ==
;The Cardboard Box
== Solution ==
The $11$ basic shapes are:
:
All of these are asymmetrical except the first $2$ shapes.
Hence another $9$ can be flipped over to make a different shape.
So there are $20$ altogether.
== Sources ==
* : Solutions: $146$. -- The Cardboard Box
* : Answers: $316$. The Cardboard Box
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/147 - The Austrian Pretzel

== by : $147$ ==
;The Austrian Pretzel
:Here is a twisted Vienna bread roll, known as a Pretzel.
:The twist, like the curl in a pig's tail, is entirely for ornament.
:The Wiener Pretzel, like some other things, is doomed to be cut up or broken, and the interest lies in the number of resultant pieces.
:
:Suppose you had the Pretzel depicted in the illustration lying on the table before you,
::what is the greatest number of pieces into which you could cut it with a single straight cut of a knife?
:In what direction would you make the cut?
=== Click here for solution ===
== Sources ==
* : Geometrical Problems: Various Geometrical Puzzles: $147$. -- The Austrian Pretzel
* : Geometrical Problems: Solid Geometry Puzzles: $317$. The Austrian Pretzel
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/147 - The Austrian Pretzel/Solution

== by : $147$ ==
;The Austrian Pretzel
== Solution ==
:The Pretzel may be divided into as many as ten pieces by one straight cut of the knife in the direction indicated in the illustration.
:
== Sources ==
* : Solutions: $147$. -- The Austrian Pretzel
* : Answers: $317$. The Austrian Pretzel
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/148 - Cutting the Cheese

== by : $148$ ==
;Cutting the Cheese
:I have a piece of cheese in the shape of a cube.
:How am I to cut it in two pieces with one straight cut of the knife
::so that the two new surfaces produced by the cut shall each be a perfect hexagon?
:
:Of course, if cut in the direction of the dotted line the surfaces would be squares.
:Now produce hexagons.
=== Click here for solution ===
== Sources ==
* : Geometrical Problems: Various Geometrical Puzzles: $148$. -- Cutting the Cheese
* : Geometrical Problems: Solid Geometry Puzzles: $318$. Cutting the Cheese
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/148 - Cutting the Cheese/Solution

== by : $148$ ==
;Cutting the Cheese
== Solution ==
:Mark the midpoints of $BC$, $CH$, $HE$, $EF$, $FG$, and $GB$.
:Then insert the knife at the top and follow the direction indicated by the dotted line.
:
:Then the two surfaces will each be a perfect hexagon.
== Sources ==
* : Solutions: $148$. -- Cutting the Cheese
* : Answers: $318$. Cutting the Cheese
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/149 - A Tree-Planting Puzzle

== by : $149$ ==
;A Tree-Planting Puzzle
:How do you plant $13$ trees so as to form $9$ straight rows of $4$ trees each?
:Here is an arrangement that gets you $8$ rows, but it is clearly suboptimal as there is a tree at the bottom which is not doing enough work.
:
=== Click here for solution ===
== Sources ==
* : Geometrical Problems: Various Geometrical Puzzles: $149$. -- A Tree-Planting Puzzle
* : Combinatorial & Topological Problems: Point Alignment Puzzles: $435$. A Tree-Planting Puzzle
Category:Henry Ernest Dudeney/Modern Puzzles
Category:Orchard Planting Problem
Henry Ernest Dudeney/Modern Puzzles/149 - A Tree-Planting Puzzle/Solution

== by : $149$ ==
;A Tree-Planting Puzzle
== Solution ==
:
== Sources ==
* : Solutions: $149$. -- A Tree-Planting Puzzle
* : Answers: $435$. A Tree-Planting Puzzle
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/14 - An Easy Settlement

== by : $14$ ==
;An Easy Settlement
:Three men, Andrews, Baker and Carey, sat down to play at some game.
:When they put their money on the table it was found that they each possessed $2$ coins only, making altogether $\pounds 1 \ 4 \shillings 6 \oldpence$
:At the end of play Andrews had lost $5$ shillings and Carey had lost sixpence, and they all squared up by simply exchanging the coins.
:What were the exact coins that each held on rising from the table?
=== Click here for solution ===
== Sources ==
* : Arithmetical and Algebraical Problems: Money Puzzles: $14$. -- An Easy Settlement
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/14 - An Easy Settlement/Mistake

