Number In Order is a puzzle original to this blog to the best of my knowledge. This puzzle genre is inspired from str8ts, which is made by Jeff Widderich.
Similar to str8ts, a cell is either black (gray) or white. In an unsolved puzzle, some white cells are blank while the others are filled with numbers. To solve the puzzle, the white cells must be all filled with some numbers to satisfy a rule.
First, define a “run of cells” to be some white cells in the same row or column such that they are all adjacent to each other, and they can’t be expanded (either by being blocked by a black cell or the edge of the grid). So, R3C1 and R3C2 is a run of 2 cells. R3C4 is a run of 1 cell, separate from R3C1 and R3C2. Let’s define the length of the longest run of cells to be x.
All the numbers in a run of cells must be consecutive integers. Moreover, each cell may only contain a number between 1 and x, inclusive. So the longest run of cells will have all the numbers from 1 to x. This effectively also stops a run of cells to have duplicate numbers.
Fill each cell with a number to satisfy the above conditions.
Walkthrough of the example
Note the R2C3-R2C4 run. R2C3 is 1, and R2C3 and R2C4 must be consecutive positive integers. So R2C4 is 2. This also makes R1C3, R3C1, and R4C2 be 2 too, since the longest run of cells is 3, not 4.
The rest are runs of three cells, which must contain all numbers from 1 to 3. Simply use the usual Latin Squares technique and the puzzle is solved.