## The puzzle: Move 5 sticks in the heads-up kite and make it nose-dive

This is the eighth matchstick puzzle with solution. **You have to move 5 matchsticks in the figure of kite to make it upside down as if diving down. In how many ways can you do it?**

*Recommended time is 20 minutes.*

You should enjoy solving this unusual puzzle.

By chance if you can't find the solution in time or feel curious about how we have solved the puzzle by step by step analysis, then you should go ahead and go through the solution.

### Solution to the puzzle: Move 5 sticks in the figure of kite to make it diving down nose first

#### Analysis of the structure and knowing precisely what you have to do

Do we know exactly which figure of kite we have to form finally? The **first step** should be then to *form this final figure*. It would be the same kite, but flipped upside down by 180 degrees.

In solving earlier matchstick puzzles except one, *you didn't know the final solution figure at the beginning*. In this aspect you have apparently an advantage—*you know the final solution figure here at the start itself*.

Your only job is then to **identify 5 sticks** from the problem figure and move the sticks suitably to transform the heads up kite on the left to the solution figure of nose down kite on the right. Easy?

Let's see.

You find quickly that common stick concept or counting stick technique will not help you at all. To solve this puzzle you have to adopt a different approach.

#### Solution using End State Analysis Approach

In 5 square puzzle solutions earlier we have used this very powerful approach to compare promising possible final figures with the problem figure and select the promising figure that had maximum similarity with the problem figure.

We compared the End state and the Initial state first to identify the most promising End state, and then found out how we could transform the selected possible final figure from the problem figure.

In our problem now, we know the final solution figure. So *how and why* would we use the End State Analysis?

In End State analysis the *key action is comparison of two matchstick figures* for judging maximum similarity between the two. The *assessment is done visually*.

Here also we will assess the similarity between the two figures—the final solution and the problem. But we will assess **how many sticks** **are common** between the two **when they are superimposed in various positions**.

That's the *key action precisely specified*. We won't assess the similarity with respect to number of squares in same position, but *we'll assess instead, the number of sticks in same position by superimposing the two figures in promising possible ways.*

#### Structural Analysis for finding Promising Ways of Superimposition—Axis of Symmetry

While analyzing the structure of the kite, you would surely notice that on both sides of an imaginary vertical line passing exactly through the middle of the figure, the two parts of the kite are mirror images of each other. This is the line of symmetry.

Why do we talk about this line anyway? The reason lies in its great use of fixing the relative position of the two figures when superimposed.

Let's show the* axes of symmetry* of the two figures first.

The *blue vertical line* and the *brick-red vertical line* are the axes of symmetry of the problem figure and the solution figure respectively.

You may imagine to hold the solution figure by its axis of symmetry fixed to the kite body, and move it leftwards to examine the effect of superimposing it on the problem figure (assuming that the line is a fixed part of kite body with all the sticks also fixed with each other).

#### First promising superimposition of the two figures

At the very first attempt, you would superimpose the four squares, the body of the kites, on one another. The effect is shown below with the sticks of the solution faded and the two figures separated a little for ease of visualization.

You have superimposed the tail-joint of the nose-diving solution kite figure with the nose-joint of the puzzle kite figure so that the body of the two figures consisting of two sets of four squares are superimposed on one another.

How many sticks failed to match? It is four on the left of common axes of the two figures plus four on the right of the axes—a total of eight sticks.

Just move sticks 4 and 8 to form the tail of the solution figure (stick positions 3 and 7), stick 2 to stick position 1 and lastly stick 5 to stick position 6. You get the nose-diving kite. In total you have moved 4 sticks.

Instead of solving the given puzzle, you have solved a new puzzle,

Move 4 sticks in the figure of kite to make it diving down nose first.

So you have to *separate the two axes of symmetry when superimposing* and make a second attempt.

The second fact you discover as,

Number of sticks unmatched in the superimposed two figures must be 10, so that moving 5 sticks of puzzle figure to 5 new positions of the solution figure (with rest of the sticks of the puzzle figure kept untouched) you can form the nose-diving kite.

#### Second promising superimposition of the two figures

This time you are more experienced and coincide the tail-joint of the solution figure with the first right-bottom joint of the puzzle figure. The second superimposed figure is shown below.

Now the number of unmatched sticks as a whole is 10. This should solve the puzzle.

And indeed it is so. Move sticks 2, 3 and 4 to positions 7, 8 and 9, stick 1 to position 6 and lastly stick 5 to position 10 to complete the tail of the nose-diving kite. You have moved exactly 5 sticks to 5 new positions to transform the puzzle kite to the nose-diving solution kite.

Mark that the two axes of symmetry are separated from each other as expected.

Now the second part of the puzzle—how many such solutions are possible?

#### One more solution to the nose-diving kite puzzle

At this point you are fully experienced on the structure of the kite figures, the process of superimposition and formation of the solution figure. You can easily answer that, only one more solution to the puzzle is possible by equivalent left positioning of the red axis of symmetry of the solution figure.

You should be able to form this second solution yourself easily now.

### Summary

Though we have shown the process of superimposing the nose-diving kite on the puzzle kite in the figures easily, when you actually solve the puzzle, it won't be so easy for you—the sticks won't be fixed together in a figure—you have to either place the sticks one by one alongside matching sticks or better still, carry out the superimposition in your mind. With practice that shouldn't be difficult.

The second interesting point to mark in this puzzle is the use of the same End State Analysis approach, but not counting similar closed shapes of squares (or triangles), but counting unmatched sticks instead.

This is customizing or fine-tuning the same method slightly for solving a different type of problem.

We do this often in real life problem solving also.

### End note

Lastly, **to solve matchstick puzzles you don't need to know maths or any other subject**—you just have to identify key patterns and use your inherent analytical reasoning skills to home in to the solution with assurance and speed.

*The way to the solution, the approach, the thinking are more important than the solution itself. The concepts and methods stay with you and are enriched as you proceed to solve more and more problems.*

And you can take even a short break of fifteen minutes to create a new puzzle of your own and spend the time solving it. If you do it regularly it will sharpen your pattern based problem solving skill, an extremely valuable skill.

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