11x11 Queens Puzzle #88
Difficulty: medium. Place 11 queens on this 11x11 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 25-step reasoning is written out below.
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Step-by-step solution for puzzle #88
What follows is a complete solve of puzzle #88, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 11 queen placements and 14 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the pairing argument on columns.
Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a region.
Show the full walkthrough (19 steps) — contains the solution
Step 1. Slate cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 9 squares in all.
Step 2. Sea green is down to 4 candidates in rows 10 and 11, columns 8–10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about sea green itself.
Step 3. The teal region has 3 squares left, spread across rows 10 and 11, columns 10 and 11. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 4. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Sea green and teal are now boxed into rows 10 and 11 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 11 squares at once.
The board after step 5 — 0 of 11 queens placed. Step 6. The khaki region has 5 squares left, spread across rows 7–9, columns 2–4. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 7. Mint and teal are now boxed into columns 10 and 11 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 13 squares at once.
Step 8. All of the amber region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 10 squares come off.
Step 9. Sea green cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 6 squares in all.
Step 10. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
The board after step 10 — 0 of 11 queens placed. Step 11. Slate cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 1 square in all.
Step 12. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
Step 13. Only row 8, column 2 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 2, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 14. Here is the pairing argument. The pale yellow and orange regions have no open squares outside columns 6 and 7 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 10 squares come off in one stroke.
The board after step 14 — 1 of 11 queens placed. Step 15. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 5; slate has one square left, so its queen takes row 5, column 1; mint has one square left, so its queen takes row 4, column 11; pale yellow has one square left, so its queen takes row 7, column 7; orange has one square left, so its queen takes row 9, column 6; teal has one square left, so its queen takes row 11, column 10; sea green has one square left, so its queen takes row 10, column 8. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 10, column 8.
Step 16. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 17. Only row 3, column 3 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 3, in blue and in the ring around that square is now dead.
Step 18. The indigo region is down to one open square, at row 1, column 4. Every colour holds exactly one queen, so that square is forced. Row 1 and column 4 are now spoken for, along with the squares touching that queen.
Step 19. Only row 2, column 9 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 9, the rest of the amber region and every square adjacent to it — diagonals included.
The board after step 19 — 11 of 11 queens placed.
That completes the board: 11 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 11x11 Queens puzzle #88
This is puzzle number 88 in the 11x11 collection, rated medium. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #88, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these eleven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #88 are: slate (3 squares), sea green (4 squares), teal (4 squares), mint (5 squares), amber (6 squares), indigo (13 squares), khaki (14 squares), orange (14 squares), red (15 squares), pale yellow (15 squares), blue (28 squares). An even split would hand every color 11 squares, so the slate region is running 8 squares under that average while the blue region is the roomiest area on the grid with 28 squares.
Every color takes one queen and one only, whatever its size — which is why the 25 square difference between slate and blue is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 2×3 box, and orange the most diffuse at 50% of its 4×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Slate covers 3 squares, spanning 3 rows and 1 column (rows 5–7, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Sea green occupies 4 squares within rows 10–11, columns 8–10.
Teal: 4 squares across rows 10–11, columns 9–11.
Mint covers 5 squares, spanning 4 rows and 2 columns (rows 4–7, columns 10–11).
Amber occupies 6 squares within rows 1–2, columns 9–11.
Indigo: 13 squares across rows 1–3, columns 3–8.
Khaki covers 14 squares, spanning 5 rows and 4 columns (rows 7–11, columns 1–4).
Orange occupies 14 squares within rows 8–11, columns 4–10.
Red: 15 squares across rows 3–6, columns 5–11.
Pale yellow covers 15 squares, spanning 5 rows and 6 columns (rows 5–9, columns 6–11).
Blue occupies 28 squares within rows 1–9, columns 1–5.
Where to start on this board
Open on the tightest color. Here that is slate, confined to 3 squares between rows 5 and 7 and columns 1 and 1. Its queen has nowhere else to go, which means those 3 rows and 1 column carry more constraint than anywhere else on #88.
Because slate stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Amber, sea green and teal are no more than two rows tall, so their queens are confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.
The tightest lines on puzzle #88
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3 and 4 are the narrowest, touched by only 3 colors. Column 2 is the tightest column, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 7 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
The middle of #88 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #88 is rated medium
Ratings track how generous the opening is. On #88 you get one region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 3 colors, and a 25-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
Stuck on #88 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
Techniques that work well at 11x11
At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #88 is column 2.
As a sanity check rather than a spoiler: the finished solution to #88 puts 4 queens on the outer border and 7 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 11x11 Queens puzzle #88?
Puzzle #88 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #88 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #88, has been verified to have exactly one valid arrangement of eleven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #88?
Start with the slate region — the most constrained color on this board, with 3 squares spanning rows 5–7 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.