Why Does This Math Puzzle Always Equal Five?

Many people are often fascinated by math puzzles as they provide a fun and intellectual challenge. One such puzzle involves picking a number, performing a series of mathematical operations, and producing a consistent result in this case, the number five. Let's break down the steps mathematically to understand why the answer is always five.

Mathematical Breakdown of the Puzzle

Let's denote the initial number by (x).

Double the number: (2x)

Add ten: (2x 10)

Divide the sum by two: (frac{2x 10}{2} x 5)

Subtract the original number (x): ((x 5) - x 5)

No matter which number (x) you choose initially, the final result will always be five. This is why the answer is five. The operations cancel out the original number and the addition of ten is halved, leaving a constant five.

Another Example with Detailed Steps

Let's consider another example with variables to further illustrate the concept:

Let (m) be any real number (or an imaginary number (text{iR})).

The premises of the problem are:

2m   10 / 2 - m  5

Let's simplify the premises:

2m   10 / 2 - m  52m   5 - m  5m   5  5m  0

This means that the initial number ((m)) can be any real or imaginary number since any (m) will simplify to five. For instance:

If (m frac{1}{2} 1 sqrt{4}), the result is verified:

2(frac{1}{2}   1   sqrt{4})   10 / 2 - (frac{1}{2}   1   sqrt{4})  51   2   2sqrt{4}   5 - frac{1}{2} - 1 - 2sqrt{4}  55  5

If (m 1), the result is also verified:

2(1)   10 / 2 - 1  52   5 - 1  56 - 1  55  5

If (m √4), the result is also verified:

2(√4)   10 / 2 - (√4)  52sqrt{4}   5 - sqrt{4}  54   5 - 2  57 - 2  55  5

Explanation Without Variables

It might also be helpful to consider the simpler explanation without variables:

Pick any number and call it (x).

Double it: (2x).

Add six: (2x 6).

Divide by two: (frac{2x 6}{2} x 3).

Subtract three: ((x 3) - 3 x).

This shows that no matter what number (x) is, the end result is always the original number. If we adjust the example slightly by changing the addition and subtraction steps, we get:

Double an initial number (x): (2x).

Add ten: (2x 10).

Divide by two: (frac{2x 10}{2} x 5).

Subtract the original number: ((x 5) - x 5).

Here, the operations cancel out, leaving the constant five.

Conclusion

To summarize, this math puzzle works because the operations are meticulously designed to cancel each other out, resulting in the number five. Whether you use variables or not, the underlying principle remains the same. This puzzle is a great example of algebraic manipulation and logical reasoning.