**Puzzle: **A matchstick puzzle is given below, move **3 matchsticks** to get **3 squares**. Give all the possibly unique solutions for the given problem.

**Solution:**

Let’s discuss a step-by-step solution to arrive at a result.

1. Count the total number of matchsticks available in the problem. In the figure, the total number of matchsticks is equal to 12 as shown below figure.

2. Now for making a square 4 sticks are used so, to form 4 squares 16 sticks would have been used but we have only 12 sticks. Here the concept of sticks common between two adjacent unit shapes comes.

3. Now to make 3 squares 4 * 3 = 12 sticks are required (at max, when sticks are not common between squares). And in this case, 12 matchsticks are available. Now, all we have to do is-

- Eliminate all 4 common sticks.
- Form 3 independent squares,
- By moving just 3 sticks.

4. **Concept of promising stick**

- Interestingly, if any one of the 12 sticks is moved first, at least two common sticks will lose their common properties. In moving any of the four common sticks, immediately 4 free-standing sticks will occur. This is impossible to accomplish in the remaining two moves. Therefore, the first stick move eliminating two common sticks cannot be a common stick.
- Eight of the remaining sticks (1, 2, 3, 12, 11, 9, 8, 7) can be moved first, as all 8 sticks share the same position, function, and location within the structure. These are the corner sticks. These are the
**Promising sticks.**

__Solution 1: Common stick elimination technique__

Number 4 cannot be split into three positive integers with a minimum value of 1. One of the three numbers has to be 2. By applying **Reasoning**, 4 common sticks can be eliminated in just 3 moves as-

1. Move stick numbered 2 first. This destroys 1 square and eliminates 2 common match sticks.

2. Move stick numbered 3. This results in two free sticks gained, two common sticks eliminated (like 5, 4 are not common sticks now) and 1 square reduced.

3. Move 3rd stick such that it eliminates 2 more common sticks and destroys 1 more square.

Note:

Any of the four sticks numbered 1,7,11,12 belonging to the two squares adjacent to the square destroyed just now cannot be removed as it would eliminate 1 more common stick, destroy 1 more square, however would create two unplaced sticks and the situation could be not possible to manipulate.

The only feasible solution for this step is to select stick 8.

Hence **3 more solutions** can be concluded by this method as shown.

**Solution 2: **If sticks 1,7,12 are considered for movement instead of 2,3,8 in the first solution.

**Solution 3:** If sticks 1,7,11 are considered for movement instead of 2,3,8 in the first solution.

**Solution 4: **If sticks 8,9,2 are considered for movement instead of 2,3,8 in the first solution.