What is a Cryptarithm?
A cryptarithm is a mathematical puzzle in which letters or symbols are used in place of digits. The aim is to find the digit represented by each letter so that the given mathematical calculation becomes correct.
Cryptarithms are also known as letter-number puzzles or alphametic puzzles.
Simple Example
A
+ A
----
8
Since:
A + A = 8
Therefore:
A = 4
So:
4 + 4 = 8
Important Rules of Cryptarithms
While solving a cryptarithm, remember the following rules:
Rule 1: Each Letter Represents a Digit
Every letter represents a digit from 0 to 9.
For example, if A = 5, then A represents the digit 5 throughout the puzzle.
Rule 2: The Same Letter Always Represents the Same Digit
If a letter appears more than once, its value remains the same everywhere.
For example:
A2
+ A2
----
64
Here, both occurrences of A must have the same value.
Since:
A = 3
The puzzle becomes:
32
+ 32
----
64
Rule 3: Different Letters Represent Different Digits
Two different letters cannot represent the same digit.
For example, if:
A = 4
then B cannot also be 4.
Rule 4: The First Letter of a Number Cannot Be Zero
The leftmost letter of a number cannot represent 0.
For example:
AB = 25 ✅ Valid
But if A = 0, then:
AB = 05 ❌ Not valid
This is because 05 is simply 5, not a two-digit number.
However, zero can appear in other positions.
For example:
A0 = 20 ✅ Valid
Rule 5: Follow the Normal Rules of Arithmetic
Cryptarithms follow the normal rules of:
- Addition
- Subtraction
- Multiplication
- Division
- Carrying
- Borrowing
How to Solve a Cryptarithm?
Follow these steps:
- Start solving from the rightmost column.
- Check whether there is a carry or borrowing.
- Find the possible value of each letter.
- Remember that different letters must have different values.
- Substitute the values in the original puzzle.
- Verify the final answer.
Solved Examples of Cryptarithms
Example 1: Simple Cryptarithm
A
+ A
----
6
Solution:
A + A = 6
2A = 6
Therefore:
A = 3
Verification:
3 + 3 = 6 ✅
Example 2: Two-Digit Cryptarithm
A2
+ A2
----
64
Solution
Start with the ones place:
2 + 2 = 4
Now solve the tens place:
A + A = 6
2A = 6
Therefore:
A = 3
So:
32
+ 32
----
64
Answer: A = 3 ✅
Example 3: Cryptarithm with Carry
A5
+ A5
----
70
Solution
Start with the ones place:
5 + 5 = 10
Write 0 and carry 1 to the tens column.
Now:
A + A + 1 = 7
2A + 1 = 7
2A = 6
A = 3
Therefore:
35
+ 35
----
70
Answer: A = 3 ✅
Example 4: Find the Missing Letter
A8
+ A7
----
95
Solution
Start with the ones place:
8 + 7 = 15
Write 5 and carry 1.
Now solve the tens place:
A + A + 1 = 9
2A + 1 = 9
2A = 8
A = 4
Therefore:
48
+ 47
----
95
Answer: A = 4 ✅
Example 5: Find Two Letters
A6
+ 2B
----
73
Solution
Start with the ones place:
6 + B = 3
Since 6 + B must end in 3, there must be a carry.
Therefore:
6 + B = 13
So:
B = 7
Carry 1 to the tens place.
Now:
A + 2 + 1 = 7
A = 4
Therefore:
46
+ 27
----
73
Answer: A = 4 and B = 7 ✅
Example 6: Different Letters in Two Numbers
AB
+ AC
----
BD
One possible solution is:
23
+ 24
----
47
Therefore:
A = 2
B = 3
C = 4
D = 7
All the letters represent different digits.
Verification:
23 + 24 = 47 ✅
Example 7: A Cryptarithm with More Than One Solution
AB
+ BA
----
99
We can write:
AB = 10A + B
and
BA = 10B + A
Therefore:
10A + B + 10B + A = 99
11A + 11B = 99
11(A + B) = 99
Therefore:
A + B = 9
Possible solutions are:
- A = 1, B = 8 → 18 + 81 = 99
- A = 2, B = 7 → 27 + 72 = 99
- A = 3, B = 6 → 36 + 63 = 99
- A = 4, B = 5 → 45 + 54 = 99
Thus, some cryptarithms may have more than one possible solution.
Important Tips and Tricks
1. Always Start from the Right
In addition and subtraction cryptarithms, begin with the ones place and move towards the left.
2. Look for a Carry
If the sum of two digits is 10 or more, remember to carry 1 to the next column.
3. Look for Repeated Letters
A repeated letter always has the same value.
4. Eliminate Impossible Values
If a digit has already been assigned to one letter, it cannot be assigned to another letter.
5. Check the First Letter
The first letter of a number cannot be zero.
6. Always Verify Your Answer
Replace all letters with the digits you have found and check the complete calculation.
Uses and Benefits of Cryptarithms
Cryptarithms are not just mathematical puzzles. They help students develop several important skills.
1. Logical Reasoning
Students learn to use clues and conditions to reach the correct answer.
2. Problem-Solving Skills
Cryptarithms encourage students to try different approaches and eliminate impossible answers.
3. Number Sense
Students develop a better understanding of numbers and their relationships.
4. Understanding of Place Value
Solving cryptarithms strengthens the understanding of ones, tens, hundreds and other place values.
5. Better Understanding of Carrying and Borrowing
Students practise the concepts of carrying and borrowing in an interesting way.
6. Critical Thinking
Students analyse each clue carefully before assigning a digit to a letter.
7. Concentration and Patience
Cryptarithms require careful observation and step-by-step thinking.
Practice Questions
Solve the following cryptarithms:
Question 1
A
+ A
----
8
Question 2
A4
+ A4
----
68
Question 3
A5
+ A5
----
90
Question 4
A7
+ A6
----
93
Question 5
A8
+ 2B
----
75
Question 6
AB
+ BA
----
88
Answers to Practice Questions
1. A = 4 → 4 + 4 = 8
2. A = 3 → 34 + 34 = 68
3. A = 4 → 45 + 45 = 90
4. A = 4 → 47 + 46 = 93
5. A = 4, B = 7 → 48 + 27 = 75
6. A + B = 8 → Multiple solutions are possible, such as 17 + 71 = 88 or 26 + 62 = 88.
Conclusion
A cryptarithm is an interesting mathematical puzzle in which letters represent digits. To solve a cryptarithm successfully, students must understand place value, follow the rules of arithmetic, carefully check carries and use logical reasoning.
The most important rules to remember are:
Same letter = Same digit
Different letters = Different digits
The first letter cannot be zero
Always start solving from the rightmost column
With regular practice, cryptarithms can make mathematics more interesting while improving logical thinking and problem-solving skills.
