A hare jumps over a 35 square grid starting from 1 never visiting a square twice. It jumps only vertically or horizontally. Place 5 jump counts...Read on...

A hare jumps from square 1 to square 35 on a 5-row by 7-column 35-square grid. It can jump to any adjacent square, either horizontally or vertically, but cannot revisit any square. Along its path, the hare marks certain squares with the numbers 7, 14, 21, 28, and 35 to indicate the number of jumps it has made up to those points. These squares are marked with circles, as shown in the puzzle figure.

**Your Challenge:** Determine the exact positions of these five numbers and complete the hare's jump sequence from 1 to 35.

**Recommended time to solve:** 30 minutes.

**Difficulty Level:** This puzzle is not easy. Solving it requires pattern recognition, strategy definition based on positional configuration, promising path analysis, and step-by-step logical deduction. If you find this puzzle challenging, you may want to try the simpler Jumping Hare on the 20-Square Grid Puzzle first.

### Solution to the Hare Jumping Across a 35-Square Grid Puzzle

#### Step 1: Analyze the Path Requirements Between Key Points

To solve this puzzle, we first need to understand the constraints and relationships between the marked squares.

**Final Destination (35):**One of the five marked circles will represent the hare's final jump, which must finally have a single direct path leading to it.**Intermediary Points:**The remaining four circles must have at least two feasible paths leading to or from them.**Specific Intermediary (7):**One circle will represent the 7th jump and will need only one path connecting it to another circle. Rest three must have at least two paths with other circles.

#### Step 2: Define a Promising Path Strategy

We'll define a **"promising path"** as one that **lies on the periphery of the grid.** This strategic choice helps keep central squares free, allowing for more flexible path selection later on. By focusing on these peripheral paths, we can avoid potential conflicts and increase our chances of finding a valid solution quickly.

**Strategic choice:** Once a promising path between two key points (circles) is identified, further analysis is continued with the promising path in place, ignoring other feasible paths, if any.

#### Step 3: Map Out Promising Paths Between Circles: First look

To facilitate analysis,

- Each square is identified by its row number appended with its column position. For example, the square located at Row 1 and Column 4 would be identified as R1C4.
- The circles are numbered 1 to 5.

Numbered circles are in the figure below.

**Key Observations on Circle Locations:**

**Circles 4 and 5**are far apart with no direct paths between them. Each must connect to one of the remaining circles (1, 2, or 3).**Circles 1, 2, and 3**are clustered together. with**no promising feasible path**between each other**except one between circle 1 and 3:**(1 → R1C5 → R1C6 → R1C7 → R2C7 → R2C6 → R2C5 → 3). These six squares also form one of the two feasible paths from circle 5.

**Conclusion: ***This promising path on the periphery should be a part of the solution either to connect 5 or to connect 3.*

#### Step 4: Analyze Paths from Isolated Circle 4

To avoid immediate complexity, Circle 4 is taken up first because the other isolated Circle 5 has one of its feasible paths to Circle 3, which conflicts with the path from Circle 1 to Circle 3 that also uses the same promising peripheral six adjacent squares on the top right corner of the grid.

**Detailed analysis of paths from isolated Circle 4:**

- A promising peripheral path between 1 and 4: (4 → R4C1 → R3C1 → R2C7 → R1C1 → R1C2 → R1C3 → 1). This path is accepted as a part of the solution for now as per strategy.
- With no path between Circles 4 and 3, the only feasible second path from Circle 4 to Circle 2 is identified and it fits snugly with the first promising path between Circles 4 and 1.
- The second path between Circles 4 and 2: (2 → R2C2 → R3C2 → R4C2 → R4C3 → R5C3 → R5C2 → 4). This new path, joining the first path from Circle 4 to Circle 1, occupies all squares on the left side three columns of the grid, leaving no gaps and interfering with no other centrally placed squares.

This forms a very promising chain of two paths and should be a part of the solution. It also has two advantages:

- One square, R3C3, remains free for a second path leading to Circle 2.
- One square, R1C5, remains free for a second path going out from Circle 1.

Below is the chain of two paths shown marked with stars, as well as the six adjacent squares on the top right corner of the grid identified as a source of a potential promising path.

#### Step 5: Analyze Feasible Promising Paths from Circles 5 and 3

The situation is now quite clear to reach the solution.

- Circle 5 has one promising path to Circle 1 and a second feasible path to Circle 2, with no path to Circle 3. With the promising paths already chosen, there is no other path from Circle 3 except a path using the most promising adjacent block of six squares on the top right corner of the grid.

So, we allocate this promising block of six squares for an outgoing path from Circle 1 (1 → R1C5 → R1C6 → R1C7 → R2C7 → R2C6 → R2C5 → 3) to Circle 3. It extends the chain of existing two paths to a chain of three paths (2 → 4 → 1 → 3). Circle 3 becomes the terminal circle of the 35th jump.

It only remains to:

- Find a feasible path from Circle 5 to Circle 2 marking Circle 5 as the location for the 7th jump, and
- Find a path from a suitable starting point 1 to reach Circle 5 in 7 jumps.

It is easy to resolve these two paths, and the complete solution is shown in the figure below.

#### Postscript

Without resorting to special pattern identification, strategic approach, use of the promising path concept, and step-by-step logical analysis, this puzzle would have been hard to solve.

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