Can three cover it all?
Cover every empty square with the reach of just three queens.
The clock starts when you reveal the board.THE THINKING CHESSBOARD
Three queens.
Every square.
A small board.
A surprisingly big idea.
4 rows. 8 columns. One lovely problem.AT YOUR OWN PACE
A brain snack, with room to think.
THAT’S YOUR AHA MOMENT
A little board. A big win.
Why your arrangement works
Each queen takes care of her whole row and column. The diagonals do the delicate work: reaching the gaps in rows without a queen.
See why it works ↓Keep your time
THE IDEA BEHIND THE ANSWER
Watch the gaps disappear.
Three kinds of reach.
One idea to unpack.
Reached Still a gap24 / 32
The board keeps your queens fixed. Only the highlighted reach changes.
01
Start with the rows.
WHAT TO NOTICE
ONE MORE AHA
Could two queens do it?
There are four rows. Two queens can sit in at most two of them, so at least one row must have no queen.
On that row, each queen can reach at most three squares: one along her column and one along each diagonal.
2 queens×3 squares=6 at most
••••••××
Even if all six reached squares are different, two remain. Overlaps and board edges can only leave more gaps.
But the row has eight squares.
Two queens cannot be enough. This three-queen arrangement shows that three is the minimum.
The idea to take away
Instead of checking every possibility, find a limit that rules a whole group out. Here, one empty row tells us something about every possible two-queen arrangement.