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Consecutive Numbers Grid Puzzle

The cells in this 4×4 grid puzzle contain the numbers 1-16 each once. There are two rules to the arrangement of the numbers:

1. Any two consecutive numbers must share a row/column (1 and 16 should be considered consecutive).
2. No row/column can contain three consecutive numbers.

Fill in the remaining numbers.

Solution

One super helpful insight for solving this puzzle is that numbers that are 2 apart cannot be in the same row/column. For example, 2 must be in the same row or column as both 1 and 3. If 1 and 3 were to be in the same row, 2 couldn’t share a column with both of them (can’t be in two place at once), and 2 can’t be in the same row as 1 and 3 (violates the 2nd rule of no three consecutive numbers), so this scenario can never occur.

With this implicit rule, one possible way to solve this is as follows:

1. The column with 4 and 7 must contain (3 or 5) and (6 or 8).
• But since (5 and 7) and (4 and 6) cannot be in the same row/column, this column must contain 3 and 8.
• But since 8 cannot be in the same row/column as 10, we know exactly where 3 and 8 must go.
2. 9 must be at the intersection of 8 and 10.
3. 11 must share a row/column with 10, but can’t share a column with 9 or 13, so it must be in the first column.
4. 2 must be the remaining number in the row with 11, 3, and 10.
5. 12 must be at the intersection of 11 and 13.
6. 1 must be at the intersection of 2 and 16.
7. 5 must be the remaining number in the row with 1, 4, and 16.
8. 6 must be the remaining number in the column with 12, 11, and 5.
9. 14 must be the remaining number in the column with 13, 2, and 1.
10. 15 must be the remaining number.