== Source Work ==
:
:Arithmetical and Algebraical Problems
::Money Puzzles
:::$14$ -- An Easy Settlement
== Mistake ==
:Three men, Andrews, Baker and Carey, sat down to play at some game.
:When they put their money on the table it was found that they each possessed $2$ coins only, making altogether $\pounds 1 \ 4 \shillings 6 \oldpence$.
:At the end of play Andrews had lost $5$ shillings and Carey had lost sixpence, and they all squared up by simply exchanging the coins.
:What were the exact coins that each held on rising from the table?
The solution offered:
:At the start of play
::Andrews held a half-sovereign and a shilling,
::Baker held a crown and a florin,
::and Carey held a double florin and a half-crown.
:After settlement,
::Andrews held double florin and florin,
::Baker the half-sovereign and half-crown,
::and Carey held crown and a shilling.
:Thus Andrews lost $5 \shillings$, Carey lost $6 \oldpence$, and Baker won $5 \shillings 6 \oldpence$.
:The selection of the coins is obvious, but their allotment requires a little judgment and trial.
There are in fact $4$ distinct solutions.
The full analysis can be found here.
== Sources ==
* : Arithmetical and Algebraical Problems: Money Puzzles: $14$. -- An Easy Settlement
Category:Mistakes/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/150 - Counter Solitaire

== by : $150$ ==
;Counter Solitaire
:This simplification of the board of the old game of Solitaire lends itself to many entertaining little pastimes of patience.
:Copy the simple diagram on a sheet of paper or cardboard and use sixteen counters, numbered and placed as shown.
:The puzzle is to remove all but one counter by a succession of leaps.
:A counter can leap over another adjoining it to the next square beyond, if vacant,
::and in making the leap you remove the one jumped over.
:But no leap may be made in a diagonal direction.
:
:The following is a solution in eight moves:
::$5 - 13$, $(6 - 14, 6 - 5)$, $16 - 15$, $(3 - 13, 3 - 6)$, $2 - 10$, $(8 - 7, 8 - 16, 8 - 3)$, $(1 - 9, 1 - 2, 1 - 8)$, $(4 - 12, 4 - 1)$
:This means that $5$ leaps over $13$ and $13$ is removed, then $6$ leaps over $14$ and $14$ is removed, and so on.
:The leaps within a bracket count as one move, because the leaps are made with the same counter in succession.
:It will be seen that No. $4$ makes the last leap.
:Now try to find a solution, in seven moves, in which No. $1$ makes the last leap.
=== Click here for solution ===
== Sources ==
* : Moving Counter Problems: $150$. -- Counter Solitaire
* : Geometrical Problems: Moving Counter Puzzles: $368$. Counter Solitaire
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/150 - Counter Solitaire/Solution

== by : $150$ ==
;Counter Solitaire
== Solution ==
Play as follows:
:$2 - 10$
:$4 - 12$
:$6 - 5$
:$3 - 6$
:$7 - 15$
:$8 - 16 - 7 - 14 - 3$
:$1 - 9 - 2 - 11 - 8 - 13 - 4$
== Sources ==
* : Solutions: $150$. -- Counter Solitaire
* : Answers: $368$. Counter Solitaire
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/151 - Sinking the Fishing-Boats

== by : $151$ ==
;Sinking the Fishing-Boats
:There are forty-nine fishing-boats in the North Sea.
:How could an enemy ram and sink the lot in twelve straight courses,
:starting at $A$ and finishing up at the same place?
:
=== Click here for solution ===
== Also see ==
* Definition:Queen's Tour
== Sources ==
* : Moving Counter Problems: $151$. -- Sinking the Fishing-Boats
* : Combinatorial & Topological Problems: Route & Network Puzzles: $416$. Sinking the Fishing-Boats
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/151 - Sinking the Fishing-Boats/Historical Note

