Codebusters - Cryptarithm

A cryptarithm is an arithmetic equation written with letters instead of digits. Each letter stands for one decimal digit, the same letter always means the same digit, and different letters mean different digits. Your job is to find the digit behind every letter. The question then gives a word or phrase written in digits, and you decode it with the mapping you found. It is a logic puzzle solved with column arithmetic, carries and elimination, not letter frequencies.

In the 2027 rules, cryptarithms are a Regional code type in both Division B and Division C, and because State and National exams may use every Regional type, they can appear at every level.

What the rules cover

The 2027 rules describe two tasks: find each letter's digit in a base-10 (decimal) equation, then decode a word or phrase with that mapping. They do not limit the number of letters, the number of addends or the operation, so addition is the classic form, but read every question for the operation it uses.

The event format is the same in both divisions: teams of up to 3, about 50 minutes, a written or computer-based exam. Teams bring only writing utensils. No notes, no calculator of any kind and no other tools are allowed beyond what the Event Supervisor hands out. The resource sheet (Morse table, letter frequencies, Porta table, Baconian mappings, and an Atbash table in Division B only) holds nothing for cryptarithms, so all the arithmetic is done by hand.

The two layers of a cryptarithm

The equation layer is the puzzle itself, such as a sum of two words that equals a third word. The rules you must respect are:

• Every letter stands for one digit from 0 to 9.
• Each letter has exactly one digit, and no two different letters share a digit.
• The equation must be true when the letters are replaced by their digits.

The phrase layer is the answer. The question gives a word or phrase as a string of digit groups, and each digit points to a letter through the mapping you found. Decoding means reversing that lookup.

Because there are only ten digits, a puzzle with ten distinct letters uses every digit exactly once.

How to solve one step by step

1. Align the addends. Write each addend right-justified, with the result below. Label the columns from the right (units, tens, hundreds), and leave a row above the top addend for carries.
2. Write each column as an equation. Use the column rule below, with letters rather than guessed digits.
3. Start with forced values. The leftmost result letter is usually a carry. With two addends, that carry is always 1.
4. Work right to left. The carry out of one column is the carry into the column on its left. Each new column adds one more constraint.
5. Track the digits. Cross off each digit as you place it, so no letter can take a digit that another letter already holds.
6. Check the whole sum. Recompute the addition with the full mapping before you decode anything (see the casting-out-nines check below).
7. Decode the phrase. Read each digit group through the inverse mapping.

Worked example: SEND + MORE = MONEY

Solve the equation, then decode the phrase.

Step 1: the leading letter. MONEY has five digits and the addends have four, so M is the carry out of the thousands column. Two four-digit numbers sum to less than 20,000, so the carry is 1 and M = 1.

Step 2: the thousands column. The column reads S + M + c₃ = O + 10. Substituting M = 1 gives S + c₃ = O + 9. If O = 1, then S + c₃ = 10 and M and O would both be 1, which is not allowed. So O = 0 and S + c₃ = 9.

Step 3: the hundreds column. The column reads E + O + c₂ = N + 10c₃. Since O = 0, E + c₂ = N + 10c₃. If c₃ = 1, then E = 9 and N = 0, but O is already 0. So c₃ = 0. That makes S = 9 from Step 2, and E + c₂ = N. N cannot equal E, so c₂ = 1 and N = E + 1.

Step 4: the tens column. The column reads N + R + c₁ = E + 10c₂ = E + 10. Substituting N = E + 1 gives R + c₁ = 9. S is already 9, so R cannot be 9. That forces c₁ = 1 and R = 8.

Step 5: the units column. The column reads D + E = Y + 10. The digits still free are 2 through 7, and N = E + 1 means E is at most 6. Testing each value of E:

• E = 2: N = 3, and the largest free D is 7, so D + E is at most 9 and never reaches 10.
• E = 3: N = 4, and only D = 7 reaches 10, which gives Y = 0, already held by O.
• E = 4: N = 5. D = 6 gives Y = 0 and D = 7 gives Y = 1, both taken; D = 2 or 3 stays below 10.
• E = 5: N = 6, and D = 7 gives Y = 2. This works.
• E = 6: N = 7. D = 4 gives Y = 0 and D = 5 gives Y = 1, both taken; D = 2 or 3 stays below 10.

