Enigma 1416: Filling in the squares
From New Scientist #2576, 4th November 2006
Susan and Paul have a new game which is played on a 4×4 array of small squares.
Two dark lines though the centre of the array, one horizontal and one vertical, divide the array into four 2×2 “regions”. If you select a square in the array, then its “environment” is the seven squares which are in the same row, column or region as the selected square.
Susan begins a game by writing 1, 2, 3 or 4 in some of the squares in the array. Paul then takes over. If he can find an empty square whose environment contains three of the four numbers then he writes the fourth number in that empty square. Paul repeats this operation on the resulting array and he does that as many times as he can. If, when Paul has finished, they find that every square of the array is full then they say that Susan made a “wise” beginning.
On Monday they played a game for which Susan made a wise beginning. On Tuesday they decided she would begin by filling one fewer squares than on Monday. After exhaustive searching they found that, with that restriction, Susan was unable to make a wise beginning.
How many squares did Susan fill on Monday?