== Historical Note on by : $151$ - Sinking the Fishing-Boats ==
This solution appears both in 's and 's .
 claims to have first given the puzzle in $1908$, but whether he copied it from of vice versa has not been determined.
Note that this puzzle is also a solution for the queen's tour on a $7 \times 7$ chessboard in $12$ moves.
, in his $1968$ repackaging , reports on:
:'s proof that $2 n - 2$ straight line segments are necessary for a closed path on all squares
:'s proof that, for a square array of $n$ dots on a side, as few as $2 n - 2$ straight line segments can be used to draw through them all, for $n > 2$
:'s proof that $2 n - 2$ straight line segments are sufficient for a closed path on all such squares where $n > 3$
He also reports on the sizes of squares meeting the restriction that none of the segments go outside the borders.
== Sources ==
* 
* 
* : Answers: $416$. Sinking the Fishing-Boats
Category:Historical Notes
Henry Ernest Dudeney/Modern Puzzles/151 - Sinking the Fishing-Boats/Solution

== by : $151$ ==
;Sinking the Fishing-Boats
== Solution ==
:
== Historical Note ==
== Sources ==
* : Solutions: $151$. -- Sinking the Fishing-Boats
* : Answers: $416$. Sinking the Fishing-Boats
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/152 - A New Leap-Frog Puzzle

== by : $152$ ==
;A New Leap-Frog Puzzle
:Make a rough board, as shown, and place seventeen counters on the squares indicated.
:The puzzle is to remove all but one by a series of leaping moves, as in draughts or solitaire.
:A counter can be made to leap over another to the next square beyond, if vacant, and you then remove the one jumped over.
:It will be seen that the first leap must be made by the central counter, No. $9$, and one has the choice of $8$ directions.
:A continuous series of leaps with the same counter will count as a single move.
:It is required to take off $26$ counters in $4$ moves, leaving the No. $9$ on its original centre square.
:Every play must be a leap.
:
=== Click here for solution ===
== Sources ==
* : Moving Counter Problems: $152$. -- A New Leap-Frog Puzzle
* : Geometrical Problems: Moving Counter Puzzles: $369$. A New Leap-Frog Puzzle
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/152 - A New Leap-Frog Puzzle/Solution

== by : $152$ ==
;A New Leap-Frog Puzzle
== Solution ==
Make the following plays:
:$9$ over $13$, $14$, $6$, $4$, $3$, $1$, $2$, $7$, $15$, $17$, $16$, $11$.
:$12$ over $8$.
:$10$ over $5$ and $12$.
:$9$ over $10$.
== Sources ==
* : Solutions: $152$. -- A New Leap-Frog Puzzle
* : Answers: $369$. A New Leap-Frog Puzzle
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/153 - Transferring the Counters

== by : $153$ ==
;Transferring the Counters
:Divide a sheet of paper into six compartments, as shown in the diagram,
::and place a pile of $15$ counters, numbered consecutively $1$, $2$, $3$, $\ldots$, $15$ downwards, in compartment $A$.
:
:The puzzle is to transfer the complete pile, in the fewest possible moves, to compartment $F$.
:You can move the counters one at a time to any compartment,
:but may never place a counter on one that bears a smaller number than itself.
:Thus, if you place $1$ on $B$ and $2$ on $C$, you can then place $1$ on $2$, but not $2$ on $1$.
=== Click here for solution ===
== Sources ==
* : Moving Counter Problems: $153$. -- Transferring the Counters
* : Geometrical Problems: Moving Counter Puzzles: $370$. Transferring the Counters
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/153 - Transferring the Counters/Solution

== by : $153$ ==
;Transferring the Counters
== Solution ==
First we establish that if you have $n$ empty squares, you can place a pile of $n$ counters on one of those squares from another square in $2 n - 1$ moves.
This is trivially done by:
:placing each of the $n$ counters on an arbitrary empty square, taking $n$ moves
:placing $n - 1$ of these counters on the $n$th counter in numerical order.
So:
Make a pile of $5$ counters, numbers $1$ to $5$, on $B$ in $9$ moves.
Make a pile of $4$ counters, numbers $6$ to $9$, on $C$ in $7$ moves.
Make a pile of $3$ counters, numbers $10$ to $12$, on $D$ in $5$ moves.
Make a pile of $2$ counters, numbers $13$ and $14$, on $E$ in $3$ moves.
Place number $15$ on $F$ in one move.
Place $13$ and $14$ on $F$ in $3$ moves.
Place $10$ to $12$ on $F$ in $5$ moves.
Place $6$ to $9$ on $F$ in $7$ moves.
Place $1$ to $5$ on $F$ in $9$ moves.
Total: $49$ moves.
== Sources ==
* : Solutions: $153$. -- Transferring the Counters
* : Answers: $370$. Transferring the Counters
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/154 - The Way to Tipperary

