Henry Ernest Dudeney/Modern Puzzles/169 - The Card Pentagon/Solution

Modern Puzzles by Henry Ernest Dudeney: $169$

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.

Deal the cards $1$, $2$, $3$, $4$, $5$ in the manner indicated by the dotted lines

(that is, drop one at every alternate angle in a clockwise direction round the pentagon),

and then deal the $6$, $7$, $8$, $9$, $10$ in the opposite direction, as shown,

taking care to start with the $6$ on the correct side of the $5$.

The pips on every side add to $14$.

If you deal the $6$, $7$, $8$, $9$, $10$ in the first manner and the $1$, $2$, $3$, $4$, $5$ in the second manner,

you will get another solution, adding up to $19$.

Now work with the two sets of numbers $1$, $3$, $5$, $7$, $9$ and $2$, $4$, $6$, $8$, $10$ in the same way

and you will get two more solutions, adding, respectively, to $16$ and $17$.

There are $6$ solutions in all.

The last two are peculiar.

Write in, in the same order, $1$, $4$, $7$, $10$, $13$ and $6$, $9$, $12$, $15$, $18$;

also write in $8$, $11$, $14$, $17$, $20$ and $3$, $6$, $9$, $12$, $15$.

Then deduct $10$ from every number greater than $10$.