Place six numbers 1, 2, 3, 4, 5, 6 in six circles as in the figure. Only condition is: no two consecutive numbers can be placed in two connected circles.

**Recommended time to solve**: 20 minutes.

Do you find it difficult? Just try to solve the riddle without first discovering the special patterns inherent in the puzzle and decide whether is easy or difficult. Random approach would take a lot of time.

Let me tell you a secret:

Most, if not all, problems can be solved elegantly and efficiently if you can discover and inventively use special patterns hidden in the problems.

### Solution to the Place Six Numbers in Six Circles Riddle

After trying to place the six digits in the circles for a few minutes without any attempt to discover the special inherent patterns, I realized this common tendency to solve a problem using random attempts won't work.

#### Step 1: Identification of the crucial property to analyze in the riddle

What is the **crucial property** of the six numbers and the six circle figure?

It can be expressed in a single word ADJACENCY.

*Consecutive numbers means adjacent numbers. The six circles that are connected with each other are adjacent. And the numbers must be placed in the circles so that no two numbers are placed in adjacent or connected circles.*

The puzzle dictates that you can't place two adjacent numbers in two adjacent circles.

This is why **analysis of adjacency** in both the set of numbers and the six-circle figure is so crucial.

#### Step 2: Discovering the special patterns inherent on the set of six numbers and the six-circle figure

Taking up the six numbers to analyze their adjacency, I am a bit surprised to discover that the end numbers 1 and 6 are different from the other four in-between numbers with respect to adjacency.

Each of 1 and 6 is adjacent to only ONE number, whereas each of the other four numbers 2, 3, 4 and 5 are adjacent to TWO numbers.

Likewise, when I looked at the six-circle figure more closely, I was not surprised to find,

Only two circles labeled 1 and 4 in the figure below are connected to

maximum number of 4 circles. Each of theother four circles are adjacent (or connected) to just three circles.

Adjacency varies in both set of six numbers and the six circles in the puzzle figure.

#### Step 3: Using the special patterns to place the first two numbers in two circles perfectly

At this point I used my **common sense reasoning:**

I must put the two numbers 1 and 6 with LOWEST adjacency in two circles labeled 1 and 4 with HIGHEST adjacency.

This maximizes the overall ease of placing the rest four numbers 2, 3, 4 and 5 in suitable circles because,

- Each of 1 and 6 are adjacent to only one number: increased flexibility in further placement, and,
- The selected two circles labeled 1 and 4 with maximum four connections are taken care of (each of rest four circles has three connected circles): overall flexibility of further placement increased.

This is the only way the flexibility of placement of the numbers can be maximized.

Still a decision is to be made: which of 1 and 6 to place in which circle!

But that is easy to decide. From the point of view of adjacency, the numbers 1 and 6 are equivalent to each other as are the two circles labeled 1 and 4. I can place any of 1 and 6 in any of the two selected circles.

I decide to place number 1 in the circle labeled 1 and the number 6 in the circle labeled 4. Figure follows.

#### Step 4: Placing the rest four numbers 2, 3, 4 and 5: Solution to the riddle

Placing 2 is easy. It can be placed only in the circle labeled 6 not connected to circle 1, the home of number 1. Out of 5 possible connections to a circle, the circle 1 has 4 connected circles and one is left for the number 2 consecutive to number 1.

Likewise, number 5 consecutive to number 6 can only be placed in circle 5 not connected to circle 4, the home of number 6.

Left are numbers 3 and 4. Number 3 cannot be placed in circle 2 as it is adjacent to the circle 6 where number 2 is placed. Number 3 can be placed only in circle 3 and finally, number 4 be placed in circle 2. No violation of the adjacency condition.

Final solution below.

No single assumption. All decisions based on sound logic. Step by step goal directed assured approach. Based on special pattern identification and inventive use of the pattern. Not difficult as it seems at first.

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