EdgeNumsSolving EdgeNums step by step
This tutorial solves one 5×5 puzzle move by move. Every step shows the grid as it stands and explains why exactly that cell is next.
The rules in three sentences
- Every row and every column contains each number from 1 to 4 exactly once, plus one empty cell.
- The edge number is the first number visible from that side. It sits in the first or second cell, and only the empty cell may come before it.
- Every puzzle has exactly one solution, and you find it without guessing.
The puzzle
We start with an empty 5×5 grid. The numbers on the outside are the edge numbers; a side without a number gives no clue. Wanted are the numbers 1 to 4 plus one empty cell per row and column.
The solution move by move
Edge number (left) into the first cell
Edge number left, row 2: 3. The 3 belongs in the 1st or 2nd cell; column 2 is ruled out (edge number 3 below column 2 requires 3 to be in the 1st or 2nd cell from the bottom), so the highlighted cell remains.
Almost every puzzle starts like this: two edge numbers together rule out a cell.
Edge number (left) into the second cell
Edge number left, row 5: 3. The 3 belongs in the 1st or 2nd cell; column 1 is ruled out (3 is already present in column 1), so the highlighted cell remains.
Edge number (top) into the first cell
Edge number top, column 1: 2. The 2 belongs in row 1 or 2; row 2 is ruled out (it is already occupied), so the highlighted cell remains.
Edge number (top) into the second cell
Edge number top, column 2: 2. The 2 belongs in row 1 or 2; row 1 is ruled out (2 is already present in row 1), so the highlighted cell remains.
Edge number (bottom) into the last cell
Edge number bottom, column 3: 1. The 1 belongs in row 4 or 5; row 4 is ruled out (edge number 1 left of row 4 requires 1 to be in the 1st or 2nd cell from the left), so the highlighted cell remains.
Edge number (bottom) into the second-to-last cell
Edge number bottom, column 1: 1. The 1 belongs in row 4 or 5; row 5 is ruled out (1 is already present in row 5), so the highlighted cell remains.
Edge number (left) forces an empty cell
Edge number left, row 5: 3. The 3 is already in column 2. All cells before it must be empty, including the highlighted one.
That locates the empty cell without a single number being involved.
Column almost complete
In column 1 only the highlighted cell is still free. That is where the 4 belongs.
Edge number (right): first cell
Edge number right, row 5: 4. The empty cell lies further left, so the 4 goes directly in the highlighted cell.
Row almost complete
In row 5 only the highlighted cell is still free. That is where the 2 belongs.
Edge number (top) forces an empty cell
Edge number top, column 2: 2. The 2 is already in row 2. All cells above it must be empty, including the highlighted one.
Edge number (right): first cell
Edge number right, row 1: 3. The empty cell lies further left, so the 3 goes directly in the highlighted cell.
From here the puzzle almost fills itself in.
The rest
After these twelve moves the remaining 13 cells are routine: every row and column is missing only one or two values, and those follow from what is already there. This is what the finished solution looks like.
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The techniques at a glance
These patterns come back in every puzzle, in larger grids up to 8×8 as well. The complete list of techniques is in the app under Rules.
Edge number into the first or second cell
At most the empty cell may come before the edge number, because any number in front of it would itself be the first one visible. Two cells per edge number, never more choice than that. This is the lever for almost every move.
If the edge number sits in the second cell, the first one is empty
The reverse of the basic rule. As soon as it is clear that the edge number is not on the outside, the outer cell is that row's empty cell.
Edge number forces an empty cell
If you find the edge number somewhere else in the row, everything between the edge and that number must be empty. Since there is only one empty cell per row, its position is then fixed.
Two edge numbers cross
A row and a column claim the same cell. If only one of the two demands can hold, the cell is blocked for the other number. The first move above comes about exactly like this.
Only position in the row or column
One number is still missing, and in the whole row exactly one cell is left where it can go: everywhere else it is already in the column. In the example the 1 belongs in the highlighted cell of row 1.
Row or column almost complete
With four of five cells filled, the last one follows from what is left. After every move it pays to check which row or column has just been completed. In row 5 only the 2 is missing.