Crack the hidden code of missing number puzzle to test your pattern identification and math reasoning Skills. Can you solve it in more than one way?

This type of puzzles test your inventive thinking skill as well. After all, inventive thinking is: ability to discover possibilities that are not easy to get hold of.

## The missing number puzzle with a well-hidden code

If,

**4 + 1 = 53**

**3 + 2 = 51**

**4 + 3 = 71**

** 5 + 2 = 73,**

then,

**6 + 3 = ???**

*You have good logical thinking skills if you spot the hidden operation in each equation and crack the code in 2 minute's time.*

### How to crack the hidden code of the missing number puzzle: Step by step logical approach

#### Property of the missing number puzzle: Analysis

- One rule must satisfy each equation.
- The digits are not coded, otherwise there would have been more variables than equations making the puzzle unsolvable.
**Conclusion:**each digit in the equations is the actual face value of the digit (for example, 2 is 2 and 6 is 6).- The key lies in an operation on the left hand side of each equation.

#### Focus is on the left hand side part: Logic Analysis and Math Reasoning

**Logic says,**

- Usually the arithmetic operations are not tinkered with. The result on the right side of an equation should be the actual value of the number. We'll make these two assumptions to start with.
**Conclusion:***Under these assumptions*, the key operation must be**multiplying the first term by a unique multiplier and the second term by a second unique multiplier in all four equations.**

#### Finding the two multipliers and crack the code: Isolating the second multiplier to two values

With a bit of trial and error, it would have been dead easy to find the pair of multipliers, but I am all for finding the key using basic math knowledge and reasoning.

We'll use our **basic knowledge of factors and multiples** to analyze the equations for finding the pair of multipliers. **We'll analyze the first two equations to start with.**

**4 + 1 = 53**

**3 + 2 = 51**

- In the second equation,
*the result being divisible by 3*,**the multiplier of the second term must also be divisible by 3**. This conclusion is based on basic factors and multiples concept. The second multiplier is then one of 3, 6 or 9. - But, from the first equation, the
**second multiplier can't be even as the result is odd.** **Conclusion:**The**second multiplier must be one of 3 or 9.**

#### Pinpointing the second multiplier as well as the first: Solution

Now we will *checkout the third equation for the validity of second multiplier values.*

**4 + 3 = 71**

**Observation:****If the second multiplier is 3**,**result**71 - (3 x 3) =**62**is**not divisible by 4.****Conclusion:**If at all our assumptions made at the very beginning were correct, the**second multiplier must be 9.**

From first equation,

**First multiplier** = (53 - 9) / 4 = **11.**

Let us see *whether the pair of multipliers 11 and 9 satisfies the rest of the equations:*

3 x 11 + 2 x 9 = 33 + 18 = 51

4 x 11 + 3 x 9 = 44 + 27 = 71

5 x 11 + 2 x 9 = 55 + 18 = 73.

**The rule works for all four equations.** This must be **a key rule** (*there might be another key rule, would you try at this point?*).

**Solution:** 6 + 3 => 6 x 11 + 3 x 9 = 66 + 27 = **93.**

In this solution, **I have used logic and math reasoning skills.**

Combine logical thinking with Systematic approach, and you would get the solution with confidence.

Now let me tell you the answer to the little riddle I posed just a moment ago.

To tell you the truth, *I had ignored the sanctity of equation and addition symbols altogether and formed the two digits of the number on the right by a simple operation on the two terms on the left.*

#### The second method to crack the code

First **add the two terms on the left to form the first digit of the term on the right** of equation symbol.

Next **subtract second term from the first to form the second digit of the term on the right**.

**First equation:** (4 + 1) = 5 and (4 - 1) = 3 forming result 53.

**Second equation:** (3 + 2) = 5 and (3 - 2) = 1 forming result 51.

**Third equation:** (4 + 3) = 7 and (4 - 3) = 1 forming the result 71, and,

**Fourth equation:** (5 + 2) = 7 and (5 - 2) = 3 forming the result 73.

**Solution:** **Fifth equation:** (6 + 3) = 9 and (6 - 3) = 3 forming the **result 93.**

In this second solution instead, **I have used my pattern discovery skills.**

This way of solving a problem in more than one way I call **Many ways technique**. *This is a powerful and valuable general problem solving approach.*

The **biggest advantage of solving a problem in more than one way** (if you are not in an emergent situation) is:

It enriches different sets of problem solving skills in your brain increasing your overall problem solving ability.

Examples of applying many ways technique are many in our large collection of problems solved.

*I am happy if you have solved this cute little riddle taking the two different approaches yourself.*

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