Connect 7 dots with 6 straight lines so that each line connects 3 dots. Challenge is in placement of the dots. Think out-of-the-box.

### The out-of-the-box thinking puzzle of connecting 7 dots with 6 lines

Place 7 dots in such a way that the dots can be connected by 6 straight lines, but each line must connect 3 dots.

Time to solve **20 minutes.**

Give it a good try. It will be a great pleasure if you can solve it yourself.

### Solution to the Connect seven dots with six lines puzzle

Realization dawns quickly that the puzzle cannot be solved easily. Learning more of the problem in uncertain situations needs **well-judged trials.**

*A well thought out trial and its analysis is equivalent to the method of prototyping used frequently in design of new products.*

Avoiding random trial and error, first task is to identify the main reason behind the difficulty of the solution and what is the necessary property of the solution to overcome the difficulty.

**Observation:**The number of lines and the number of dots are very close together—too few dots to be connected by too many lines (with the additional condition of each line connecting three dots).

**Realization:**

**Prime objective:**The dots must be as compactly arranged as possible with most efficient use of the connecting lines. No line should connect less than 3 dots.

With compactness in mind, the first very **compact dot arrangement formed as the first trial:**

This must be one of the most compact arrangement of 7 dots connected by straight lines each connecting 3 dots. But, the arrangement is one line short—5 lines connecting the dots—not the solution.

A **second trial with most compact dot placement:**

This also is one line short.

**Realization:** Conventional approach won't work. A more complex **inventive approach is needed** for solving this hard puzzle.

#### Deeper analysis based on inventive property analysis technique

When faced with a barrier that can't be overcome in any conventional way, usually the **inventive problem solving technique of property analysis produces effective breakthrough.**

In this method, a critical analysis of the properties of the problem is carried out

to discover a property that is all along assumed as taken for granted.

Checking what are known:

- All 7 dots must be connected.
- The dots must be connected by 6 straight lines.
- Each straight line must connect 3 dots.

These are the inviolable puzzle conditions.

Are we missing anything? Any property of the main object, a straight line, that we have taken for granted?

**A new angle explored:**

- We have assumed all the lines to be on a plane surface. But, this a property that we can't change.

Any other taken-for-granted property of the lines we can think of? Insistent probing on a single issue produces results.

Yes,

we have assumed the lines to be non-overlapping and nothing stops us to use overlapping lines.By now, it is a certainty that non-overlapping 6 straight lines cannot connect 7 dots with 3 dots on each line—overlapping straight lines must be used to arrive at the elusive solution.

This is the critical breakthrough idea. But, the solution is not yet easy to visualize.

A few trials with overlapping lines and analysis of the results should clear up the situation.

#### First impulse: Use maximum overlapping

All 7 dots placed on a single straight line has maximum overlapping.

How many distinct straight lines connecting three dots in this arrangement with maximum ovelap?

To count, the dots are numbered. The number of distinct straight lines connecting 3 dots are,

**1-2-3**,**2-3-4**,**3-4-5**,**5-6-7**,**1-3-4**,**1-3-5**,**1-3-7**and so on.

The distinct lines are way more than 6. This is the number of combinations of 3 objects chosen out of 7. Mathematically the number is, ^{7}C_{3} = 35.

We must reduce the number of dots in an overlapped line.

**Reducing overlapped number of dots to the maximum extent,** analyze first, the **4 dots overlapped on a single straight line.**

The distinct lines connecting three dots are: **1-2-3**, **2-3-4** and **1-3-4**, just 3. What about the remaining 3 dots? Remaining 3 dots may be placed on a 4th straight line. **Result:** in total two lines short.

Critical realization:6 lines connecting 3 dots eachcan only be formed by two sets of 4 dots, each placed on one straight line.

But, this will make number of dots to 8, one more than what we have.

#### Form the final question, discover the answer and arrive at the solution

**Final probing question:** how to use two sets of 4 dots each placed on one straight line AND **reduce the total number of dots needed by one?**

**Second breakthrough idea:**Just*connect two sets of 4 dots on a line at ONE DOT.*The*common dot participates in both the 4 dot sets.*

Seven dots connected by 6 distinct straight lines with each line connecting 3 dots solution:

The inventive

property change analysis produced the desired critical breakthrough.Analysis of the property of overlapping and its change helped to proceed further, but a second breakthrough idea was needed for the solution.

It's a truly unique out of the box riddle.

### More puzzles to enjoy

From our large collection of interesting puzzles enjoy: * Maze puzzles*,

**Riddles**,

*,*

**Mathematical puzzles***,*

**Logic puzzles***,*

**Number lock puzzles***,*

**Missing number puzzles***,*

**River crossing puzzles***and*

**Ball weighing puzzles***.*

**Matchstick puzzles**You may also look at the full collection of puzzles at one place in the **Challenging brain teasers with solutions: Long list.**

*Enjoy puzzle solving while learning problem solving techniques.*