Join the nine dots in a 3-by-3 grid by 4 straight lines without raising the pencil. Solve in 30 mins. Learn how to solve the tough puzzle from the solution.
Connect the nine dots riddle
Join the nine dots in a 3-by-3 two-axis grid by 4 straight lines without visiting a dot more than once and without raising your pencil from the paper. The nine dots are on a plane surface.
Time to solve: 30 minutes.
The nine dots in a 3-by-3 two-axis grid shown.
Hint: The puzzle is hard and demands out-of-the-box thinking.
Solution to Connect the nine dots riddle: Analysis and strategy decision
Soon you figure out that the puzzle is hard. To solve, you may choose either trial and error or systematic reasoning.
As trial and error is arbitrary with no certainty of result within a reasonable time, it is rejected.
First step in any serious problem-solving is to make a quick trial, analyze the results of the trial and make a key conclusion.
Solution to connect the nine dots riddle: first trial and analysis
As a first trial, the nine dots are joined in the simplest manner violating no condition.
What clues can we get from this simple trial?
Observation: To connect the nine dots, three independent straight lines each joining 3 dots are used. But to connect the 3 lines, 2 more straight lines are used. The joining is invalid with 5 straight lines.
Reasonable conclusions that may be made,
Conclusion 1: Two groups of lines to analyze—primary lines joining the dots and lines joining the primary lines. These are secondary lines. In the trial, we used 3 primary lines and two secondary lines.
Conclusion 2: Secondary lines do not contribute to joining more dots and so are wasteful. Number of secondary lines should be least. It should preferably be zero.
Observation: A property of the connections not considered is the direction of the lines. All 5 lines in the figure are directed vertically or horizontally with no oblique line.
Conclusion 3: Whichever way you join the dots with these mutually perpendicular sets of lines aligned with the two axes, solution cannot be reached. At least one line must be oblique. Such an oblique diagonal reduces the number of wasteful secondary lines by 1.
To try out the new ideas, we need a second experiment.
Solution to connect the nine dots riddle: Second trial and analysis of its results
It is identified that the most efficient oblique line joins three dots,
Conclusion 4: One primary line in the solution must be one of the two diagonals.
Where should the joining pencil move after joining three diagonally placed dots?
Conclusion 5: It can only be along one of the two adjoining mutually perpendicular outer boundary sides of the grid. This directly contributes to a better situation by reducing the number of secondary lines (joining two primary lines) by 1.
The result of the actions shown.
Conclusion 6: The two lines, one diagonal and the other a vertical line, together join 5 dots and this combination must be a part of the final solution.
Point is, how to connect the remaining 4 dots and join the resulting lines with the two existing lines?
Let us complete this second trial by joining the rest 4 dots in a simple way. Result shown.
Total number of lines is now 6, again an invalid solution. So obvious ways of joining the dots cannot give you the solution. We must discover an innovative new way.
Property analysis of connections to identify a taken-for-granted property: Connect the nine dots riddle
When a problem demands out-of-the-box idea to fix, usually a property of the primary entity in the obstacle (in our problem, a joining line) remains invisible. As a result, choice of considering the invisible property is overlooked. If the obstacle could solely be fixed by adjusting (or changing) this overlooked property, the problem will prove impossible to solve (as we have ignored the property).
Let us list out the properties of our main entity, the joining line.
- First property: A joining line must be straight. No curved line can be used. It is a puzzle condition. We cannot use this property for solution—obviously we must use straight lines.
- Second property: All joining lines are to be drawn on a plane surface. This is true, as the nine dots are on a plane surface. This cannot also be used to break the barrier.
Any other property of the joining lines? Did we take any property for granted and forgotten that we can change the property?
Do we have to draw a line always inside the 3-by-3 grid-box? Is it a must? Insistent questioning brings out the new idea. The property that remained invisible is identified now,
- Property taken-for-granted: The joining lines to be drawn only inside the 3-by-3 grid.
Following shows what is meant by this,
Lines in the solution are assumed to be restricted to the grid-box.
Can’t a joining line pass over a dot at the border and continue outside the grid-box to connect with a second extended line! Well, surely it can. Nobody prohibits is.
More importantly,
If two primary lines continue beyond the grid-box and join each other, it saves a precious extra line for linking the two.
This must be the sought after change in property. And it should be the key pattern as well.
Key pattern: At least two lines must continue beyond the last dot joined and move out of the grid. Number is two as a line continued beyond the grid must return inside the grid by joining with a second beyond the grid line.
The key pattern shows the way forward, but not directly the solution.
Solution to connect the nine dots riddle: Joining the remaining two pairs of dots
For convenience of understanding, the two lines joining 5 dots in the 9 dots grid shown with the dots labeled.
The pencil starts from dot 1, moves over 5 dots and stops at dot 3. Five dots are connected without violating a condition and four dots stay unconnected.
The analysis and joining action now are segment by segment, joining the 4 remaining dots by independent line segments first and then connecting the segments.
How many line segments are needed to join these 4 dots?
Conclusion 7: Only way to join two pairs of dots is by two independent line segments. The number cannot be 1 or 3.
Joining the two remaining pairs of dots shown,
Last challenge is to connect the four lines.
Solution to Connect the nine dots riddle: Final stage of joining the line segments
Observation: Nine dots are already joined by four line segments. Quota of line segments exhausted. The segments must be joined using no more line segments.
How to connect, line segment 2-4 with line segment 8-7?
Conclusion 8: Extend line segment 2-4 and line segment 8-7 till the two meet at a point outside the grid-box.
The two pairs of connected line segments are to be joined now.
Better expressed, the problem reduces to connecting dot 2 with dot 3 using no more line segment. And that is easy.
Conclusion 9: Extend line segment 4-2 and line segment 6-3 to meet at a point outside the grid-box.
That’s the solution finally.
All four connecting lines move out of the grid-box to get joined with each other. Truly an out-of-the-box solution.
Where should the connecting pencil start? Two choices.
Connecting the dots may start at dot 1 and end at dot 8 OR it may start at dot 8 and end at dot 1. Both are same.
An alternative is to start by connecting the diagonal 3-5-7.
This looks to be a second solution. On second thoughts, if you rotate one of the two, it coincides perfectly with the other.
That means, only one rotationally unique solution possible.
The critical breakthrough in this hard puzzle is provided by property analysis trying to find the taken-for-granted, invisible attribute or property. This is a powerful inventive technique.
Try the inventive technique when you face with an impenetrable barrier while solving a hard problem, in academics or in real-life.
End note
The classic riddle is one of a kind with no easy intuitive solution at all. Solution by random trial is harder, if not impossible. Solution needs elusive out-of-the-box thinking sought after all over the world. In this solution, the inventive Property Change Analysis Technique is used to arrive at the out-of-the-box unthought-of possibility.
Know how to solve difficult problems easily without wasting time on random attempts
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