Henry Ernest Dudeney/Puzzles and Curious Problems/Combination and Group Problems

Henry Ernest Dudeney: Puzzles and Curious Problems: Combination and Group Problems

$273$ - City Luncheons

The clerks attached to the firm of Pilkins and Popinjay arranged that three of them would lunch together every day at a particular table

so long as they could avoid the same three men sitting down twice together.

The same number of clerks of Messrs. Radson, Robson, and Ross decided to do precisely the same, only with four men at a time instead of three.

On working it out they found that Radson's staff could keep it up exactly three times as many days as their neighbours.

What is the least number of men there could have been in each staff?

$274$ - Halfpennies and Tray

What is the greatest number of halfpennies that can be laid flat on a circular tray

(with a small brim to prevent overlapping the edge)

of exactly $9$ inches in diameter, inside measurements?

No halfpenny may rest, however slightly, on another.

Of course, everybody should know that a halfpenny is exactly one inch in diameter.

$275$ - The Necklace Problem

How many different necklaces can be made with $8$ beads, where each bead may be either black or white,

the beads being indistinguishable except by colour?

$276$ - An Effervescent Puzzle

In how many ways can the letters in the word $\text {EFFERVESCES}$ be arranged in a line without two $\text E$s ever appearing together?

Of course, two occurrences of the same letter, such as $\text {F F}$, have no separate identity,

so that to interchange them will make no difference.

When the reader has done that, he should try the case where the letters have to be arranged differently in a circle, with no two $\text E$s together.

We are here, of course, only concerned with their positions on the circumference, and you must always read in a clockwise direction.

$277$ - Tessellated Tiles

Here we have $20$ tiles, all coloured with the same four colours.

The puzzle is to select any $16$ of these tiles that you choose and arrange them in the form of a square,

always placing same colours together -- white against white, red against red, and so on.

$278$ - The Thirty-Six Letter Puzzle

If you try to fill up this square by repeating the letters $A$, $B$, $C$, $D$, $E$, $F$,

so that no $A$ shall be in a line across, downwards, or diagonally, with another $A$,

no $B$ with another $B$, no $C$ with another $C$, and so on,

you will find that it is impossible to get in all the $36$ letters under these conditions.

The puzzle is to place as many letters as possible.

$279$ - Roses, Shamrocks, and Thistles

Place the numbers $1$ to $12$ (one number in every design) so that they shall add up to the same sum in the following $7$ different ways --

viz., each of the two centre columns, each of the two central rows,

the four roses together, the four shamrocks together, and the four thistles together.

$280$ - The Ten Barrels

A merchant had ten barrels of sugar, which he placed in the form of a pyramid, as shown.

Every barrel bore a different number, except one, which was not marked.

It will be seen that he had accidentally arranged them so that the numbers in the three sides added up alike --

that is, to $16$.

Can you arrange them so that the three sides shall sum to the smallest number possible?

Of course the central barrel (which happens to be $7$ in the diagram) does not come into the count.

$281$ - A Match Puzzle

The $16$ squares of a chessboard are enclosed by $16$ matches.

It is required to place an odd number of matches inside the square so as to enclose $4$ groups of $4$ squares each.

There are $4$ distinct ways to do this, up to reflection and rotation.

$282$ - The Magic Hexagon

In the diagram it will be seen how the numbers from $1$ to $19$ are arranged so that all $12$ lines add up to $23$.

Six of the lines are the six sides, and the other six lines radiate from the centre.

Can you find a different arrangement that will still add up to $23$ in all the $12$ directions?

$283$ - Pat in Africa

Many years ago, when the world was different, a team of explorers consisting of $5$ men from Western Civilization and $5$ natives

fell into the hands of a hostile local chief, who, after receiving a number of gifts, consented to let them go,

but only after half of them had been flogged by the head of the security services.

The Westerners cruelly hatched a plot to make the flogging fall upon the $5$ natives.

They were all to be arranged in a circle, and Pat, in position no. $1$, was given a number to count round and round in the clockwise direction.

