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Letter sums

SEND + MORE = MONEY: each letter is a different digit. Work out what one letter must be — longer words, three addends, subtraction, products.

  • Calculation
  • Untimed
  • Endless levels

How it plays

A sum is written in letters. Each letter stands for one digit, different letters for different digits, and no number starts with 0. You are asked for one letter: find the digit it must be.

Start where the sum gives itself away — a carry out of the leftmost column can only be 1; a column whose letters repeat pins a digit. Each puzzle is checked by trying every assignment, and the letter asked for has the same digit in every one that works.

Levels lengthen the words without end and bring in a longer addend, three addends, subtraction and multiplication by one letter.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises column arithmetic run backwards — reasoning about carries and digits. It is practice at these puzzles, not a measured gain in reasoning elsewhere.

Learn

How to get better at letter sums: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Find the letter you are asked for and look only at the columns it sits in. You do not need the whole solution, only the digit that letter must be.
  2. Check the left end first. If the answer is longer than every number being added, its first letter is a carry, and a carry is small.
  3. Then check the right end. The units column has no carry coming in, so it is the one column you can read exactly.
  4. Work column by column from whichever end gave something away, writing each carry above the next column. Stop as soon as the asked letter is pinned.

Techniques and tricks

  • The carry out is 1

    Two numbers added can carry at most 1 out of the leftmost column. So a result one letter longer than both addends starts with 1.

    e.g. ML + YY = MMH: M is the carry out, so M = 1 (15 + 99 = 114 is one way it works).

  • Three addends carry at most 2

    Three digits plus a carry come to at most 9 + 9 + 9 + 2 = 29, so with three words added a new leading letter is 1 or 2.

  • A letter plus X gives itself

    If a column reads X + Y = Y, then X adds nothing: X = 0 with no carry in, or X = 9 with a carry of 1 in. In the units column there is never a carry in, so X = 0.

    e.g. RL + EF = FF: in the units, L + F ends in F, so L = 0.

  • Turn subtraction into addition

    A − B = C is the same puzzle as B + C = A. Rewrite it in your head and use the addition clues: the top number’s first letter is often a carry out of B + C.

  • Last-digit table for ×

    For a word times one letter, the units digit of the product is the units of the last letter times the multiplier. List which digit pairs give the result’s last letter.

    e.g. If C × D ends in C, then C is 0, or 5 with D odd, or C is even and D = 6.

  • Size from length

    Count digits. A 6-letter word times one letter giving 7 letters means the product passed 1,000,000; a 6-letter result means it did not. That bounds the leading letters.

  • Rule out, then confirm

    With options shown, test each candidate digit in the one column that holds the asked letter. A digit that breaks that column, or repeats another letter’s digit, is gone.

As the levels climb

  • The kinds of sum unlock in a fixed order: two words added, a longer and a shorter word, three words added, subtraction, then a word times one letter, all five by level 7.
  • Between each new kind, the words grow a letter. They keep growing without end after the last kind arrives, so a long puzzle is the same column reasoning repeated, not a new method.
  • Three words added makes the leftmost carry 1 or 2, not just 1. Re-check the carry rule every time the shape changes.
  • Subtraction and multiplication arrive late. For subtraction, rewrite it as an addition straight away; for multiplication, start at the units and carry the tens of each column’s product into the next.

Traps to avoid

  • A neighbouring digit, one up or one down. These come from a carry forgotten or added in the column you used. Write the carries down, even in your head.
  • The digit of another letter in the same column. Each letter is a different digit, so if your answer equals a letter you have already pinned, something is wrong.
  • 0, 1 and 9 among the options. They are the usual guesses, and they are right only when a carry or a “gives itself” column forces them. Check the column, not the habit.
  • A leading 0. No number starts with 0, so the first letter of any word of two or more letters is never 0. That alone rules out an option often.

How to practise

  • Leave the Time pressure setting Off at first: these are puzzles, and the time is better spent finding the column that gives the letter away.
  • Use “Kinds of sum” to practise one shape at a time, starting with two words added; add subtraction and “times one letter” only once the addition clues are automatic.
  • Raise “Word length at level 0” once short sums feel easy, so you meet long words without climbing through the short ones again.
  • Switch Answer by to “Typing the digit” to stop yourself testing options and make yourself derive the digit from the columns.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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