Henry Ernest Dudeney/Puzzles and Curious Problems/Moving Counter Problems

Henry Ernest Dudeney: Puzzles and Curious Problems: Moving Counter Problems

$245$ - Magic Fifteen Puzzle

This is Loyd's famous $14$-$15$ puzzle,

in which you were asked to get the $14$ and $15$ in their proper order by sliding them about in the box.

It was, of course, impossible of solution.

I now propose to slide them about until they shall form a perfect magic square

in which the four columns, four rows and two diagonals all add up to $30$.

It will be found convenient to use numbered counters in place of the blocks.

What are your fewest possible moves?

$246$ - Transferring the Counters

Place ten counters on the squares of a chessboard as here shown,

and transfer them to the other corner as indicated by the ten crosses.

A counter may jump over any counter to the next square beyond, if vacant,

either horizontally or vertically, but not diagonally,

and there are no captures and no simple moves -- only leaps.

Not to waste the reader's time it can be conclusively proved that this is impossible.

You are now asked to add two more counters so that it may be done.

If you place these, say, on $\text {AA}$, they must, in the end, be found in the corresponding positions $\text {BB}$.

Where will you place them?

$247$ - The Counter Cross

Arrange twenty counters in the form of a cross, in the manner shown in the diagram.

Now, in how many different ways can you point out four counters that will form a perfect square if considered alone?

Thus the four counters composing each arm of the cross, and also the four in the centre, form squares.

Squares are also formed by the four counters marked $\text A$, the four marked $\text B$, and so on.

And in how many ways can you remove six counters so that not a single square can be so indicated from those that remain?

$248$ - Four in Line

Here we have a board of $36$ squares, and four counters are so placed in a straight line

that every square of the board is in line horizontally, vertically, or diagonally with at least one counter.

In other words, if you regard them as chess queens, every square on the board is attacked by at least one queen.

The puzzle is to find in how many different ways the four counters may be placed in a straight line so that every square shall thus be in line with a counter.

Every arrangement in which the counters occupy a different set of four squares is a different arrangement.

Thus, in the case of the example given, they can be moved to the next column to the right with equal effect,

or they may be transferred to either of the two central rows of the board.

This arrangement, therefore, produces $4$ solutions by what we call reversals or reflections of the board.

Remember that the counters must always be disposed in a straight line.

$249$ - Odds and Evens

Place eight counters in a pile on the middle circle so that they shall be in proper numerical order, with $1$ on the top and $8$ on the bottom.

It is required to transfer $1$, $3$, $5$, $7$ to the circle marked "Odds", and $2$, $4$, $6$, $8$ to the circle marked "Evens".

You can only move one counter at a time from circle to circle, and you must never place a number on a smaller number,

nor an odd number and an even number together on the same circle.

What are the fewest possible moves?

$250$ - Adjusting the Counters

Place $25$ counters in a square in the order shown.

Then it is a good puzzle to put them all into regular order so that the first line reads $1 \ 2 \ 3 \ 4 \ 5$, and the second $6 \ 7 \ 8 \ 9 \ 10$,

and so on to the end, by taking up one counter in each hand and making them change places.

The puzzle is to determine the fewest possible exchanges in which this can be done.

$251$ - Nine Men in a Trench

Here are nine men in a trench.

No. $1$ is the sergeant, who wishes to place himself at the other end of the line -- at point $1$ --

all the other men returning to their proper places at present.

There is no room to pass in the trench, and for a man to climb over another would be a dangerous exposure.

But it is not difficult with these three recesses, each of which will hold a man.

How is it to be done with the fewest possible moves?

A man may go any distance that is possible in a move.

$252$ - Black and White

Place four light and four dark counters alternately in a row as here shown.

The puzzle is to transfer two contiguous counters to one end and then move two contiguous counters to the vacant space,

and so on until in four such moves they form a continuous line of four dark counters followed by four light ones.

Then try this variant.

The conditions are exactly the same, only in moving a contiguous pair you must make them change sides.

How many moves do you now require?

$253$ - The Angelica Puzzle

Draw a square with three lines in both direction and place on the intersecting points eight lettered counters as shown in our illustration.

The puzzle is to move the counters, one at a time, along the lines from point to vacant point until you get them in the order $\text {ANGELICA}$ thus:

$\begin{array} \\ A & N & G \\ E & L & I \\ C & A & .\end {array}$

Try to do this in the fewest possible moves.

$254$ - The Flanders Wheel

Place eight lettered counters on the wheel as shown.

Now move them one at a time along the line from circle to circle

until the word $\text {FLANDERS}$ can be correctly read round the rim of the wheel as at present,

only that the $\text F$ is in the upper circle now occupied by the $\text N$.

Of course two counters cannot be in a circle at the same time.

Find the fewest possible moves.

$255$ - A Peg Puzzle

The diagram represents a square board with $49$ holes in it.

There are $10$ pegs to be placed in the positions shown,

and the puzzle is to remove only $3$ of these pegs to different holes,

so that the ten shall form $5$ rows with $4$ pegs in every row.

Which three would you move, and where would you place them?

$256$ - Catching the Prisoners

Make a rough diagram on a sheet of paper, and use counters to indicate the two warders (marked as $W$) and the two prisoners (marked as $P$).

At the beginning the counters must be placed in the squares shown.

The first player moves each of the warders to an adjacent cell, in any direction.

Then the second player moves each prisoner to an adjoining cell;

and so on until each warder captures his prisoner.

If one warder makes a capture, both he and his captive are out of the game, and the other player continues alone.

You may come to the conclusion that it is a hopeless chase, but it can really be done if you use a little cunning.

$257$ - Five Lines of Four

The diagram shows how ten counters may be placed on the points of the grid where the lines intersect,

so that they form five straight lines with four counters in every line.

Can you find a second way of doing this?

$258$ - Deploying Battleships

Ten battleships were anchored in the form here shown.

The puzzle is for four ships to move to new positions (the others remaining where they are)

until the ten form five straight rows with four ships in each row.

How should the admiral do it?

$259$ - Flies on Window Panes

The diagram represents a window with $81$ panes.

The dots represent nine flies, on as many panes,

and no fly is in line with another one horizontally, vertically, or diagonally.

Six of those flies are very torpid and do not move,

but each of the remaining three goes to an adjoining pane.

And yet, after this change of station, no fly is in line with another.

Which are the three lively flies, and to which three panes (at present unoccupied), do they pass?

$260$ - 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?