== by : $154$ ==
;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?
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $154$. -- The Way to Tipperary
* : Combinatorial & Topological Problems: Route & Network Puzzles: $411$. The Way to Tipperary
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/154 - The Way to Tipperary/Solution

== by : $154$ ==
;The Way to Tipperary
== Solution ==
:
The Irish Sea is the line marked red.
The thick lines illustrate the best (as in shortest) route.
== Sources ==
* : Solutions: $154$. -- The Way to Tipperary
* : Answers: $411$. The Way to Tipperary
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/155 - Marking a Tennis Court

== by : $155$ ==
;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?
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $155$. -- Marking a Tennis Court
* : Combinatorial & Topological Problems: Route & Network Puzzles: $412$. Marking a Tennis Court
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/155 - Marking a Tennis Court/Solution

== by : $155$ ==
;Marking a Tennis Court
== Solution ==
:
The $10$ labelled points are vertices of degree $3$, and therefore odd.
Hence we need to run over at least $4$ lines twice.
The shortest possible double tracking is running $BE$, $CD$, $GH$ and $FJ$ twice.
Hence the route to take:
:$ADCDKGHGABEBCFJFEHK$
or its reverse.
There are minor variants:
:By symmetry, $EH$, $AB$, $CF$ and $JK$ could be the lines to double track, then we start at $G$ and end at $D$, or the other way about
:We could run over the double-tracked lines when we encounter the other end of them when we reach them, for example: $ADKGABCDCFEBEHGHJFJK$
and so on.
== Sources ==
* : Solutions: $155$. -- Marking a Tennis Court
* : Answers: $412$. Marking a Tennis Court
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions
Henry Ernest Dudeney/Modern Puzzles/156 - Water, Gas and Electricity

== by : $156$ ==
;Water, Gas and Electricity
:I think I receive, on an average, about ten letters a month from unknown correspondents respecting this puzzle which I published some years ago under the above title.
:They invariably say that someone has shown it to them who did not know the answer, and they beg me to relieve their minds by telling them whether there is, or is not, any possible solution.
:As many of my readers may have come across the puzzle and be equally perplexed, I will try to clear up the mystery in a more complete way than I have done in .
:First of all here is the thing, as I originally gave it, for their consideration.
: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.
=== Click here for solution ===
== Sources ==
* : Unicursal and Route Problems: $156$. -- Water, Gas and Electricity
* : Combinatorial & Topological Problems: Route & Network Puzzles: $413$. Water, Gas and Electricity
Category:Henry Ernest Dudeney/Modern Puzzles
Henry Ernest Dudeney/Modern Puzzles/156 - Water, Gas and Electricity/Historical Note

== Historical Note on by : $156$: Water, Gas and Electricity ==
, on publishing this puzzle in his , claims that the proof of the impossibility of a non-tricksy solution to this puzzle is:
:for the first time in a book.
Whether this is true or not remains to be investigated.
This is now known in the field of graph theory as the utilities problem, and this particular non-planar graph is known as the Thomsen graph.
== Sources ==
* : Solutions: $156$. -- Water, Gas and Electricity
* : Answers: $413$. Water, Gas and Electricity
Category:Historical Notes
Henry Ernest Dudeney/Modern Puzzles/156 - Water, Gas and Electricity/Solution

== by : $156$ ==
;Water, Gas and Electricity
== Solution ==
There are some tricksy solutions which involve running the pipes underneath the actual houses, such as the below, which apply to the letter of the law:
:
but the fact is that if you treat the houses and pipes as vertices and edges, the problem has no solution.
== Proof ==
The graph as defined is the complete bipartite graph $K_{3, 3}$.
The problem is equivalent to attempting to prove that $K_{3, 3}$ is a planar graph.
However, from Kuratowski's Theorem, $K_{3, 3}$ is non-planar.
== Historical Note ==
== Sources ==
* : Solutions: $156$. -- Water, Gas and Electricity
* : Answers: $413$. Water, Gas and Electricity
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions