How to solve symbol sums (cryptarithms)
Symbol sums: start at the units column
In a column addition, every symbol hides one digit and no two symbols share one. Work right to left, carry 0 or 1, and stop the moment the asked symbol is pinned.
Updated
A short addition written in symbols — say ▲■ + ■▲ = ●● — where each symbol stands for one digit, no two symbols share a digit, and the question asks what digit ONE named symbol represents. The puzzle is built so that the asked symbol has exactly one possible digit, however many ways the rest could fall.
Units first, carries always
Work the columns from the RIGHT. The units column constrains its symbols first, and each solved column feeds a carry into the next. In an addition of two numbers a carry is only ever 0 or 1 — writing it down at each column boundary costs a second and prevents the classic drift.
Leading symbols are never zero
The first symbol of any number in the sum cannot stand for 0. That single fact often eliminates half the candidates for a column.
Repeats are the levers
One symbol appearing in several places is worth more than three appearing once: everywhere it occurs it is the SAME digit, so each occurrence is another equation. Look for a symbol that appears in both an addend and the result — those columns solve first.
Stop when the asked symbol is pinned
You rarely need the full solution. The moment the queried symbol's digit is forced, answer — but answer THAT symbol. The other symbols' digits are all on the option list, and 'solved the puzzle, reported the wrong symbol' is this family's classic own-goal.
The strategy
- Circle the queried symbol before touching the arithmetic.
- Units column; write the carry explicitly.
- Use leading-symbol-isn't-zero to prune.
- Check each candidate digit against every column its symbol touches.
- Answer the queried symbol, not the one you solved last.
Common traps
- A forgotten carry — or a carry of 2, which addition of two numbers cannot produce.
- Two symbols assigned the same digit.
- Reporting a different symbol's digit.
- A leading zero passed without notice.
Speed · ~70s/item: Roughly 40–110 seconds. If the units column resists, try the leftmost column instead — leading digits and the final carry often pin it faster.
Try it yourself: 4 practice questions
Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.
Each symbol stands for a different digit, and no two symbols share one. What digit does the asked symbol stand for?
Leading symbols are not zero. Which digit is ●?
Reveal the answer
The answer is E — 0.
Show why — the worked reasoning
Each symbol stands for a different digit, and no two symbols share one. What digit does the asked symbol stand for?
Leading symbols are not zero. Which digit is ▲?
Reveal the answer
The answer is C — 0.
Show why — the worked reasoning
Each symbol stands for a different digit, and no two symbols share one. What digit does the asked symbol stand for?
Leading symbols are not zero. Which digit is ●?
Reveal the answer
The answer is E — 9.
Show why — the worked reasoning
Each symbol stands for a different digit, and no two symbols share one. What digit does the asked symbol stand for?
Leading symbols are not zero. Which digit is ■?
Reveal the answer
The answer is B — 9.
Show why — the worked reasoning
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Questions people ask
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