Equal Sum

Equal Sum Lines (sometimes called Equal Sum Regions or Equal Sum Line Regions) are blue dashed lines. The line is split by (3x3) box lines. Each slip creates new line region, every region on same line has same sum.

On image above three cells in box 1 have the same sum as digit in box 2.

On image above green must sum to blue and orange to yellow.

If line is split multiple times it has multiple regions, even if they are in same (3x3) box.

In this case on image above 3 regions are marked by green background, yellow and orange. Even though orange and green are is the same box, they are different regions. All three of the regions must sum to same value.

As an example following image shows one of the valid solutions:

Equal Sum Solving Strategies

If line has regions with single cell, those cells act as palindrome, they must be same digit because they must have same sum.

On image below we can remove candidates for 2, 3 and 9 because one of the single cell regions cannot contain those values.

If region has 2 cells, it must have at least sum of 3, and if it has 3 cells, it must have sum of at least 6. In following example sum must be 9 because region with 3 digits cannot have 1, hence they must be 2, 3 and 4, which sum to 9.

In some extreme cases we can look at the minimum and maximum sum of each region. On image below minimum sum in box 1 is 15 (1+2+3+4+5) and minimum sum in box 2 is 3 (1+2). That means overall minimum sum for the line is greater of the two, in this case 15 (because we cannot put 3 in box 1 in 5 cells).

We can also look for the maximum sum. For box 1 maximum sum is 35 (5+6+7+8+9) and for box 2 is 17 (8+9). Overall maximum sum is 17 (because we cannot but 35 in box 2 in two cells).

Knowing the minimum (15) and maximum (17) sums we can remove candidates from cells on the line.