Short Answer

We check correctness after substitution by putting the assigned digits into the original equation and verifying if both sides are equal. Each letter must keep the same digit everywhere, and all rules like unique digits and no leading zero must be followed.

If the equation becomes true and all conditions are satisfied, the solution is correct. If not, we revise the values and check again using logical steps.

Detailed Explanation:

Checking correctness

After assigning digits to letters in a cryptarithm or substitution puzzle, it is very important to check whether the solution is correct. This step ensures that the values we have found actually satisfy the given equation and follow all the rules. Without checking, even a small mistake can lead to a wrong answer.

Substitution in equation

The first step is to replace all letters with their assigned digits in the original equation. After substitution, the equation should look like a normal mathematical problem. Then we solve it using basic arithmetic rules.

Matching both sides

After solving, we must check if both sides of the equation are equal. If the left side and right side give the same result, it means the substitution is correct. If they are not equal, then there is an error in the assigned values.

Maintaining same values

Each letter must have the same value everywhere in the equation. While checking, we must ensure that no letter has been given different values in different places. Consistency is very important for correctness.

Checking digit uniqueness

We must verify that no two letters have the same digit. If any digit is repeated for different letters, the solution is incorrect. This rule must always be satisfied.

Verification methods

There are different logical methods that help in verifying the correctness of substitution.

Check place value

We should check each column based on place value, such as units, tens, and hundreds. Each column must follow correct arithmetic rules. This helps in identifying small mistakes easily.

Verify carry

Carry must be checked carefully. If there is a carry in addition, it must be correctly transferred to the next column. If the carry is wrong, the entire solution becomes incorrect.

Reverse calculation

Sometimes, we can check correctness by reversing the operation. For example, if the puzzle involves addition, we can verify by subtraction. This confirms whether the values are correct.

Logical consistency

All logical conditions given in the puzzle must be satisfied. If any condition is violated, the solution must be rejected. Logical consistency ensures accuracy.

Re-evaluate errors

If the solution is wrong, we must go back and find where the mistake happened. Then we correct it and check again. This process continues until the correct solution is found.

Practice importance

With practice, checking becomes faster and more accurate. It helps in avoiding mistakes and improves confidence in solving puzzles.

Conclusion

Checking correctness after substitution is an important step in solving cryptarithms. By verifying the equation, ensuring unique digits, and checking place value and carry, we can confirm whether the solution is correct. This step ensures accuracy and strengthens logical reasoning skills.