So E = 5, N = 6, D = 7, Y = 2. A full search of all digit assignments finds this as the only valid mapping when no number may begin with 0.

The mapping is S = 9, E = 5, N = 6, D = 7, M = 1, O = 0, R = 8, Y = 2. Check the sum: 9567 + 1085 = 10652. Casting out nines confirms it. The digit sums are 27, 14 and 14, so the addends contribute 27 + 14 ≡ 5 (mod 9), and 10652 has digit sum 14 ≡ 5 (mod 9). The two sides match.

Encoding and decoding a phrase

Encoding a word with the mapping above gives SEND = 9 5 6 7 and MORE = 1 0 8 5. Decoding runs the other way: a question that gives 9 5 6 7 wants each digit looked up in the inverse table (9 is S, 5 is E, 6 is N, 7 is D), so the answer is SEND. Treat each digit group as a separate word.

How the exam scores a cryptarithm

Cryptarithm scoring is stricter than most Codebusters questions. Most answers earn full credit with two or fewer errors, but under section 4.e of the 2027 rules a cryptarithm needs zero errors for full credit. Every error costs 100 points, and the penalty cannot exceed the question's value. In Division C the same zero-error rule also covers Aristocrat questions that ask for the keyword or key phrase.

For a hypothetical 500-point cryptarithm:

The 8-error row matches the rule's own example.

Special bonus. Up to three questions that are not Aristocrats, Patristocrats or Xenocrypts can be marked special bonus. Nothing in the 2027 rules excludes a cryptarithm, so one can be marked. New for 2027, at least one special bonus question must use the Homophonic cipher (last season that slot went to the 5x5 Checkerboard), so at most two slots are left for other types such as a cryptarithm. Solving one, two or three of them with no penalty points is worth 150, 400 or 750 points. Bonus credit needs a solve with no penalty points, which for a cryptarithm means zero errors.

Timing bonus. The timing bonus is 2 × (600 − seconds) and is based only on the first question, which is always an Aristocrat with spaces. It stops at 600 seconds (10 minutes). A cryptarithm never affects it directly. For example, a first question solved in 6 minutes (360 seconds) earns 480 points.

Solving tactics under tournament conditions

• Eliminate aggressively. Keep a list of used digits and a candidate list for each letter. One column can rule out most candidates.
• Use the ten-letter count. If the equation has ten distinct letters, every digit is used exactly once, so the leftover digit belongs to the last unplaced letter.
• Cast out nines. If A + B = C, then the digit sum of A plus the digit sum of B is congruent to the digit sum of C modulo 9. It does not prove a mapping, but it catches most addition slips. Calculators are not allowed in Codebusters, so practice column addition and this check by hand until both are fast.
• Decode last. Decode only after the mapping passes both checks. A decode built on a wrong mapping can look plausible and still be wrong.
• Budget your time. The exam runs about 50 minutes and a cryptarithm does not touch the timing bonus. If a few columns place no new letter, mark it and come back. A partial decode never scores below 0, so write your best answer before time runs out.

Common pitfalls

How tournaments test this

There are no archived cryptarithm questions on scio.ly yet, so this guide does not give counts for how often they appear or how they are worded. What the 2027 rules fix is the task: a base-10 equation, then a word or phrase decoded from the mapping, scored by errors in that decoded word or phrase. Read the packet carefully for the operation (usually addition), the number of addends, and the exact answer format it wants. Practice on the puzzles below and on archived Codebusters exams on the scio.ly site.

Practice

Each puzzle has exactly one valid mapping under the no-leading-zero convention (checked by exhaustive search). Solve the equation, then decode the digit string using the mapping. Answers are in the callout that follows.

1. BASE + BALL = GAMES. Decode: 8 9 4 5 5
2. CROSS + ROADS = DANGER. Decode: 7 6 5 3 3
3. COOK + BOOK = BAKED. Decode: 1 3 3 6 7 2

References and further reading

• Science Olympiad, Division B and Division C Codebusters rules (2027 rules manual) on soinc.org. Read sections 2, 3.e, 3.f, 3.h, 3.i and 4 for resources, the code types, the timed question, special bonus and scoring.
• Wikipedia, Verbal arithmetic, an overview of cryptarithm puzzles.
• Wikipedia, Casting out nines, the digit-sum check.