Badshah

Read all the constraints together

Before placing a crown, check its row, its column, its coloured region, and the immediately neighbouring cells. A square can look safe within one region while touching a ruler across its boundary. Equally, a distant crown can rule it out by occupying the same column. Passing one check does not make the square available if another check fails.

Distinguish an empty square from a possible square. An unmarked cell may already be excluded by something you have placed. Marking supported exclusions helps you see the remaining candidates, but an X is only as reliable as the reasoning behind it. When a region looks nearly empty, check whether its apparent restriction comes from the rules or from a tentative mark you made earlier.

Find a placement that is forced

Scan for a row, column, or region with just one valid candidate left. That square must contain its ruler, provided the earlier placements and exclusions are sound. After placing it, work through the newly excluded cells and scan again. A single placement can create another forced choice some distance away.

Do not confuse the most appealing candidate with the only candidate. If two squares remain possible and neither violates a rule, you have not yet proved which one to use. Look at the intersecting rows, columns, and nearby regions for more information. A useful pause is to ask what would rule out one of those squares. That directs your attention toward a specific missing deduction rather than an unsupported guess.

Use a region confined to one line

Sometimes a region has several candidates, but all of them lie in the same row. You do not yet know the crown's exact square, but you do know which row it must occupy. Other regions therefore cannot place their crowns anywhere else in that row. The same reasoning applies when all candidates lie in one column.

This deduction concerns the entire set of remaining candidates, not just the ones you happen to be considering. Check that you have accounted for every square in the region. A region with candidates across two rows does not reserve both rows: it needs only one crown. Avoid extending a valid single-line deduction into a stronger claim that the board has not established.

• Which deductions a board is checked against

Follow the toy board's exclusions

The illustration below demonstrates one placement and its immediate consequences. It is a teaching board, separate from a daily challenge. Start at the displayed crown and follow its row and column. Those cells cannot hold another ruler. Next identify the rest of its region, then the cells touching it, including the diagonal neighbours.

Several explanations can rule out the same square. You only need one valid explanation to exclude it, but noticing the overlap helps you read the board accurately. The X marks show these immediate exclusions; they are not a claim that every other square will eventually contain a crown. To continue, examine the candidates that remain in each row, column, and region. The next placement needs its own reason.

Revisit a contradiction at its source

If a region has no candidates or two crowns conflict, stop adding marks. Work backward to the placement or exclusion that created the problem. Check whether an X was a deduction or merely an idea you meant to revisit. Clearing unrelated areas can make the board harder to understand without addressing the original mistake.

For example, imagine you marked a cell because you expected a crown in a neighbouring square. If that crown was never forced, the exclusion depended on an assumption. Reconsider both together. Keep the deductions you can still explain independently, and use Undo when it helps you return to an understandable position. A contradiction is a reason to inspect your reasoning; it is not evidence that a rule has stopped applying.

Make a hint into a useful explanation

When you cannot find the next deduction, a hint can direct your attention to a productive area. Before asking again to apply it, inspect that area yourself. Count candidates, look along the relevant lines, and consider which restriction might make a square impossible. The focus is an opportunity to understand what you missed.

After the hint applies a deduction, trace its consequences rather than immediately requesting another. Which neighbouring region changed? Did a row lose all but one candidate? At the end, notice which type of deduction helped most. Recognizing that pattern gives you a concrete place to look when the next court initially appears to offer too many choices.