Welcome back, everyone! In today’s blog post, we’re diving into an intriguing question from a past St. Olave’s 11+ paper. This one revolves around a number machine designed by Raj. Unlike typical number machines, this one’s a bit unusual because we don’t know what it does exactly. Our task is to figure it out using given examples.

## Understanding the Number Machine

Raj has created a number machine where two numbers go in, and an answer pops out. We’ve got two examples to help us decipher its function:

Example 1:

- Input: 3 and 8
- Output: 25

Example 2:

- Input: 6 and 7
- Output: 43

But the trick lies in understanding how these outputs are derived from the inputs.

### Initial Thoughts and Attempts

Let’s first try to add the numbers:

- (3 + 8 = 11) -> But we get 25. What if we double 11 and add something?
- Similarly, (6 + 7 = 13) -> But the output is 43.

Another approach is multiplication followed by addition. Notice 25 is close to (3 \times 8) and 43 is close to (6 \times 7):

“25 is quite close to (3 times 8=24), so maybe add 1?”

Let’s establish a general formula: if (a) and (b) are the input numbers, our output could be (a \times b + 1).

### Validating the Hypothesis

To prove this, let’s see if the formula holds for both examples:

- (3 times 8 + 1 = 24 + 1 = 25)
- (6 times 7 + 1 = 42 + 1 = 43)

It works!

## Applying the Formula

Now our challenge is to fill in missing numbers using this formula. Consider the diagram from the transcript where ( 4 ) and ( 9 ) are inputs, and we need to find the output:

- (4 \times 9 + 1 = 36 + 1 = 37)

We’ll use another example to explore our reasoning further.

### Solving for Unknown Inputs

In a case where the output is given as 15 and one input number is 2, we find the other input, denoted (b):

- The equation is (2 times b + 1 = 15)
- Subtract 1 from both sides: (2 times b = 14)
- Divide both sides by 2: (b = 7)

Checking our result:

- (2 \times 7 + 1 = 14 + 1 = 15)

### Simplifying with Algebra

Using algebraic methods proves beneficial:

- Equation: (2b + 1 = 15)
- Solve step-by-step:
- Subtract 1: (2b = 14)
- Divide by 2: (b = 7)

## Visualizing the Concepts

#### Example Calculation

`Input: 4, Formula: 4 x 9 + 1 = 37`

## Enhancing Problem-Solving Skills

Working through number machine problems hones both algebraic and logical reasoning skills. Try more such questions to sharpen your mathematical abilities!

### Final Thoughts

The puzzle-like nature of number machines is fascinating and enriching. They challenge us to think critically and apply mathematical principles creatively.

Remember to look for patterns, validate your hypotheses, and enjoy the process of discovery:

“Add, multiply, subtract, even square sometimes and see if that comes close.”

### Challenge Yourself

Take on additional number machine problems and see how this understanding helps you:

- What if one input and the output are given?
- Could there be different operations involved?
- How quickly can you identify the pattern?

Feel free to share your experiences and let us know how the video has aided your learning.