Codebreaking toolkit

The letter E

HintSay any ordinary sentence aloud and listen for the vowel that turns up in almost every word — it is not A, I or O.

The whyIn ordinary English, roughly one letter in eight is an E, and no other letter comes close, so the commonest symbol in a long enough message is the safest first guess a codebreaker can make. After E, the usual order runs T, A, O, I, N — the string codebreakers remember as ETAOIN. The word "long enough" matters: a short message can easily break the pattern by chance, which is why the Challenge sets its early cryptograms as whole paragraphs rather than single lines.

A Caesar shift

HintNamed after the Roman general who used it to write to his commanders.

The whyHe reportedly slid by three, so A became D. Only 25 of the 26 slides change anything at all, which is why trying every one of them in turn is a practical attack — codebreakers call that brute force.

the

HintLook at almost any English sentence and count which short word appears most often. It is not a noun, and it is not a name.

The whyIt is by some distance the commonest word in written English, and the commonest run of three letters too. Pin it down and you have three letters of your key for the price of one — and every "t", "h" and "e" elsewhere in the message falls out with it. That is why codebreakers count groups of two and three symbols as well as single ones: a tally of pairs and triples carries far more information than a tally of letters alone.

same

HintShuffling a pack of cards does not add a card or take one away.

The whyDraw a bar chart of the letter counts and a transposition leaves it untouched — the bars are identical to those of ordinary English, even though the message reads as nonsense. That single check is the fastest way to sort the two great families apart: if the chart looks English, nothing has been swapped and the letters have merely been moved; if the chart looks flat or lopsided, the letters themselves have been changed.

shifts; symbol; order

HintEach family does exactly one job — slide along, swap for a stand-in, or rearrange. Name the missing word for each in turn.

The whyNearly every cipher in the Challenge is one of these three, or two of them stacked on top of each other. Naming the family first is what turns a wall of capitals into a plan: a shift is cracked by trying all of them, a substitution by counting symbols, a transposition by looking for the shape of the grid the letters were written into. Guessing the family wrong costs an evening, so make the letter counts tell you before you start.

The frequencies get flattened

HintOne plain letter now hides behind several disguises at once, so no single symbol stands out from the crowd.

The whySpread an E across four different disguises and each one carries only a quarter of the evidence, so the tall bar that gave the game away in a substitution simply is not there. The counts of a Vigenère message look far more level than English does. That is the whole point of a polyalphabetic cipher, and it is why the first job is no longer counting letters but finding out how long the keyword is.

Their highest common factor

HintThe keyword must fit a whole number of times into every gap, so hunt for what all the gaps share.

The whyA repeat usually means the same plain word met the same part of the keyword twice, so the keyword fits exactly into the gap between them. Do that for several repeats and the keyword length has to divide every gap — so the factor they all share is your best estimate of it. This is called Kasiski examination, published in 1863, and it broke a cipher that had stood for three hundred years. Once you know the length, the message splits into that many separate Caesar shifts, and each one falls to plain letter counting.

The keyword

HintSender and receiver must already share it, and it is usually something short enough to remember.

The whyWrite it under the message over and over — LEMONLEMONLEMON — and each letter tells you how far to slide the letter above it. Because it repeats, the shifts run in a cycle whose length is exactly the number of letters in it, and that cycle is the crack a codebreaker prises open. A longer one is safer for the sender: the longer the cycle, the fewer repeats an attacker gets to measure.

Modular arithmetic

HintA clock face does the same thing: four hours after eleven is not fifteen.

The whyNumber the letters A = 0 up to Z = 25, add the shift, and take the remainder after dividing by 26 — that is the whole of a Caesar cipher in one line. The alphabet is treated as a circle rather than a line, so nothing ever falls off the end. Every cipher that shifts, including Vigenère, rests on this idea, and it is also why decoding is just shifting the other way round the same circle.

26

HintCount from A through to Z once, and give the answer for English rather than for any other language.

The whyIt is a two-way table: pick your row by the keyword letter, your column by the plain letter, and read the cipher letter where they meet. Because it is square, the same grid decodes as well as encodes.

Part A is easier

HintThe first of each weekly pair also hands you something useful for its harder twin.

The whyThe Challenge is run by the University of Southampton's School of Mathematics: eight to ten challenges, each split into a part A and a harder part B, published week by week around a story. Part A is the gentler cryptogram and usually gives you information you need for part B, which is scored on both accuracy and speed. You may enter on your own or as a team of any size, and you need to be in full-time school-level education to qualify for the prizes. The 2026 competition starts on 8 October.

transposition

HintNothing was swapped for anything else, so the only thing that can have happened is a rearrangement.

The whyCount first, then decide. Matching counts mean the letters are all still there and only their positions changed, so look for the shape they were written into — a grid filled along the rows and read down the columns, or the other way about. The number of columns is usually small, so a patient search of the sensible ones gets there. Skipping this one check is the classic way to waste an evening hunting for a substitution key that never existed.