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