Matchstick puzzle, Move 3 sticks in tic-tac-toe figure to form 3 perfect squares

Submitted by Atanu Chaudhuri on Fri, 19/07/2019 - 16:50

Tic-Tac-Toe move 3 matches to make 3 squares

Matchstick puzzle on Tic-Tac-Toe: Move 3 matches to make 3 squares

In the tic-tac-toe puzzle figure, move 3 matches to make 3 squares.

How many unique solutions can you find?

Recommended time: 5 minutes.

Tic-Tac-Toe move 3 matches to make 3 squares puzzle

Give it a try. It will surely be fun.

Hint: Use your analytical reasoning instead of random trial and error.

We'll go ahead to solve the puzzle step by step by analyzing the puzzle structure and making conclusions based on common stick constraint and a series of questions, analysis and answers.

Solution: Move 3 matches to make 3 squares tic-tac-toe matchstick puzzle: Structural analysis, Common stick analysis, Chain of reasoning by Question analysis answer

The following is tic-tac-toe figure with each stick numbered uniquely so that we will be able to explain clearly our analysis and required actions.

Tic-Tac-Toe move 3 matches to make 3 squares sticks numbered

First step in solving any matchstick puzzle is to count number of sticks in the puzzle figure. It is exactly 12.

Question 1: How many matchsticks do you require to make 1 square?

Answer: It is 4.

Question 2: How many sticks would you need to make 3 INDEPENDENT squares.

Answer: 2: To make 3 independent squares you need exactly 12 matchsticks that you already have.

As you know that if there were a single stick common between two squares need for sticks would have reduced by 1 to 11, you can make your first conclusion. This is a key conclusion or key pattern identification based on common stick concept.

Conclusion 1: The three squares in the solution figure won't have any common stick between any two of them.

Note: This is what we call key pattern identification as well as precise requirement specification for the final solution. Subsequent analysis and decisions will follow from this result.

Chain of reasoning based on common stick constraint and Identification of sticks to move

Look at the puzzle figure and take stock.

Observation: There are four nearly complete squares in the tic-tac-toe. Each has three sides already in place (three times four makes up the 12 sticks). Only 1 stick is required to close the fourth side and complete a square in each.

These are,

I: Sticks 1, 5, 3.

II: Sticks 4, 8, 12.

III: Sticks 11, 7, 9.

IV: Sticks 10, 6, 2.

Conclusion 2: Some of these four one side only open nearly complete squares must be completed to make the final solution figure. This is an obvious conclusion.

Now ask yourself the question,

Question 3: What would be the situation if any one of these nearly complete squares is completed?

Just assume, the square with sides 10, 6, 2 is completed by moving stick 1. Any problem you observe?

Answer 3: Yes there will be a serious problem.

It will create a common stick 6 with the existing central square made up of sticks 5, 6, 7, 8.

An example is shown for making the consequence clear. This is not trial and error. This is the well-used and highly valuable management technique of Consequence analysis.

Tic-Tac-Toe move 3 matches to make 3 squares puzzle invalid square

That's why you have to make the second important conclusion,

Conclusion 2: To make the solution you want, the central square MUST be destroyed by selecting one or more than one stick among 5, 6, 7, 8 for moving.

As all the four nearly complete squares are STRUCTURALLY EQUIVALENT, you may choose any of the four.

Action 1: So you choose stick 5 with full confidence that this must give you a solution. And move it straightaway to complete one of three remaining nearly complete squares, say square with three sides 11, 7, 9.

The result of the action is shown below.

Tic-Tac-Toe move 3 matches to make 3 squares puzzle intermediate solution

Well, well, well. You have got the solution in your pocket with this single confident action.

The two sticks 1 and 3 are hanging, ready to be moved to complete the other two squares, the square with three sides 10, 6, 2 and the square with three sides 4, 8, and 12.

Following is the result of these actions. We have kept the original sticks that we have moved, but made them faded to give you a very clear idea on how you have created the solution figure from the starting figure.

Tic-Tac-Toe move 3 matches to make 3 squares puzzle solution

Consider that you could have selected any of the other three one-side-open-three-stick components to form three perfect squares. But the result would have been same if you would just rotate the figures suitably.

So we conclude,

These three other solutions would be rotationally same with the solution you have created. That means, the solution is rotationally unique.

You may try for any other solution figure. You won't get any more as you know your reasoning has been bullet-proof.

End note

We have approached the problem systematically and analytically, and not in any random way. This is what we call systematic approach to problem solving.

Generally, systematic approach to problem solving depends heavily on identification of key patterns, creation of effective methods and deductive reasoning to move towards the solution without any confusion as well as in minimum number of steps, if possible.

This is systematic problem solving.

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.