2.3 Cryptarithmetic (Harfli İşlemler), Letter-Digit Coding (Şifreler) & Symbolic Operators

Cryptarithmetic (Harfli İşlemler), Letter-Digit Coding (Şifreler) & Symbolic Operators

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1. Cryptarithmetic (Harfli İşlemler / Kriptaritmetik)

In a cryptarithmetic puzzle, base-10 digits ($0, 1, 2, \dots, 9$) in a vertical addition, subtraction, multiplication, or division layout are replaced by capital letters ($A, B, C, K, L, M$) or geometric icons (▲, ■, ●). Three universal conventions govern these problems:

1. Bijection Rule: Identical letters represent the exact same digit everywhere in the operation; distinct letters represent distinct digits (unless the problem stem explicitly allows equal values).
2. No Leading Zero Rule: The leftmost digit of any multi-digit number can never be $0$ (for example, in $\overline{ABC}$, $A \in \{1, 2, \dots, 9\}$).
3. Base-10 Carry (Elde) Bounds:
   • Adding two single digits ($X + Y$, plus a possible incoming carry of $1$) has a maximum sum of $9 + 9 + 1 = 19$, so the outgoing carry to the next column left is at most $1$.
   • Adding three single digits ($X + Y + Z$, plus an incoming carry of at most $2$) has a maximum sum of $9 + 9 + 9 + 2 = 29$, so the outgoing carry is at most $2$.

Place-Value Decomposition (Basamak Çözümlemesi)

Whenever letters appear symmetrically across units, tens, and hundreds columns, expand each multi-digit integer in powers of $10$ rather than guessing column by column:

• $\overline{AB} = 10A + B$ and $\overline{BA} = 10B + A \implies \overline{AB} + \overline{BA} = 11(A + B)$ and $\overline{AB} - \overline{BA} = 9(A - B)$
• $\overline{AA} = 11A$ and $\overline{AAA} = 111A = 3 \times 37 \times A$ (Note the prime factor $37$—it can simplify $\overline{BBB}$ multiplication puzzle!)
• $\overline{ABC} + \overline{BCA} + \overline{CAB} = 111(A + B + C)$

Units-Digit Invariants (Birler Basamağı Sabitleri)

Look for self-reproducing digits in the units column:

• Additive Zero Invariant: $A + B \equiv A \pmod{10} \implies B = 0$ (if there is no incoming carry) or $B = 9$ (if there is an incoming carry of $1$ from the column to the right).
• Idempotent Square Digits: $B \times B \equiv B \pmod{10} \implies B \in \{0, 1, 5, 6\}$ because $0^2 = 0$, $1^2 = 1$, $5^2 = 25$, and $6^2 = 36$.

Worked Example 1: Carry-Bound & Place-Value Addition

In the base-10 vertical addition $\overline{AB} + \overline{BC} = \overline{BCB}$, $A, B,$ and $C$ are distinct digits. Find $A + B + C$.

1. Apply the Leading Carry Bound: $\overline{AB}$ and $\overline{BC}$ are two 2-digit numbers, so their sum $\overline{BCB}$ is a 3-digit number less than $199$. Therefore, the hundreds digit $B$ must equal $1$ ($B = 1$).
2. Substitute $B = 1$ and expand by place value:

$$\overline{A1} + \overline{1C} = \overline{1C1} \implies (10A + 1) + (10 + C) = 100 + 10C + 1$$

1. Simplify the linear Diophantine equation:

$$10A + 11 + C = 101 + 10C \implies 10A - 9C = 90 \implies 10(A - 9) = 9C$$

Since $A \in \{1, \dots, 9\}$ and $C \in \{0, \dots, 9\}$, the only solution is $A = 9$ and $C = 0$ (checking: $91 + 10 = 101$). 4. Compute the sum: $A + B + C = 9 + 1 + 0 = 10$.

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2. Letter-to-Number and Symbol Coding (Şifreler)

Column-Frequency Matching

Never try to match entire words by eye. Instead, count how many times each character appears in Column 1 (first letter) and Column $n$ (last letter):

1. Count First-Column Multiplicities: If the five words start with letters having frequency distribution $(2, 2, 1)$—meaning two words start with $K$, two start with $M$, and one starts with $S$—then the unique singleton starter $S$ is identified immediately by finding the single numeric code whose first digit appears only once!
2. Read Off the Entire Singleton Row: Once the unique word is matched to its unique code, you instantly know the digit for every letter in that word.
3. Cross-Reference Last-Column or Repeated Internal Letters: Use a word with repeated letters (such as MASA, which has identical 2nd and 4th letters) to lock in the remaining ambiguity and eliminate wrong options.

Worked Example 2: Cracking a 5-Word Coding Matrix

Five words (KASA, KIRA, SIRA, MARK, MASA) are paired in scrambled order with the five numeric codes 4232, 4752, 3752, 8254, 8232. What is the code for KIRA?

1. Inspect First Letters (K, K, S, M, M) vs. First Digits (4, 4, 3, 8, 8): S is the only first letter appearing once, and 3 is the only first digit appearing once. Thus S = 3 and SIRA = 3752, which immediately gives I = 7, R = 5, and A = 2!
2. Inspect Last Letters (A, A, A, K, A) vs. Last Digits (2, 2, 2, 4, 2): Four words end in A = 2, while only MARK ends in K. The only code ending in 4 is 8254, so MARK = 8254, giving M = 8 and K = 4.
3. Assemble KIRA: Since K = 4, I = 7, R = 5, A = 2, the code for KIRA is 4752.

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3. Custom Symbolic Operators (Sembolik İşlemler)

Unlike standard algebraic operators where the formula for $a \star b$ is explicitly given, Sembolik İşlemler in the aptitude section give three numerical input-output equations and ask you to discover the hidden formula $f(a, b)$ before evaluating a fourth expression.

When testing candidate rules for $a \star b = c$, check these four families in order:

1. Linear & Bilinear Combinations: $a \cdot b \pm (a + b)$, $a \cdot b \pm (a - b)$, or $m \cdot a + n \cdot b$.
2. Asymmetric Power Rules: $a^2 \pm b$, $a^2 \pm b^2$, $a^b \pm b$, or $(a \pm b)^2$.
3. Rational Quotients: $\frac{a + b}{a - b}$ or $\frac{a \cdot b}{a + b}$.
4. Positional Digit Concatenation: Writing $a + b$ next to $a - b$ (for example, $7 \star 3 = 104$ because $7 + 3 = 10$ and $7 - 3 = 4$ concatenated give 104).

Worked Example 3: Deducing a Hidden Binary Operator

Given the three operational examples:

$$6 \blacktriangle 2 = 34, \qquad 5 \blacktriangle 3 = 22, \qquad 7 \blacktriangle 4 = 45$$

Find the value of $8 \blacktriangle 5$.

1. Analyze the first pair ($6, 2 \to 34$): Notice that $34$ is just below $6^2 = 36$, specifically $6^2 - 2 = 34$.
2. Verify on Example 2 ($5, 3 \to 22$): $5^2 - 3 = 25 - 3 = 22$ (Confirmed).
3. Verify on Example 3 ($7, 4 \to 45$): $7^2 - 4 = 49 - 4 = 45$ (Confirmed).
4. Evaluate target ($8 \blacktriangle 5$): Using $a \blacktriangle b = a^2 - b$, we obtain $8^2 - 5 = 64 - 5 = 59$.

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Cryptarithmetic & Coding Quick-Reference Table

Review the Method

Custom symbolic operators (Sembolik İşlemler) hide a two-input formula a ★ b = f(a, b) that must be inferred and verified across all three given numerical examples