Henry Ernest Dudeney/Modern Puzzles/Combination and Group Problems

Henry Ernest Dudeney: Modern Puzzles: Combination and Group Problems

$166$ - Picture Presentation

A wealthy collector had ten valuable pictures.

He proposed to make a presentation to a public gallery, but could not make up his mind as to how many he would give.

So it amused him to work out the exact number of different ways.

You see, he could give any one picture, any two, any three, and so on, or give the whole ten.

$167$ - A General Election

In how many different ways may a Parliament of $615$ members be elected if there are only $4$ parties:

Conservatives, Liberals, Socialists, and Independents?

You see you might have $\text C. 310$, $\text L. 152$, $\text S. 150$, $\text I. 3$;

or $\text C. 0$, $\text L. 0$, $\text S. 0$, $\text I. 615$;

or $\text C. 205$, $\text L. 205$, $\text S. 205$, $\text I. 0$; and so on.

The candidates are indistinguishable, as we are only concerned with the party numbers.

$168$ - The Magisterial Bench

A bench of magistrates consists of two Englishmen, two Scotsmen, two Welshmen, one Frenchman, one Italian, one Spaniard, and one American.

The Englishmen will not sit beside one another, the Scotsmen will not sit beside one another, and the Welshmen also object to sitting together.

Now, in how many different ways may the ten men sit in a straight line so that no two men of the same nationality shall ever be next to one another?

$169$ - The Card Pentagon

Make a rough pentagon on a large sheet of paper.

Then throw down the ten non-court cards of a suit at the places indicated in the diagram,

so that the pips on every row of three cards on the sides of the pentagon shall add up alike.

The example will be found faulty.

$170$ - A Heptagon Puzzle

Using the fourteen numbers, $1$, $2$, $3$, up to $14$, place a different number in every circle

so that the three numbers in every one of the seven sides add up to $19$.