Combination Pruning: Step-by-Step Guide

When runs have multiple possible combinations, comparing them often reveals digits that never appear. Remove those digits and the puzzle tightens quickly, opening the door to advanced deductions.

Prerequisites

Cross Sums

How to recognize it

  • List all valid combinations for a run’s sum given its remaining empty cells.
  • Compare those combinations with crossing runs to see which digits never appear.

Step-by-step walkthrough

  1. Generate the combinations. For a four-cell run totalling 21 you might have {1,4,7,9}, {1,5,6,9}, {2,3,7,9}, {2,4,6,9}, {2,5,6,8}, {3,4,5,9}, {3,4,6,8}.
  2. Mark which digits appear in every combination. If 9 is present everywhere, that digit is locked into the run.
  3. Cross-check with intersecting runs and eliminate digits that never align—if column candidates forbid 9 in one position, adjust the combinations accordingly.
  4. Repeat the process after each placement; shrinking the list often exposes new forced digits.

Why it works

Any digit that never shows up in the remaining valid combinations cannot belong in that run. Removing it sharpens the candidate grid and often creates new sum singles.

Tools for organization

  • Group combinations by the highest digit to see whether 8 or 9 is mandatory.
  • Highlight digits that appear in only one position—these act like hidden singles within the run.
  • Use the candidate mode to mirror your paper notes and avoid losing track of pruned digits.

Try it now

Take a 24-in-4 run with intersections allowing {1,3,5}, {2,3,5}, {4,5,7}, {3,4,8}. Enumerate every combination, remove those violating any column, and notice how only one set survives.

Practice routine

Save a screenshot of a tough puzzle and revisit it the next day. Fresh eyes help you spot digits that never appear, reinforcing the pruning mindset.

Common pitfalls

  • Forgetting to update combinations after each placement.
  • Tracking combinations without noting which digits never appear—write them down.
  • Eliminating a digit from one run without checking whether it breaks the crossing run’s totals.

Next technique

Ready for the next level? Study Locked Sets to corral candidates across multiple runs and Residual Sum Forcing to balance the remaining totals.