[list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
[*] Previous problem for the grid of $5\times 5$ lattice.[/list]
To solve this problem, we need to determine the minimum number of tests required to demonstrate that all nodes in a power grid are connected. We will approach this problem by considering the properties of the graph formed by the nodes and wires.

### 1. Understanding the Problem
The power grid is represented as a $3 \times 3$ lattice with 16 nodes. Each node is connected to its adjacent nodes by wires. We need to ensure that there is a path of intact wires between any two nodes.

### 2. Graph Representation
The grid can be represented as a graph where:
- Nodes represent the vertices of the lattice.
- Edges represent the wires connecting the nodes.

### 3. Minimum Spanning Tree (MST)
To ensure all nodes are connected, we need to check if the graph is connected. A connected graph with $n$ nodes has at least $n-1$ edges. This is the property of a Minimum Spanning Tree (MST).

### 4. Number of Tests
To verify that the graph is connected, we need to perform tests to check the connectivity between nodes. The minimum number of tests required is equal to the number of edges in the MST, which is $n-1$.

For a $3 \times 3$ lattice:
- The number of nodes, $n = 16$.
- The minimum number of tests required is $n-1 = 16-1 = 15$.

### 5. Verification
We need to ensure that each test checks the connectivity between two nodes. If we perform 15 tests and find that all nodes are connected, we can conclude that the entire grid is connected.

### 6. Generalization for $5 \times 5$ Lattice
For a $5 \times 5$ lattice:
- The number of nodes, $n = 36$.
- The minimum number of tests required is $n-1 = 36-1 = 35$.

### Conclusion
The least number of tests required to demonstrate that all nodes are connected in a $3 \times 3$ lattice is 15. For a $5 \times 5$ lattice, the least number of tests required is 35.

The final answer is $\boxed{15}$ for a $3 \times 3$ lattice and $\boxed{35}$ for a $5 \times 5$ lattice.