Codebusters - Checkerboard

The 5x5 checkerboard is a grid cipher. A keyed 5x5 square holds the alphabet, each row and each column carries a label, and every plaintext letter is replaced by its row label followed by its column label. Under the 2027 rules it appears in both divisions: at Regional level you are given the Polybius key and decode, and at State and National level you must also crack a message built from two 5-letter keywords and an unknown Polybius key, using a crib of at least 5 characters. Learn the lookup first, then learn how to recover the grid and its labels.

What the rules cover

The 2027 rules manual lists the checkerboard at both levels in both divisions. Paraphrased:

Division B and Division C treat this cipher identically. Other 2027 facts that matter:

• Team and time. Teams of up to 3 students, with about 50 minutes for the whole exam.
• Resources. Teams bring writing utensils only. No notes, no binder and no calculator of any class are allowed. The supervisor's resource sheet carries the Morse code table, English and Spanish letter frequencies, the Porta table and Baconian mappings (Division B also gets an Atbash table). It has no Polybius square, so you carry the method in your head.
• Cribs. For the State and National checkerboard, the crib's position on the ciphertext is clearly marked, so you never have to hunt for where it goes.
• Special bonus. Up to three questions that are not Aristocrats, Patristocrats or Xenocrypts may be marked as special bonus questions, so a checkerboard can be one. The 2027 rules guarantee a Homophonic bonus question, not a checkerboard one.
• Scoring. Two or fewer errors in the decoded plaintext is full credit; each further error costs 100 points, and a question never goes below zero.

How the cipher works

A Polybius square is a 5x5 grid with 25 cells. The alphabet has 26 letters, so one pair shares a cell; the usual choice is to merge I and J. The rules do not name the merged pair, so follow your packet.

The Polybius key is a keyword that fills the grid: write its letters once each, left to right and top to bottom, then fill the remaining cells with the unused letters in alphabetical order. With the keyword CIPHER the square is:

Every letter now has an address: its row and its column. The simplest labels are the digits 1 to 5, which is the easiest way to learn the lookup.

Each letter becomes exactly two symbols, with no separators and no variable-length codes. The ciphertext therefore has an even length and splits into pairs from the first symbol. That is the most useful structural fact about this cipher.

Encrypting by hand

1. Write the plaintext in capitals with no spaces, and replace J with I if the square merges them.
2. Find each letter in the square.
3. Write its row label, then its column label.
4. Join the pairs.

Decrypting by hand

1. Check the symbol count is even. If not, you misread a symbol.
2. Split into pairs from the first symbol.
3. Read the first symbol of each pair as the row, the second as the column, and look up the cell.
4. Join the letters and add word breaks by reading the result aloud.

Letter labels from two keywords

The 2027 rules describe the full cipher with three keys: two 5-letter keywords and a Polybius key. The Polybius key fills the grid, one 5-letter keyword labels the rows and the other labels the columns. Five letters per keyword means one label per row or column, so each keyword needs five different letters. Each plaintext letter becomes a pair of letters.

Keep the CIPHER grid and label the rows LIGHT and the columns WORDS:

Decrypting works as before: first letter is the row label, second is the column label. If a letter is in both keywords, go by its position in the pair.

The keyword-letter rule

New bold text in the 2027 rules requires every letter used in the row and column keywords to appear in the ciphertext. In practice, a test writer must choose a plaintext that touches every row and every column at least once. The earlier message ATTACKATDAWN would not qualify under these labels: none of its letters sits in column R, so the R of WORDS never appears.

The rule matters to you because it hands you the label letters for free. List the distinct first letters of the pairs and you have the row keyword's letters; list the distinct second letters and you have the column keyword's letters. In the CROSSTHERIVERATNOON ciphertext the first letters are G, H, I, L, T and the second letters are D, O, R, S, W. What remains is putting each set in the right order.

Cracking the square with a crib

At State and National level nothing but the crib is given. Three facts make the square recoverable. The same letter always gets the same pair. Different letters never share a pair. Pairs that share a first letter lie in the same row, and pairs that share a second letter lie in the same column.

Take the ciphertext above with the crib CROSS marked over the first five pairs (the minimum crib length).

Work outward from this:

1. Read the label sets. By the keyword-letter rule, the row letters are G, H, I, L, T and the column letters are D, O, R, S, W. Each set is five letters, as it must be.
2. Anagram the keywords. G, H, I, L, T gives LIGHT. D, O, R, S, W gives both WORDS and SWORD, so the crib and the grid must decide.
3. Use the crib's lines. C, R and O all sit in column W, in rows L, I and H. S shares row H with O but sits in column R. Those letters must sit in the order the grid's fill allows.
4. Use the Polybius key's fill. After the keyword, the grid runs in alphabetical order, so long runs of letters in alphabetical order with only keyword letters skipped (such as K L M N, or Q S T U with R skipped) should appear along rows. If a label order breaks those runs, it is probably wrong.
5. Test by decoding. With columns ordered WORDS the message reads CROSSTHERIVERATNOON. With SWORD, the same ciphertext decodes to IAQTTUECAPWCABUGQQG, which is nonsense. Decode the first ten pairs under each candidate before committing.

Once the labels are ordered, fill the grid from the known pairs and decode the rest.

The idea of an address made of two coordinates goes back to the historical Polybius square, which arranged the Greek alphabet in a grid:

Tournament tactics

Regional questions are lookups. Find which labels mark the rows, check the count is even, split into pairs, and decode. Write each pair above its letter so a teammate can check you. The two-error tolerance absorbs one or two misreads, but not more.

State and National questions are reconstructions. Follow the five steps above, and reject a guess the moment it puts two letters in one cell or one letter in two cells. Write the full square once it is complete, then decode the rest in one pass.

Split the job: one teammate segments pairs while another anagrams the label sets.

Common mistakes

How tournaments test this

No archived or question-bank items for this cipher are available yet, so this section relies on the 2027 rules. Expect a given-key decode at Regionals and possibly a crib-based crack at State and National. Scoring uses the standard two-error tolerance. The first question is always an Aristocrat, so this cipher is never the timed question.

Division and level notes

• Division B and Division C share the same checkerboard items: B 3.e.xi and 3.f.v, C 3.e.x and 3.f.vi.
• Regional: the Polybius key is given, so the work is decoding.
• State and National: two hidden 5-letter keywords and a hidden Polybius key, with a marked crib of at least 5 characters.
• Devices: none. Writing utensils only, no calculator of any class.

Practice ciphertexts

Practices 1 and 2 use the CIPHER square with digit labels; practice 3 uses LIGHT rows and WORDS columns. Every answer was checked by re-encrypting the plaintext.

References and further reading

• Science Olympiad, 2027 Codebusters rules (Division B and Division C manuals): soinc.org.
• SciOly Wiki, Codebusters: scioly.org/wiki/index.php/Codebusters.
• Wikipedia, Polybius square: en.wikipedia.org/wiki/Polybius_square, for the historical background.
• David Kahn, The Codebreakers (1967), the standard history of cryptanalysis.