Solving the Two-Digit Number Puzzle with Math and Programming

When working with mathematical riddles and puzzles, particularly those involving two-digit numbers, the challenge can be both engaging and educational. This article explores a specific problem: What is the two-digit number if the number is subtracted from its reversed number has a result of 63? Here, we will solve this problem step by step using both algebraic methods and programming techniques.

Algebraic Solution

Let's denote the two-digit number as 10a b, where (a) is the digit in the tens place and (b) is the digit in the ones place. The reversed number would then be 10b a.

Step-by-Step Algebraic Solution

According to the problem, when we subtract the original number from its reversed number, we have:

(10b a - (10a b) 63)

Simplifying this equation:

[10b a - 10a - b 63]

[9b - 9a 63]

[b - a 7]

This means that the digit (b) is 7 greater than the digit (a).

Since both (a) and (b) must be single digits (0 to 9), we can determine possible values for (a) and (b). The valid pairs are:

When (a 0), then (b 7). This is not a valid two-digit number (07 is not a valid two-digit number). When (a 1), then (b 8). This is a valid two-digit number, which is 18. When (a 2), then (b 9). This is a valid two-digit number, which is 29. When (a geq 3), (b) would exceed 9, which is not valid.

Therefore, the valid two-digit numbers that satisfy the condition are 18 and 29. Let's verify the results:

(81 - 18 63) (92 - 29 63)

Hence, the two-digit numbers that satisfy the given condition are 18 and 29.

Brute Force Solution Using Programming

The programming solution above shows a brute force approach using the J programming language to solve the problem. Here are the steps:

Translate the two-digit number as 10a b. Create a list of all two-digit numbers (70 to 99). Subtract the reversed number (10b a) from the original number (10a b) and check if the result is 63.

The output indicates there are three answers: 70, 18, and 29.

Verifying the Brute Force Solution

Let's verify these solutions using the same verification step as before:

(92 - 29 63) (81 - 18 63) (29 - 29 0) (This is incorrect, indicating a potential issue with the brute force method or the J programming language output.) (70 - 70 0) (Again, incorrect, further indicating a potential issue with the brute force method or the J programming language output.)

Thus, the correct two-digit numbers that satisfy the condition are 18 and 29.

General Problem Solving Techniques

Both the algebraic and programming solutions highlight the importance of logical reasoning and verification. Understanding the underlying algebraic relationships and writing clear, testable programs can help solve complex mathematical problems. For beginners, these techniques can be instrumental in developing problem-solving skills.

Mental exercises like solving puzzles and brain teasers not only entertain but also help keep the mind sharp and engaged.