Cross Sums: Step-by-Step Guide

Cross sums leverage overlapping runs. When an across run and a down run intersect, the digits allowed in one can shrink the options of the other, often forcing eliminations and revealing placements.

Prerequisites

Sum Singles

How to recognize it

  • An across run and a down run share a cell with limited overlapping digits.
  • Eliminating digits from one run immediately restricts the partner run.

Step-by-step walkthrough

  1. List the candidates for each run separately. Example: across 17-in-3 → {1,7,9}, {2,6,9}, {3,5,9}, {4,5,8}.
  2. Intersect those sets. If the down run only permits {2,5,8}, the shared cell must be one of {2,5,8}.
  3. Remove any digits from the across list that do not appear in the intersection (1,7,9 in this case) and update candidates across the run.
  4. Recalculate each run’s remaining totals—many times the narrowed combination produces a new sum single elsewhere.

Why it works

Runs must use unique digits. If an across run can only use {1,3} in a shared cell and the down run can only use {3,5}, the intersection must be 3, allowing you to prune those digits from the rest of each run.

Intersections to prioritize

  • Short + long: a two-cell run crossing a five-cell run often pins down one digit quickly.
  • High totals: intersections containing 8 or 9 drastically change the rest of the run’s options.
  • Duplicate totals: when two runs share the same sum, cross-checking prevents mirrored mistakes.

Try it now

Sketch a small grid with a 16-in-3 across run crossing a 23-in-4 down run. Enumerate candidates, intersect them, and note how the overlap determines which digits disappear from each run.

Practice routine

Play a daily puzzle and log every intersection that removes at least one digit. Tracking your successes cements the habit of scanning cross sums before branching into harder deductions.

Common pitfalls

  • Ignoring that eliminating a digit from one run can change the possible combinations dramatically.
  • Forgetting to check both directions—across and down—in each intersection.
  • Overlooking that a forced digit in the intersection also locks digits out of the remaining cells in both runs.

Next technique

Once you are comfortable trimming candidates with cross sums, take the next step with Combination Pruning and Residual Sum Forcing for late-game breakthroughs.