## The 7 squares matchstick puzzle should be easy if you apply reasoning based on matchstick puzzle concepts

The 7 squares matchstick puzzle: Move 2 matches to make 7 squares in the puzzle figure. Time 10 mins. Apply reasoning based on matchstick puzzle concepts.

#### The 7 squares matchstick puzzle

In the matchstick puzzle figure, move 2 matches to make 7 squares with each matchstick part of a square and no stick kept hanging.

**Time to solve:** 10 minutes.

First try to solve yourself in your own way. It is quite possible that you would be able to solve this not so difficult stick puzzle within 10 minutes.

Our solution follows a *step by step reasoning approach* to arrive at the answer figure quickly with minimum guessing attempts.

### Move 2 matches to make 7 squares matchstick puzzle: First stage of solution to specify action requirements - Problem analysis and selection of most appropriate technique

In solving a matchstick puzzle (or solving any puzzle or problem), you would decide first which technique (or strategy) would be most effective in solving the puzzle. This decision would invariably be based on **broad and quick analysis of the problem** evaluating the strategy or technique that would produce best results.

This is the important first stage of any systematic approach to problem solving.

For this puzzle, based on matchstick concepts (we assume you have some experience) you would draw your conclusions quickly,

- The
**Common stick analysis technique**won't be applicable for this puzzle as this has*many squares of different sizes with few common sticks.* - You won't select also the
**End state analysis technique**of*comparing the possible final figure and the puzzle figure to select the possible final figure that has maximum similarity*with the puzzle figure. This is because, imagining the possible final figures itself is not an easy task that can be done quickly.

**Note:** Both these techniques proved to be very effective in solving many of the earlier stick puzzles. To know more about these techniques, you may like to go through the following two,

**End state analysis technique:** **5 squares to four squares in 2 stick moves.**

**Common stick analysis technique: ****Make 5 squares from 6 in 2 stick moves - 6 square matchstick puzzle.**

But yes, you can analyze the number of closed shapes at start and at end. This is **shape analysis technique**. You select it as the most appropriate approach for this puzzle.

It is easy to draw the **first conclusion** from this analysis,

The target figure would have 7 squares—an increase of 2 squares from the puzzle figure.

And the **first action you can specify** from the puzzle description as easily,

To create 2 more new squares in 2 stick moves, effectively you have to create 3 new squares by moving 2 sticks.

The **reason behind this action requirement is**—by moving 2 sticks you would destroy at least 1 square, and so you have to create 3 new squares to create 7 squares in the final figure.

Now you realize that by moving 2 sticks, you might destroy 2 squares, not 1. In that case, you would have to create 4 new squares by moving two sticks.

With this realization, you define the stick movement action requirement with more stringency as,

To create 2 more new squares in 2 stick moves, you would

destroy just 1 squareand create 3 new squares by moving the 2 sticks.

**Till now** you have analyzed the puzzle broadly and drawn the conclusions without looking at the puzzle figure.

The conclusions and action requirements would be true for any matchstick puzzle that needs increasing number of closed shapes by 2 in 2 stick moves. These are independent of any specific puzzle.

This is why **shape analysis technique is ABSTRACT** and hence has more power and applicability.

Now only you

analyze the puzzle figure in detailsand realize thatsome of the sticks are similar to each other with respect to their equivalent positions in the whole structure of sticks.

### Move 2 matches to make 7 squares matchstick puzzle: Second stage of solution to narrow down stick move analysis - Analyze given puzzle figure to Classify types of sticks for movement

A **natural technique** that is used in many problem situations is **Classification into types technique**.

Realizing that **some of the sticks occupy exactly same equivalent positions in the figure**, you decide to apply this powerful technique on the puzzle figure.

If you study the effect of moving

any of these sticks of equivalent positions,you would know what would happen if you move any other stick equivalent to stick you have studied. Effectively,by testing one stick in an equivalent class you study all sticks in the class—a great reduction in time to test.

**Note:** Technical name for these equivalent classes is **Equivalence Classes.**

**Positional equivalence of sticks is the classification criterion** for this puzzle figure.

The following figure shows classification of the sticks into equivalent types.

