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Amusements in Mathematics

SOLUTIONS 301-430
301.—THE EIGHT STARS.—solution
The solution of this puzzle is shown in the first diagram. It is the only possible solution
within the conditions stated. But if one of the eight stars had not already been placed as
shown, there would then have been eight ways of arranging the stars according to this
scheme, if we count reversals and reflections as different. If you turn this page round so
that each side is in turn at the bottom, you will get the four reversals; and if you reflect
each of these in a mirror, you will get the four reflections. These are, therefore, merely
eight aspects of one "fundamental solution." But without that first star being so placed,
there is another fundamental solution, as shown in the second diagram. But this
arrangement being in a way symmetrical, only produces four different aspects by reversal
and reflection.
302.—A PROBLEM IN MOSAICS.—solution
The diagram shows how the tiles may be rearranged. As before, one yellow and one
purple tile are dispensed with. I will here point out that in the previous arrangement the
yellow and purple tiles in the seventh row might have changed places, but no other
arrangement was possible.
303.—UNDER THE VEIL.—solution
Some schemes give more diagonal readings of four letters than others, and we are at first
tempted to favour these; but this is a false scent, because what you appear to gain in this
direction you lose in others. Of course it immediately occurs to the solver that every
LIVE or EVIL is worth twice as much as any other word, since it reads both ways and
always counts as 2. This is an important consideration, though sometimes those
arrangements that contain most readings of these two words are fruitless in other words,
and we lose in the general count.
The above diagram is in accordance with the conditions requiring no letter to be in line
with another similar letter, and it gives twenty readings of the five words—six
horizontally, six vertically, four in the diagonals indicated by the arrows on the left, and
four in the diagonals indicated by the arrows on the right. This is the maximum.
Four sets of eight letters may be placed on the board of sixty-four squares in as many as
604 different ways, without any letter ever being in line with a similar one. This does not
count reversals and reflections as different, and it does not take into consideration the
actual permutations of the letters among themselves; that is, for example, making the L's
change places with the E's. Now it is a singular fact that not only do the twenty word-
readings that I have given prove to be the real maximum, but there is actually only that
one arrangement from which this maximum may be obtained. But if you make the V's
 
 
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