The Empty Rectangle Technique, Explained

Picture the empty rectangle

Take a single digit and look at one three by three box. If every remaining candidate for that digit in the box fits inside one row and one column of the box, then the other four cells hold no candidate for it at all. Those four empty cells form a small rectangle, which is where the name comes from, and the live candidates trace an L along the shared row and column. The point where that row and column meet is called the intersection cell, and it does the heavy lifting later. On its own, though, the empty rectangle proves nothing. You have to combine it with a strong link somewhere else on the board, meaning a row or column where the same digit has only two candidate cells left, a conjugate pair. One end of that strong link has to land on the empty rectangle own row or column. With the L shape and the strong link pointing at each other, the digit is squeezed out of a single cell at the far crossing, and that squeezed cell is your elimination.

A worked example

Suppose the digit is six. Inside the centre box, which spans rows four to six and columns four to six, the only sixes sit along row five and column five, so the four corner cells r4c4, r4c6, r6c4 and r6c6 are all empty of six. That is your empty rectangle, and its intersection sits at r5c5. Now look outward across the rest of the grid and find that column eight has six as a candidate in just two cells, r5c8 and r2c8, forming a clean conjugate pair on the digit six. Notice that one end of that pair, r5c8, lies on the empty rectangle row, row five. This is the exact configuration the technique needs: a box whose candidates for one digit hug a single row and column, plus a conjugate pair on that same digit reaching in along the box row. With both pieces in place, the elimination is fixed by the geometry, and it falls on the cell where the box column, column five, crosses the row of the pair far end, row two.

Following the logic

To see why r2c5 must lose its six, assume the opposite and watch it break. Suppose r2c5 were a six. Then column five would already own a six, so inside the centre box the six could not sit anywhere in column five and would be forced onto row five instead. That would place row five own six inside the box, which means r5c8 could not also be six. But r5c8 and r2c8 are a conjugate pair, so if r5c8 is not six then r2c8 must be. Now you have a six at r2c5 and a six at r2c8, two sixes in row two, which is illegal. The assumption has collapsed into a contradiction, so r2c5 simply cannot be six, and you erase that candidate. This is the whole charm of the empty rectangle: a single lopsided box and one conjugate pair combine to delete a candidate from a cell that neither of them touches directly, reaching across the grid through pure logic.

How to find one

1. Scan box by box for a digit whose candidates hug a single row and a single column, leaving four empty corners.
2. Mark the intersection cell where that row and column meet inside the box.
3. Look outward along the box row and its column for a conjugate pair of the same digit, a line where it has only two candidate cells.
4. The elimination lands where the box line crosses the row or column of the pair far end, so erase the digit there.

Empty rectangle versus the fish

The empty rectangle covers ground similar to an X-Wing or a Skyscraper, but it is often easier to spot, because you only have to notice one lopsided box rather than scan many parallel lines at once. It also turns up more frequently than the rarer fish patterns like the Swordfish or Jellyfish, so the effort of learning it pays off on ordinary hard boards rather than only on exotic ones. If you already know the single-digit family, treat the empty rectangle as a close, friendly cousin you can reach for the moment a box looks unbalanced. The fastest way to make it second nature is to hunt for it deliberately on medium and hard puzzles until the L shape jumps out on its own. Shapedoku shows the same nine values as glowing shapes rather than digits, which makes a lopsided box genuinely easy to see, since a run of one colour and form along a single row and column stands out at a glance. Turn on notes, look for that telltale L, confirm the conjugate pair, and the elimination follows almost by itself.