KNS Puzzles offers the Fillomino puzzles in the following formats:
4 x 4 - Level 1 - Level 2
5 x 5 - Level 1 - Level 2
6 x 6 - Level 1 - Level 2
7 x 7 - Level 1 - Level 2
9 x 9 - Level 1
Complete the grid such that every row and column contains the numbers 1 to the size of the grid. The arrows on the grid are less-than and greater-than signs. e.g. 1 < 4, 3 < 5, 2 > 1.
This is the start of the puzzle, we're given the <3>s as initial clues.
There is only one way to fulfil these two squares 3 > 2 > 1.
This could be either 3 > 1, or 3 > 2, but we've already got the <2> in Column 2, so it must be 3 > 1.
In Row 1 the remaining square must be <4>. In Column 2 the remaining square must also be <4>.
As we've already got the <3> in Column 4 and the <4> in Row 4, there is only one way to complete these two squares, 2 > 1.
The only missing number for Row 4 is <3>.
The <1> in Row 2 can only go here (as there is already a <1> in Column 1 and Column 3).
The <4> in Column 4 must go here.
The <3> from Row 3 must go here (as there is already a <3> in Column 1).
The puzzle completes as both Row 3 and Column 3 have one remaining number. Which leaves the last <4> to complete Row 2 / Column 1.
The completed puzzle.