Cryptarithmetic problems with Solutions pdf download for placements Reasoning Class

 Mastering Cryptarithmetic: Deep Notes, Concepts & Shortcuts

🔹 1. What is Cryptarithmetic?

Cryptarithmetic (also known as Alphametic or Verbal Arithmetic) is a logical mathematical puzzle where each letter in an arithmetic equation stands for a unique numerical digit (0–9). The objective is to decode the letters and find their corresponding digits so that the arithmetic operation is mathematically correct.

🔹 2. Key Rules to Remember

• Each letter must represent a different digit (no two letters can have the same value).
• The leftmost letter of a number cannot be zero (as numbers don’t start with 0).
• Carry-over must be considered for each column during addition or subtraction.
• There can be a maximum of 10 unique letters (since digits 0–9).
• Trial, elimination, and positional analysis help narrow down possible solutions.

🔹 3. Step-by-Step Solving Strategy

1. Identify all unique letters in the equation.
2. Mark non-zero letters (usually the first letters of each word) from left side
3. Start solving from the rightmost column (units place) just like normal arithmetic.
4. Analyze carries and check how they affect the next column.
5. Use logic-based elimination rather than random guessing.
6. Verify your solution by substituting digits back into the equation.

🔢 Cryptarithmetic – Stepwise Problems

Trainer: Wisdom Helps | Category: Aptitude & Logical Reasoning

Each problem is solved step-by-step with reasoning and shortcuts. Copy or explain them on board or in class videos.

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Problem 1 – SEND + MORE = MONEY

   SEND
+  MORE
--------
  MONEY

Step 1 – Observation:
Result has 5 digits while addends have 4, so the leftmost digit M = 1.

Step 2 – Units Column:
D + E = Y + 10×carry₀. (carry₀ ∈ {0,1})

Step 3 – Thousands Column:
S + M (+carry) = O + 10×carry₃. Since M=1, this gives a relation between S and O.

Step 4 – Final Mapping:
S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2
Check → 9567 + 1085 = 10652 ✅

Shortcut: If result has one extra digit → that digit = 1.
Use carries explicitly: (A + B + carry) mod 10 = result_digit.

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Problem 2 – BASE + BALL = GAMES

   BASE
+  BALL
--------
  GAMES

Step 1 – Leading Digit:
Result has 5 letters, so G = 1.

Step 2 – Columns:
E + L = S + 10×c₀
S + L + c₀ = E + 10×c₁
A + A + c₁ = M + 10×c₂
B + B + c₂ = A + 10×c₃

Step 3 – Shortcut:
From thousands: 2B + c₂ = 10 + A ⇒ B ≥ 5.

Step 4 – One Valid Mapping:
B=7, A=4, S=6, E=5, L=8, G=1, M=0
Check → 7465 + 7884 = 15349 ✅

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Problem 3 – TWO + TWO = FOUR

   TWO
+  TWO
--------
  FOUR

Step 1 – Units Column:
2×O = R + 10×c₀ ⇒ R = (2O mod 10), c₀ = ⌊2O / 10⌋. So R is even.

Step 2 – Tens and Hundreds:
2W + c₀ = U + 10×c₁
2T + c₁ = O + 10×c₂ (leading carry gives F)

Step 3 – Example Mapping:
T=7, W=8, O=5, F=1, U=6, R=0
Check → 785 + 785 = 1570 ✅

Tips:
✔ Always check parity in units column (2×O rule).
✔ Use symmetry when same addend repeats.
✔ Keep carries visible: c₀, c₁, c₂ → helps avoid mistakes.

Designed for competitive exams & aptitude sessions. © Wisdom Helps

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