Answer :
The number of
different ways is 63,504.
The
general formula
for such
arrangements, when the number of letters in the sentence is 2n
+ 1, and
it is a palindrome without diagonal readings, is [4(2^{n}
 1)]^{2}.
I think it will be
well to give here a formula for
the general
solution
of each of the four most common forms of the diamondletter
puzzle.
By
the word "line" I mean the complete diagonal.
Thus in A, B, C, and D,
the
lines respectively contain 5, 5, 7, and 9 letters.
A has a
nonpalindrome
line (the word being BOY), and the general solution for such cases,
where
the line contains 2n
+ 1 letters, is 4(2^{n}
 1).
Where the line is a
single palindrome, with its middle letter in the centre, as in B, the
general formula is [4(2^{n}
 1)]^{2}.
In cases C and D we
have double palindromes, but these
two
represent very
different
types. In C, where the line contains 4n1
letters, the general expression is 4(2^{2n}2).
But D is by far the
most difficult case of all.
I had better here
state that in the diamonds under
consideration (i.) no
diagonal readings are allowed—these have to be dealt with
specially in
cases where they are possible and admitted; (ii.) readings may start
anywhere; (iii.) readings may go backwards and forwards, using letters
more than once in a single reading, but not the same letter twice in
immediate succession.
This last condition
will be understood if the
reader glances at C, where it is impossible to go forwards and
backwards
in a reading without repeating the first O touched—a
proceeding which I
have said is not allowed.
In the case D it is very different, and this
is
what accounts for its greater difficulty.
The formula for D
is this:
where the number of
letters in the line is 4n
+ 1. In the example given
there are therefore 400 readings for n
= 2.
