Code and ciphers: Julius Caesar, the Enigma and the internet

4.2 (Number of possible transposition boxes)
In the particular case of nine letters in three columns the possibilities
are:
7, 1, 1; 1, 7, 1; 1, 1, 7;
6, 2, 1; 6, 1, 2; 2, 6, 1; 2, 1, 6; 1, 6, 2; 1, 2, 6;
5, 3, 1; 5, 1, 3; 3, 5, 1; 3, 1, 5; 1, 5, 3; 1, 3, 5;
5, 2, 2; 2, 5, 2; 2, 2, 5;
4, 4, 1; 4, 1, 4; 1, 4, 4;
4, 3, 2; 4, 2, 3; 3, 4, 2; 3, 2, 4; 2, 4, 3; 2, 3, 4;
3, 3, 3.
A total of 28 (no column of the box is allowed to have 0 letters).
This is a particular case of a more general problem:
In how many ways can n be represented as the sum of k positive
integers when the order of the integers is relevant?
It can be shown (see M18) that the number is
(n � 1)!
(k � 1)!(n � k)!
Putting n�9 and k�3 gives us
8! 8�7
�
2!6! 2
�28.
When n�35 and k�5 the corresponding figure is (34�33�32�31)/24
�46 376.
4.3 (Boustrophedon rows in a transposition box)
Alternate vertical digraphs at the ends of the rows will be unaltered in the
cipher text.
Chapter 5
5.1 (MDTM)
The cipher text is
CFIGS FLTBC XKEEA EBHTB GLDPI
and the 5�5 substitution box is shown in Table S.3.
Solutions to problems 221