straddling checkerboard May 28, 2010
You must first choose an alphabet. Taking H and W to be the height and width of the checkerboard, the number of letters in the alphabet, N, must satisfy N = (W - 1) * H + 1. A common choice for the Latin alphabet is to add two characters (for a total of 28---I will use dot and slash) and take H = 3 and W = 10. Historically, other alphabets such as Cyrillic (with around 33 characters) have been used.
To form a straddling checkerboard, write the W - H + 1 most common letters in your language across the top row. For English, these are ETNRIOAS, which may be remembered with the phrase "a sin to err".
0 1 2 3 4 5 6 7 8 9 A S I N T O E R
The spaces (here under 4 and 9) may be under any of the digits, as long as your compatriots use the same locations.
Next, fill in the remaining H - 1 rows with the rest of your alphabet. At the beginnings of the rows, put the digits under which the spaces in the last step fell.
0 1 2 3 4 5 6 7 8 9 A S I N T O E R 4 B C D F G H J K L M 9 P Q U V W X Y Z . /
To encode a message, replace each letter with the digit of its row and column:
N O D E Y O U R H O B B I E S 3 6 42 7 96 6 92 8 45 6 40 40 2 7 1
Because the most common letters translate to only one digit, messages encoded with a straddling checkerboard are shorter than those using a Polybius square.
It still works out that you can apply transposition ciphers to the digits and then decode them back into your alphabet, as with a Polybius square. However, you may need to apply a digit of padding if the outcome ends in a 4 or 9 (or whatever your blank-digits were).