Question 2

Hidato puzzles require filling a grid with consecutive numbers, where the sequence length matches the number of cells.

Step 1 — Understanding Hidato Rules

We first need to understand what a Hidato puzzle is. In a Hidato puzzle, we fill all empty cells in a grid. We use a sequence of consecutive numbers. These numbers must be placed in adjacent cells. Adjacent means horizontally, vertically, or diagonally. The sequence usually starts from 1. The sequence usually ends at the total number of cells in the grid. Let us call the total number of cells in a grid N. So, the numbers in the grid should be 1, 2, 3, ..., N. All given numbers in the puzzle must be part of this sequence.

Step 2 — Analyzing Puzzle 1

Let us count the cells in the first grid. The grid has 4 rows and 4 columns. The total number of cells is: $4 \times 4$ $= \mathbf{16}$

$\boxed{\text{16 cells}}$ According to standard Hidato rules, the numbers in this grid should be from 1 to 16. Let us list the numbers already given in this puzzle: 1, 4, 8, 9, 14, 15, 20, 25. We see that the numbers 20 and 25 are given. However, these numbers are larger than 16. This means they cannot be part of the sequence from 1 to 16. This contradicts the basic rules of a Hidato puzzle. Therefore, Puzzle 1 cannot be solved as a standard Hidato puzzle.

Step 3 — Analyzing Puzzle 2

Let us count the cells in the second grid. We count the cells in each row. Row 1 has 5 cells. Row 2 has 6 cells. Row 3 has 6 cells. Row 4 has 5 cells. Row 5 has 5 cells. Row 6 has 5 cells. Row 7 has 5 cells. The total number of cells is: $5 + 6 + 6 + 5 + 5 + 5 + 5$ $= \mathbf{37}$

$\boxed{\text{37 cells}}$ According to standard Hidato rules, the numbers in this grid should be from 1 to 37. Let us list the numbers already given in this puzzle: 1, 2, 4, 6, 12, 13, 17, 21, 23, 26, 30, 31, 34, 36, 39. We see that the number 39 is given. However, this number is larger than 37. This means it cannot be part of the sequence from 1 to 37. This contradicts the basic rules of a Hidato puzzle. Therefore, Puzzle 2 cannot be solved as a standard Hidato puzzle.

Step 4 — Analyzing Puzzle 3

Let us count the cells in the third grid. We count the cells in each row. Row 1 has 1 cell. Row 2 has 2 cells. Row 3 has 5 cells. Row 4 has 7 cells. Row 5 has 7 cells. Row 6 has 7 cells. Row 7 has 5 cells. Row 8 has 1 cell. The total number of cells is: $1 + 2 + 5 + 7 + 7 + 7 + 5 + 1$ $= \mathbf{35}$

$\boxed{\text{35 cells}}$ According to standard Hidato rules, the numbers in this grid should be from 1 to 35. Let us list the numbers already given in this puzzle: 1, 5, 7, 9, 10, 15, 17, 23, 25, 26, 28, 29, 34, 35, 39, 42, 44, 49, 50, 54, 56. We see that many numbers are given that are larger than 35. For example, 39, 42, 44, 49, 50, 54, 56. These numbers cannot be part of the sequence from 1 to 35. This contradicts the basic rules of a Hidato puzzle. Therefore, Puzzle 3 cannot be solved as a standard Hidato puzzle.

Answer

(i) Puzzle 1 cannot be solved as a standard Hidato puzzle. (ii) Puzzle 2 cannot be solved as a standard Hidato puzzle. (iii) Puzzle 3 cannot be solved as a standard Hidato puzzle.