The number of different ways is 63,504.
arrangements, when the number of letters in the sentence is 2n
+ 1, and
it is a palindrome without diagonal readings, is [4(2n
I think it will be well to give here a formula for
of each of the four most common forms of the diamond-letter
the word "line" I mean the complete diagonal.
Thus in A, B, C, and D,
lines respectively contain 5, 5, 7, and 9 letters.
A has a
line (the word being BOY), and the general solution for such cases,
the line contains 2n + 1 letters, is 4(2n
Where the line is a
single palindrome, with its middle letter in the centre, as in B, the
general formula is [4(2n - 1)]2.
In cases C and D we have double palindromes, but these
types. In C, where the line contains 4n-1
letters, the general expression is 4(22n-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
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
This last condition will be understood if the
reader glances at C, where it is impossible to go forwards and
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
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.