402 Answers493. ARRANGING THE DOMINOES
There are just 126,720 different ways of playing those 15 dominoes, count-
ing the two directions as different.- A NEW MA TeH PUZZLE
The smallest possible number is 36 matches. We can form triangle and
square with I2 and 24 respectively, triangle and pentagon with 6 and 30,
triangle and hexagon with 6 and 30, square and pentagon with 16 and 20,
square and hexagon with 12 and 24, and pentagon and hexagon with 30 and
- The pairs of numbers may be varied in all cases except the fourth and last.
There cannot be fewer than 36. The triangle and hexagon require a number
divisible by 3: the square and hexagon require an even number. Therefore the
number must be divisible by 6, such as 12, 18,24,30,36, but this condition
cannot be fulfilled for a pentagon and hexagon with fewer than 36 matches.
495. HURDLES AND SHEEP
If the enclosure is to be rectangular,
the nearer the rectangle approaches
to the form of a square the greater
will be the area. But the greatest area
of all will always be when the hurdles
are arranged in the form of a regular
polygon, inscribed in a circle, and if
this can be done in more than one way
the greatest area will be when there
are as many sides as hurdles. Thus, the
hexagon given earlier had a greater
area than the triangle. The twelve-
sided figure or regular dodecagon
therefore encloses the largest possible
area for twelve hurdles-room for
about eleven sheep and one-fifth.
Eleven hurdles would only accommo-
date a maximum of about nine and
nine-twenty-fifths, so that twelve
hurdles are necessary for ten sheep. If