SUDOconnect

Take the solution of a normal SUDOKU, for instance this one:

In this solution not only the numbers are given, but also connections: two neighbour cells (horizontal, vertical or diagonal) are connected if their difference is at most one. The difference of a solution of a SUDOKU and a SUDOconnect is that in a SUDOconnect all cells are required to be connected to each other. The above example is not a SUDOconnect solution for several reasons. For instance, the 1 on the left bottom is not connected to any other cell, and the same holds for several other cells. But the connectedness requirement of a SUDOconnect solution is stronger than only requiring that every cell is connected to another cell. In the above example all cells of the green part in the bottom are connected to other cells, but there is no connection from these cells to the cells connected in red.
So every two cells should be connected to each other by a sequence of connections, as is the case in the next example:

At a first glance, SUDOconnect puzzles look the same as SUDOKU puzzles. But in solving them it is essential to exploit the extra connectedness requirement: typically considered as a SUDOKU, the SUDOconnect has hundreds of distinct solutions, only exactly one of which satisfies the connectedness requirement as described above. The goal of the puzzle is to find this.

The rules
Summarizing, a SUDOconnect solution satisfies:

The normal SUDOKU rules: every column, every row and every 3x3 block indicated by fat lines contains the numbers 1 to 9, all exactly once.
Every two of the 81 cells are connected by a path of connected neighbours, where two neighbours (horizontal, vertical or diagonal) are connected if their difference is at most one.

A first SUDOconnect puzzle to try is the following:

The puzzles
We developed three series of SUDOconnect puzzles, of which the first one is the simplest, and recommended to start with. In all three series the puzzles are arranged according increasing difficulty.

Warning: SUDOconnect is much harder than SUDOKU, and some puzzles are very hard.

Advice: always when having solved a SUDOconnect puzzle, start by an arbitrary cell, mark it, and mark every cell that is connected to it, and continue to mark every cell that is connected to a marked cell. If you have a right solution, at the end all 81 cells should be marked.

The SUDOconnect project is joint work of Hans Zantema and Walt van Ballegooijen; the idea of considering this connectedness requirement is due to Hans Kuiper.