Matchstick puzzle: Move 2 sticks to make 5 matchstick shapes
Move 2 sticks to make 5 matchstick shapes of squares, triangles or rhombuses. How many such new figures can you make? Total recommended time 10 minutes. No matchstick moved can be placed in a position of an already moved stick.
Try solving the puzzle. It will be interesting we can assure.
A Bonus Puzzle
With the experience of solving the first puzzle it should be easy for you to solve this variation.
Second puzzle of Move 2 sticks to make 5 matchstick shapes (puzzle figure is different from the first)
The Puzzle: In the puzzle figure below, Move 2 sticks to make 5 matchstick shapes. The shapes must be non-overlapping and no matchstick moved can be placed in a position of an already moved stick. How many such solutions can you find?
This is the second puzzle having same mixed matchstick shapes of a triangle, two squares and a rhombus. If you haven't solved the first such puzzle you may try it here.
Because of the mixed matchstick shapes, it is not useful to apply the End state analysis technique on the possible puzzle solution figures that we have used so effectively in solving the puzzle, Move 2 sticks to make 4 squares matchstick puzzle.
Also our usual method of analyzing the number of sticks together with common sticks required are of little use here.
To solve this puzzle instead, we will use,
- the most basic matchstick puzzle concepts,
- analysis of number of matchstick shapes,
- refining requirement specification, and,
- step-by-step reasoning.
If you already are aware, you may skip the concepts in the following section to move straight to the solution.
Concepts needed to solve the mixed matchstick shapes puzzles
Common stick properties
A common stick between two closed shapes,
- destroys two regular matchstick shapes if removed,
- reduces the need for number of sticks to form the two matchstick shapes independently by 1.
Diagonals of a square and a rhombus
In a regular shape like a rhombus A in the puzzle figure shown below, you can place a stick as the shorter diagonal to divide the rhombus into two triangles, increasing the number of closed shapes by 1.
Geometrically it is true that the shorter diagonal divides a rhombus into two equilateral triangles. But you won't use any geometry or math concepts in solving matchstick puzzles, isn't it?
You'll just try out the possibility by actually placing a stick to connect two nearer corners of the tilted four-sided closed shape and check for yourself whether you have got two triangles as a result.
But you can't use a stick as the diagonal of a square because a diagonal is always longer than any of its sides in a square.
Two triangles can form a rhombus but not a square
As you can place a stick as the shorter diagonal in a rhombus, effectively you combine two triangles to form a rhombus. But you cannot combine two triangles to form a square.
These concepts you can understand just by creating the shapes with sticks. You don't really need any geometry background.
We will now solve the puzzle systematically using step-by-step reasoning.
Stage 1: Understanding which two sticks can be moved: Analysis of requirements—Making initial conclusions
The puzzle figure with the sticks and shapes labeled is shown below for easier reference to the components.
The three common sticks are specially identified by tick-marks and are not numbered.
From the problem description the first conclusion can be made straightaway,
Moving 2 sticks, you have to destroy one and only one closed shape and create two new closed shapes to make the number of closed shapes 5 from 4.
This first conclusion is the outcome of analysis of number of shapes.
Think over to verify the truth or falsity of this conclusion.
An immediate second conclusion can be made is,
You cannot move any of the three common sticks as, moving such a stick will create as many as 4 hanging sticks at one go that would be completely unmanageable.
With these two conclusions, the choices of the two sticks that can be moved are narrowed down.
Solution Stage 2: Move 2 sticks to make 5 matchstick shapes Matchstick puzzle: How to create the 2 new closed non-overlapping matchstick shapes
You have reached the heart of the problem.
Make an experiment.
Pick up sticks 4 and 5. You have destroyed 1 shape as expected.
Now you have to place these two sticks to create 2 non-overlapping new shapes.
Can you do it?
Try as you may, you won't be able to do it. This is an infeasible action that won't lead you to the solution.
Let us abandon this path of picking up any pair of sticks and then try to place the sticks.
Instead, focus on how to create the two new shapes assuming that you have already selected two sticks to move.
This is a strategic approach to problem solving adopting the most promising path of analysis and action. This would lead you to the solution faster.
Now as you are forced to think how you will create the two new shapes with 2 moved sticks, you are in the right position to make the third conclusion with confidence,
With each stick moved, 1 new shape has to be created to finally create 2 new shapes with 2 free sticks moved.
You are gradually refining the requirements and as a result narrowing down the possible actions.
Solution Stage 3: Move 2 sticks to make 5 matchstick shapes Matchstick puzzle: Applying the precise requirement rules on the puzzle figure
The crucial question to answer now is,
With 1 stick how can you create a new shape, that too twice?
For convenience, the labeled puzzle figure is shown again,
With requirements of creating a new shape strictly defined, it is easy now to identify the ONLY one such possibility,
First shape creation: Place a stick to complete the triangle with stick 1 and stick 5 as the existing other two sides.
Second shape creation: Place a stick as the shorter diagonal of the rhombus A to divide it into two triangles.
Note: To avoid counting the rhombus also as a closed shape, the "non-overlapping" condition was included in the puzzle description.
So you have created two new shapes with two already moved sticks, and this is the only single such possibility.
But you haven't yet solved the puzzle fully. One more step is still to be taken to complete the solution. You have to identify the two sticks to move.
Solution final stage: Move 2 sticks to make 5 matchstick shapes Matchstick puzzle: Identifying the 2 sticks to move
Analysis: As stick 5 has to be used in forming one of the two new shapes, the square B cannot be disturbed at all.
Note:This is the reason why your initial attempt starting with stick 4 and 5 failed.
It is now easy to identify stick 8 and stick 9 to move to form the two new shapes. The solution figure is below.
Solution of puzzle Part II: How many ways can you form such 5 closed matchstick shapes
With the knowledge you have gained, solving the second part of the puzzle should take just a few seconds,
Move stick 6 and stick 7 to form the two new shapes.
The second solution figure is below.
Because of the precise conclusions you have made step by step, you could be sure that these are the only two solutions possible.
Solution to Move 2 sticks to make 5 matchstick shapes bonus matchstick puzzle
Puzzle: Move 2 matches to make 5 non-overlapping closed matchstick shapes. How many such solutions can you find?
One possible solution is shown below.
With you experienced now, we leave you to find ALL the possible solutions.
Solving a puzzle by step-by-step reasoning may diminish the joy of discovery of the solution in whatever-possible-way-I-can-solve attempts.
That's why we strongly recommend that you should try to solve the puzzle all by yourself in your own way first and then only go through the reason-based solution to enjoy best of both worlds,
A combination of intuitive and reason-based solution approach should produce a very satisfying overall experience.
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Move 2 sticks to make 5 closed shapes matchstick puzzle