UNDER THE VEIL.
(
Chessboard Problems)
[Illustration:
+---+---+---+---+---+---+---+---+
| | | V | E | I | L | | |
+---+---+---+---+---+---+---+---+
| | | I | L | V | E | | |
+---+---+---+---+---+---+---+---+
| I | V | | | | | L | E |
+---+---+---+---+---+---+---+---+
| L | E | | | | | I | V |
+---+---+---+---+---+---+---+---+
| V | I | | | | | E | L |
+---+---+---+---+---+---+---+---+
| E | L | | | | | V | I |
+---+---+---+---+---+---+---+---+
| | | E | V | L | I | | |
+---+---+---+---+---+---+---+---+
| | | L | I | E | V | | |
+---+---+---+---+---+---+---+---+
]
If the reader will examine the above diagram, he will see that I have so
placed eight V's, eight E's, eight I's, and eight L's in the diagram
that no letter is in line with a similar one horizontally, vertically,
or diagonally. Thus, no V is in line with another V, no E with another
E, and so on. There are a great many different ways of arranging the
letters under this condition. The puzzle is to find an arrangement that
produces the greatest possible number of four-letter words, reading
upwards and downwards, backwards and forwards, or diagonally. All
repetitions count as different words, and the five variations that may
be used are: VEIL, VILE, LEVI, LIVE, and EVIL.
This will be made perfectly clear when I say that the above arrangement
scores eight, because the top and bottom row both give VEIL; the second
and seventh columns both give VEIL; and the two diagonals, starting from
the L in the 5th row and E in the 8th row, both give LIVE and EVIL.
There are therefore eight different readings of the words in all.
This difficult word puzzle is given as an example of the use of
chessboard analysis in solving such things. Only a person who is
familiar with the "Eight Queens" problem could hope to solve it.
Answer:
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.
[Illustration:
_ _ I V E L _ _
E V L _ _ I _ _
L _ _ I _ _ V E
I _ V E _ _ _ L
_ E _ _ L V _ I
_ L I _ _ I _ E V
/V _ E L _ _ I _
_ I _ _ V E L _
]
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 change
places with the I's, and the L's with the E's, in the solution given,
you still get twenty readings--the same number as before in every
direction. Therefore there are two ways of getting the maximum from the
same arrangement. The minimum number of readings is zero--that is, the
letters can be so arranged that no word can be read in any of the
directions.
