Cryptarithms and challenges

A cryptarithm is an arithmetic puzzle where the digits have been replaced by letters. Each letter stands for a distinct digit. Your job: figure out the digits so the arithmetic works. They are exactly as old as written arithmetic and still appear in newspapers today.

Concept

The most famous example is

  S E N D
+ M O R E
---------
M O N E Y

Each letter is one of $0, 1, \dots, 9$. Different letters stand for different digits. Solve!

Strategy 1: Carry analysis. Look at the leftmost column. Adding two single digits (each $\le 9$) gives at most $9 + 9 + 1 = 19$. So the leftmost column has at most a carry of $1$. In the above puzzle, $S + M = MO + 10\cdot \text{carry}$, where the leading $M$ of the answer must come from a carry. So $M = 1$.

Strategy 2: Use the units column. The rightmost letters tell you the last digit of the sum directly (possibly with a carry to the next column).

Strategy 3: Distinct digits constraint. Often you can rule out a letter being a particular digit because another letter has already claimed it.

For SEND + MORE = MONEY the unique solution (which you can find with some work) is:

$S = 9, E = 5, N = 6, D = 7, M = 1, O = 0, R = 8, Y = 2.$

Verify: $9567 + 1085 = 10652$. ✓

Strategy for any cryptarithm.

1. Identify any letter forced to be $1$ (leading digit appearing as carry).
2. Identify any letter forced to be $0$ (when adding two columns gives the same letter back, with a carry).
3. Use the units column to constrain a small set of possibilities.
4. Try those possibilities; check consistency with other columns.

Another classic.

  A B C
+ A B C
+ A B C
+ A B C
-------
  B B B B

Translating: $4 \times \text{(3-digit ABC)} = \text{(4-digit BBBB)}$. So $4 \times \overline{ABC} = \overline{BBBB} = B \cdot 1111$. The right side is $B \times 1111$ which is between $1111$ and $9999$. The left side is $4 \times \overline{ABC}$ which is between $400$ and $3996$. So $B \cdot 1111 \le 3996$, giving $B \le 3$. And $\overline{ABC} = \dfrac{B \cdot 1111}{4}$; we need this to be a whole number with $A = B$? Let's check $B = 1$: $\overline{ABC} = 1111/4$, not integer. $B = 2$: $2222/4 = 555.5$, not integer. $B = 3$? Hmm , this puzzle needs more constraint or it has no solution; cryptarithms are problem-specific.

Why this is good practice. Cryptarithms force you to think in modular and digit-by-digit ways. They use every divisibility and place-value idea from this chapter. Solving them is a strong test of the toolkit.

Worked examples

Example 1. Solve:

  A B
+ B A
-----
  C C

• $\overline{AB} + \overline{BA} = (10A + B) + (10B + A) = 11(A+B)$.
• $\overline{CC} = 11 C$.
• So $11(A+B) = 11C$, giving $C = A + B$.
• For example, $A = 2, B = 5, C = 7$: $25 + 52 = 77$. ✓

Example 2. Find a digit $X$ so that

  X X X
+ X X X
-------
  6 6 6

• $2 \cdot \overline{XXX} = 666$, so $\overline{XXX} = 333$.
• $X = 3$. ✓

Example 3. Solve:

  A 1
\times  B
-----
  9 A

(Multiplication, not addition.) $\overline{A1} \times B = \overline{9A}$, so $(10A + 1) B = 90 + A$, i.e., $10AB + B = 90 + A$, so $A(10B - 1) = 90 - B$. Try $B = 9$: $A \cdot 89 = 81$, not integer. $B = 5$: $A \cdot 49 = 85$, not. $B = 3$: $A \cdot 29 = 87$, so $A = 3$. Check: $31 \times 3 = 93$. ✓ So $A = 3, B = 3$. But letters should be distinct usually , this puzzle does not have that constraint here.

Example 4. Solve $\overline{AB} \times 9 = \overline{BA}\overline{AB}$ (a 4-digit number whose first two are BA and last two are AB)?

This is harder; skip if time-pressed. Most class VIII cryptarithms only use addition.

Try it yourself

Solve each cryptarithm. Different letters represent different digits unless otherwise noted.

1. $A + A = B \cdot 4$, with $A, B$ single digits.

2.   B A
   + B A
   -----
     C A C
   

3.   1 A
   + 2 B
   -----
     A B
   

4. A B C
   \times 3
   -----
   B C A
   

5. T E N
   - O N E
   -----
   T W O
   

   (Not all puzzles have solutions; show whether this one does.)
6. Find the magic sum if a $5\times 5$ magic square uses $1$-$25$.

7.   A
   + B
   -----
   A A
   

   What does $A$ equal?

8.   X Y Z
   + X Y Z
   -------
     Y Z Z 0
   

   Find $X, Y, Z$ (this one has multiple steps).

Activity / Insight

Cryptarithm club. With a friend, take turns inventing cryptarithms based on small English words ($CAT + BAT = HAT$, $ONE + ONE = TWO$). Solve each other's puzzles. Discuss the strategies you each used. You will rediscover, on your own, that mathematicians call this kind of search-with-constraints a constraint-satisfaction problem , the same idea that powers Sudoku, scheduling apps, and even AI planning.