Solving the Puzzle to Get 24 Using 1, 3, 4, and 6 with Simple Mathematical Operations

Mathematics can often present intriguing puzzles that challenge our understanding and logic. One such classic problem is the 24 Game, a puzzle where you need to use four given numbers and simple mathematical operations to reach the target number, 24. In this article, we will discuss how to achieve 24 using the numbers 1, 3, 4, and 6 with the specified numbers used exactly once.

1. The Solution Using Division and Subtraction

Let's carefully walk through one of the methods to solve the puzzle:

Step 1: Calculate (frac{3}{4}). Step 2: Subtract the result from (1):
[text{1} - left(frac{3}{4}right) 1 - 0.75 0.25] Step 3: Divide 6 by 0.25: (6 div 0.25 24).

Thus, the expression (6 div 1 - frac{3}{4} 24) uses each number exactly once and the target is achieved through simple mathematical operations.

2. Another Valid Solution

Another valid approach is:

Step 1: Multiply 4 and 6 to get 24. Step 2: Subtract 1 from 3 to get 2. Step 3: Divide the product of 4 and 6 by the difference of 3 and 1: (frac{24}{2} 24).

This also reaches the target number 24 with the required numbers used exactly once.

3. Additional Solutions Using Advanced Operations

Let's consider a few more intricate methods to achieve 24 using the numbers 1, 3, 4, and 6:

Step 1: Calculate (1 - frac{3}{4}). (frac{1}{4}) Step 2: Divide 6 by (frac{1}{4}) which is the same as multiplying 6 by 4: (6 div frac{1}{4} 24).

This method also uses each number exactly once.

4. Using Concatenation and Factorials

For a creative solution, we can use concatenation and factorials:

Step 1: Use (sqrt{1^{0.5}}) [concatenation of 1 and square root symbol]. Step 2: Calculate (sqrt{3! 6!}) [concatenation of 3, factorial symbol, 6, and factorial symbol] Step 3: Simplify to (6!) [concatenation of 6 and factorial symbol] Step 4: Finally, simplify to (24)

The expression simplifies to 24, utilizing the numbers in a unique manner.

5. Using Infinite Roots (Optional Complexity)

For a more challenging method, you can use infinite roots:

Step 1: Use the infinite root expression: (1 sqrt[infty]{0.5 sqrt[infty]{3 sqrt[infty]{6}}). While this method is valid, it is more complex and less commonly used in puzzles like these.

Overall, there are several elegant methods to obtain 24 using the numbers 1, 3, 4, and 6 with simple mathematical operations, as demonstrated above.

Conclusion

The 24 Game remains a fascinating puzzle that encourages logical thinking and creativity. By exploring various methods, such as concatenation, factorials, and infinite roots, we can develop a deeper appreciation for the intricacies of mathematical operations. This puzzle not only tests our arithmetic skills but also sharpens our problem-solving abilities.