In the 4 square area puzzle figure, move 4 matches to form 3 square area. There must be a single area enclosed by matchsticks. Solve on 20 minutes.
The matchstick puzzle
The following matchstick configuration area equals 4 squares.
The V-shaped formation at the top reduces the top one square area to area A equal to one-half square. This reduction is compensated by the outward V-shaped extension of area exactly equal to one-half square at the bottom.
Move 4 matchsticks in the configuration to form an area equal to 3 squares.
Condition: The area must by enclosed by a single matchstick boundary.
Time to solve: 20 minutes.
This puzzle structure has an unusual formation and needs careful structural analysis for solving the puzzle quickly.
Do try before going through the solution.
Solution to move 4 matches to form 3 square area puzzle: Analysis of structure and sticks to move
The puzzle figure is shown again for convenience.
The V formation at the top equals the absence of an area equal to one-half of a square. The V-shaped extension at the bottom is also of area one-half of a square. This compensates the reduction of area at the top so that the total area of the configuration equals 4 squares.
The two V-shapes compensate each other in the configuration and are interdependent. If you disturb any match in one V shape, you must analyze its consequence in the other V shape as well.
Because of this structural coupling of effectively 4 sticks at the top and two sticks at the bottom, you cannot move any of these sticks for solving the puzzle.
Conclusion 1: Reduction of one square area must be achieved by moving matchsticks from the left and/or right sides of the puzzle configuration.
The question arises, how can we collapse the matches on the two sides to reduce the area by one square?
When you try to achieve this mentally you realize that, solution must come by moving sticks from both sides, not just from one side.
Naturally, we wonder whether the V shaped area reduction at the top can be used. It should have a great potential.
With this reasoning, we first make a simple trial by reversing the extended V at the bottom. Figure shown.
The inverted V-shaped formation at the bottom inside the middle square reduces its area by one-half square and along with the reduction at the top it effectively forms a 3 square area. But this cannot be the solution because we have achieved this configuration by moving just 2 sticks.
Now we know clearly that on any one side of the configuration we have to form such an inclined V inwards to reduce the area by one-half square.
Such a possibility indicated by two inclined dotted lines inwards on the left of the configuration is shown.
To form this inclined V inwards we need to move 2 matchsticks taking a loan of one free matchstick from the Right side.
Stick move scheme for solution: To make a move of a total 4 matchsticks, we have to move the single left side vertical matchstick and three matchsticks from the Right side. The three sticks convenient to move are the three outer sticks of the Right side square.
The stick move scheme with the sticks to move marked red is shown.
We have already decided to form an inward inclined V on the left side. What structure should we form on the Right?
Answer is clear,
New structural scheme for solution: An inclined V inwards on the left reduces area by one-half square. The three matchsticks identified for moving has also reduced the area by one square more. Total reduction till now is one and one-half square area. Just increase the area by a new externally extended inclined V with two of the three free sticks on the Right side.
The final solution figure shown.
The sticks moved are shown also in their original positions faded out. And the stick movements are identified by arrows.
Whole area equals exactly 3 squares, and it is enclosed by a single matchstick boundary.
End note
Because of the structural emphasis, the solution of this puzzle could be reached not by common concepts of matchstick puzzles such as common stick analysis. Solution depended heavily on structural reasoning. It is still reasoning, but reasoning on the configuration structure.
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