In the diagram, $W$ represents a Westerner, and $N$ represents a native.

When that number fell on a man, he was to be taken out for flogging,

while the counting went on from where it left off until another man fell out,

and so on until the five men had been selected for punishment.

If Pat had remembered the number correctly, and had begun at the right man,

the flogging would all have fallen upon the $5$ natives.

But Pat was humane at heart, and did not hold with the casual cruelty of his fellows,

and so deliberately used the wrong number and started at the wrong man,

with the result that the Westerners all got the flogging instead.

Can you find:

$(1)$ the number Pat selected, and the man he started the count at,

$(2)$ the number he had been expected to use, and the man he was supposed to have begun at?

The smallest possible number is required in each case.

$284$ - Lamp Signalling

Two spies on the opposite sides of a river devised a method for signalling by night.

They each put up a stand, like the diagram, and each had three lamps which could show either white, red or green.

They constructed a code in which every different signal meant a sentence.

Note that a single lamp on any one of the hooks could only mean the same thing,

that two lamps hung on the upper hooks $1$ and $2$ could not be distinguished from two on, for example, $4$ and $5$.

However, two red lamps on $1$ and $5$ could be distinguished from two on $1$ and $6$,

and two on $1$ and $2$ from two on $1$ and $3$.

Remembering the variations of colour as well as of position, what is the greatest number of signals that could be sent?

$285$ - The Teashop Check

We give an example of the check supposed to be used at certain popular teashops.

The waitress punches holes in the tickets to indicate the amount of the purchase.

$\boxed {\begin{array} {rcl} \\

\tfrac 1 2 \oldpence & --- & \bullet \\ 1 \oldpence & --- \\ 1 \tfrac 1 2 \oldpence & --- \\ 2 \oldpence & --- \\ 2 \tfrac 1 2 \oldpence & --- \\ 3 \oldpence & --- & \bullet \\ 4 \oldpence & --- \\ 6 \oldpence & --- \\ 7 \oldpence & --- \\ 8 \oldpence & --- \\ 1 \shillings & --- \\

& & \\

\end{array} }$

Thus, in the example, the two holes indicate that the customer has to pay $3 \tfrac 1 2 \oldpence$

But the girl might, if she had chosen, have punched in any one of three other ways --

$2 \tfrac 1 2 \oldpence$ and $1 \oldpence$, or $2 \oldpence$ and $1 \tfrac 1 2 \oldpence$, or $2 \oldpence$, $1 \oldpence$ and $\tfrac 1 2 \oldpence$

On one occasion a waitress said, "I can punch this ticket in any one of $10$ different ways, and no more."

Her coworker, whose customer owed a different amount, said, "Same here."

What were the amounts of the purchases of each of their customers?

Only one hole is allowed to be punched against any given amount.

$286$ - Unlucky Breakdowns

On a day of great festivities, a large crowd gathered for a day's outing and pleasure.

They all agreed to pile into a bunch of wagons, each of which was to carry the same number of people.

But ten of the wagons broke down half way, so each of the other wagons then had to carry one more person than had been planned.

As they were about to start back, it was discovered that $15$ more of these wagons had become unserviceable,

and so there were three more people in each working wagon on the way back than started out.

How many people were there in the party?

$287$ - The Handcuffed Prisoners

Nine dangerous convicts needed to be guarded.

Every day except Sunday they were taken out for exercise, handcuffed together in groups of three, as in the diagram:

On no day in any one week were the same two men to be handcuffed together.

If will be seen how they were sent out on Monday.

Can you arrange the nine men in triplets for the remaining $5$ days?

It will be seen that No. $1$ cannot be handcuffed to No. $2$ again, but $1$ and $3$ can subsequently be so.

$288$ - Seating the Party

On a family outing, Dora asked in how many ways they could all be seated.

There were $6$ of them, three of each gender, and $6$ seats:

one beside the driver, two with their backs to the driver, and two behind them, facing the driver,

No two of the same gender were allowed to sit side by side.

The only people who were able to drive were the men.

So, how many ways could they all be seated?