You would then **divide the sticks in three clearly independent and equivalent class types, **and simultaneously study the results of moving the sticks in each type,

Class type A:4 pairs of CORNER sticks of the large outer square: 1 pair labelled as A1, A2:No, you cannot move any of these pairs simply because by moving a pair of corner sticks, you would destroy 2 squares—the corner square and the large outer square. This wouldviolate the specific requirementsyou have drawn by shape analysis. These 8 sticks areidentified as invalid for further analysis.

Class type B:4 sticks on the MIDDLE of the 4 sides of the large outer square: 4 labelled as B1, B2, B3, B4:If you move any two of the four, you won't violate the shape analysis requirement (you would destroy just 1 square). But as you think about the result of moving 2 of these 4 sticks, you find you would alsocreate two hanging sticksunattached to any square. Thisviolates the second requirement of the puzzle description.These 4 sticks alsoare identified as invalid.

Class type C:4 pairs of INSIDE CORNER sticks of the large outer square: 1 pair labelled as C1, C2:Moving such a pair destroys just 1 inside corner square and does not create any unattached hanging sticks to be taken care of later.So you are sure that you would solve the puzzle by moving any pair of these 4.

With sticks identified for movement, the only task remains is—finding the places for the sticks moved.

### Move 2 matches to make 7 squares matchstick puzzle: Third stage of Solution - Analyze possible places for moving 1st stick and the 2nd stick for final solution

At this final stage, you further breakdown the problem into **analysis of 1st stick move** and then the **2nd stick move** and try out places where each stick can be moved for creating the solution figure of 7 squares.

This is the natural way of moving the sticks. Isn't it?

But before you start the trial, even at this late stage, you *decide to do shape analysis on number of squares again.*

To create 3 new squares by the 2 moves,

When you move the first stick and destroy 1 corner square you must create at least 1 new square. As creating 2 new squares by the first stick move is impossible here, you would specify the requirement as—you MUST create 1 new square.

Now you would get down to identifying the feasible promising options where you can place the just moved 1st stick.

Following figure shows the two feasible promising place P1 and P2 for the 1st moved stick.

First stick C1 can be moved to any one of these two. Move it to P1 and create the new square A.

Doing shape analysis again, now you can easily specify the strict requirement for the second stick move,

By the second stick move, you won't destroy any square, but you

MUST create two new squares.

**Question is:** how can you create two new squares by one stick?

**Answer is clear:** You would choose one of the remaining feasible place options identified in the 1st stick move so that an additional NEW SQUARE of size larger than the small squares but smaller than the large outer square is created.

Finding the solution is immediate—remaining place option is just 1.

Move stick C2 to place P2. Two new squares B and C are created.

The following shows the final solution figure.

Here, just solving of the puzzle has not been the goal—the goal was to **solve it with minimum guesses** or trials, and of course highlight that you can approach such a simple or more complex problem absolutely systematically with reason and appropriate techniques.

If you are curious, you may go through the possible representation of a model for general problem solving and innovation. If you compare the solution process of the puzzle and the possible model, you would find these to be quite a bit similar.

### A feasible model for general problem solving and innovation

**Step 1:** **Problem definition in precision:** Many times, especially in real life problems, this stage of clearly understanding what the problem is takes time, effort and will. For the puzzle, the problem is already precisely defined.

**Step 2:** **Problem analysis and strategy selection:** Make an overall brief analysis of the problem to identify appropriate strategy and resources that you would use for solving the problem.

**Step 3:** **Taking actions:** According to the strategy, when you analyze the problem more thoroughly and take solving actions, you would identify main barriers to solution as well as patterns and techniques to overcome the barriers.

**Step 4:** **Fine tune strategy and techniques:** In step 3, you may feel the need to repeat stage 1 to 3 by having a relook at the strategy selected and tune it.

**Step 5:** **Reach a limited number of possible solutions:** After going systematically through the steps, you would either reach the solution OR identify a small set of possible solutions.

**Step 6:** **Trial coupled with new way of thinking:** Reach the most desirable solution by planned study and trial on the set of possibilities as well as innovative breakthrough ideas.

**Innovation is inseparable from problem solving. **

Throughout the stages, unless innovative new way of thinking is active all the time, most desirable least cost solution may never be reached.

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