On a Zipper Line, the sum of the digits at equal distances from the center equals the digit in the center of the line. The Zipper Line offers various solving strategies across different difficulty levels.

Zipper Line Strategies

1. PairElimination-1

Level - Medium

In this strategy, we remove candidates from pairs that don’t sum up to the middle digit. If the middle cell of the Zipper Line is solved, candidates in equally distant pairs can be eliminated if they cannot add up to the middle digit.

Example:

In the image below, the center of the shaded Zipper Line is 9 (highlighted in blue). Candidates in equally distant cells must sum up to 9.

Equal distance pairs:

• F4 and F8
• G3 and G7
• H4 and H6

Eliminations:

1. 1 in F4 needs 8 in F8 to sum to 9. Since F8 doesn’t have 8, 1 can be eliminated from F4.
2. 6 in F4 needs 3 in F8 to sum to 9. Since F8 doesn’t have 3, 6 can be eliminated from F4.
3. 7 in F4 needs 2 in F8 to sum to 9. Since F8 doesn’t have 2, 7 can be eliminated from F4.
4. 1 in F8 needs 8 in F4 to sum to 9. Since F4 doesn’t have 8, 1 can be eliminated from F8.
5. 4 in F8 needs 5 in F4 to sum to 9. Since F4 doesn’t have 5, 4 can be eliminated from F8.
6. 7 in G3 needs 2 in G7 to sum to 9. Since G7 doesn’t have 2, 7 can be eliminated from G3.
7. 6 in G7 needs 3 in G3 to sum to 9. Since G3 doesn’t have 3, 6 can be eliminated from G7.

2. PairElimination-2

Level - Advanced

In this strategy, we remove candidates from pairs that don’t add up to any of the candidates in the middle cell.

Example:

In the image below, the candidates in the middle cell (blue highlight) of the shaded Zipper Line are 6, 7, and 8.

Eliminations:

1. 7 in B7 cannot sum with any other digit to equal 6 or 7, and would need 1 in D3 to sum to 8. Since D3 doesn’t have 1, 7 can be eliminated from B7.
2. The same logic applies to eliminate 7 from D3.

3. MiddleElimination-1

Level - Advanced

Remove a candidate from the middle cell if it cannot be formed by adding candidates from equally distant pairs. Each candidate in the middle cell must be the sum of a valid pair from the matching cells. If not, the candidate is invalid and can be eliminated.

Example:

In the image below, the candidates in the middle cell D5 (blue highlight) of the shaded Zipper Line are 6, 7, and 8.

Equal distance pairs:

• B3 and B7
• C4 and C6

Eliminations:

• C6 = 3, so it would need 3 in C4 to sum to 6, 4 in C4 to sum to 7, and 5 in C4 to sum to 8. Since C4 lacks 3 and 4, the sums 6 and 7 cannot be formed. Thus, 6 and 7 can be eliminated from D5, leaving 8 as the only valid candidate.

4. MiddleElimination-2

Level - Advanced

The digit in the middle cell must be at least the sum of: (Number of cells along the line that see each other) + (Smallest possible candidate value among those cells) - 1.

Example:

In the image below, the shaded Zipper Line has a length of 5, and all cells along the line see each other. The smallest candidate value on the line is 2.

Calculation: The center digit must be at least: 5 + 2 - 1 = 6.

Eliminations: Since D7 is the center of the line, it must be at least 6. Therefore, 2, 3, 4, and 5 can be eliminated from D7, leaving 9 as the only remaining candidate.