TodayHint

Pips July 21, 2026: Hints & Answers

Constructed by Ian Livengood, Ian Livengood, Rodolfo Kurchan, edited by Ian Livengood.

Recap

The easy board requires strategic placement of 5 dominoes to satisfy 6 regions with various constraints. The medium board adds more complexity with 8 dominoes and 8 regions. The hard board is the most challenging, with 16 dominoes and 27 regions, requiring careful planning and grouping of dominoes to satisfy the constraints.

The three boards that day

The tray and each region's rule — board data, no solutions.

Easy board · 5 dominoes · 6 regions

1–36–45–32–22–5

  • three cells: exactly 10.
  • one cell: left uncovered.
  • two cells: all matching.
  • two cells: exactly 10.
  • one cell: under 2.
  • one cell: exactly 2.

Small number of dominoes and regions makes it manageable.

Medium board · 8 dominoes · 8 regions

5–62–52–32–21–02–40–06–4

  • two cells: exactly 8.
  • 2 ×two cells: all matching.
  • one cell: exactly 2.
  • four cells: all matching.
  • three cells: all matching.
  • one cell: exactly 3.
  • one cell: under 2.

More dominoes and regions add complexity, but still solvable with careful planning.

Hard board · 16 dominoes · 27 regions

0–10–20–30–40–51–21–31–41–52–32–42–53–43–54–53–6

  • one cell: exactly 3.
  • 3 ×one cell: exactly 4.
  • 5 ×one cell: exactly 5.
  • 3 ×one cell: exactly 2.
  • 3 ×two cells: all matching.
  • 5 ×one cell: exactly 0.
  • 5 ×one cell: exactly 1.
  • one cell: left uncovered.
  • three cells: all matching.

Large number of dominoes and regions, combined with complex constraints, make it challenging.

Pips hints, one rung at a time

Six rungs, easy board first. The recap above already named the walkthroughs, so this ladder is here for anyone replaying the boards fresh rather than as a spoiler shield.

  1. How the three boards compared

    Sizes and constraint mixes, counted off the boards.

    Easy board
    5 dominoes, 6 regions (3 sum, 1 empty, 1 equals, 1 less)
    Medium board
    8 dominoes, 8 regions (4 equals, 3 sum, 1 less)
    Hard board
    16 dominoes, 27 regions (22 sum, 4 equals, 1 empty)
    Back to the board inventories
  2. Easy board: how to approach it

    Method only.

    Start by identifying regions with the fewest possible domino combinations, focusing on those with specific sum or equality constraints.

  3. Easy board: where to start

    One region named by its constraint.

    The region with the 'empty over 1 cell' constraint, as it has the most limited options.

  4. Medium board: approach and first region

    The same two steps, one difficulty up.

    Look for dominoes that can be placed in multiple regions, and prioritize regions with the most constraints.

    The region with the 'sum 2 over 1 cell' constraint, as it has limited options.

  5. Hard board: how to approach it

    Where the hard board differed in kind, not just size.

    Begin by identifying and grouping dominoes with similar values, then look for regions that can be satisfied by these groups.

  6. Hard board: where to start

    The last rung before the full walkthroughs.

    A region with a specific sum constraint over a small number of cells, such as 'sum 3 over 1 cell'.

    Open the full walkthroughs

Every Pips walkthrough from that day

All three walkthroughs, folded: each one places every domino on its board, one after another.

  • Easy board · 5 dominoes

    Easy board walkthrough

    Place the 2/2 domino first, as it's the only one that can fit in the 'equals over 2 cells' region. Then, use the 1/3 and 6/4 dominoes to satisfy the sum constraints. The 2/5 domino will help with the 'less 2 over 1 cell' region. Finally, place the 5/3 domino to complete the board.

  • Medium board · 8 dominoes

    Medium board walkthrough

    Start by placing the 0/0 domino in the 'sum 0 over 1 cell' is not present but 'sum 2 over 1 cell' region. Use the 2/2 domino in an equals region. Then, place the 5/6 and 6/4 dominoes to satisfy sum and equality constraints. Continue with the remaining dominoes, focusing on regions with specific constraints.

  • Hard board · 16 dominoes

    Hard board walkthrough

    Start by placing dominoes with low values, such as 0/1, 0/2, and 1/2, in regions with small sum constraints. Use dominoes with higher values, like 4/5 and 3/6, in regions with larger sum constraints. Continue placing dominoes, focusing on satisfying equality constraints and larger sum constraints. The 'equals over 3 cells' region will require careful planning. Use the process of elimination to find the correct placements for the remaining dominoes.

Those boards by the numbers

Date
Tuesday, July 21, 2026
Boards
3 — easy, medium, hard
Dominoes in play
29 across the three boards
Easy board
5 dominoes · 6 regions
Medium board
8 dominoes · 8 regions
Hard board
16 dominoes · 27 regions

More Pips

See the live Pips hints and answers page for the current boards, or browse every archived Pips date by month.

Try the other two puzzles