Squarecase and dominos
Bill has a "very large" sheet of squared paper and an equally large stack of dominoes. Each domino will exactly cover two adjacent squares on the paper.
Bill takes a pencil and boldens the outline of the square in the top left corner of the paper. He calls this square 'Square 1'. Of course, he can't fit a domino exactly over the square because the domino covers two squares.
Bill then boldens the outlines of the first two squares on the second row, to make an L shape and calls these 'Square 2' and 'Square 3'. He can now fit a domino exactly over either Square 1 and Square 2, or Square 2 and Square 3, but either way he has one square left over that he can't fit a domino on as he has an odd number of squares.
He then boldens the outlines of the first three squares on the third row, to make a staircase shape made out of squares (a squarecase?) and calls these Squares 4, 5 and 6. As he now has an even number of squares, he expects that he should be able to exactly lay three dominos, but finds that however he arranges the first two dominos, he is left with two non-adjacent squares.
Bill carries on adding extra rows, with an extra square on each row. Which of these rows results in a 'squarecase' which can fit a given number of dominos exactly?
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1 solution
There are at least two ways of looking at this problem.
1) Square 1 only has one adjacent square (Square 2, below it), so a domino must cover both Square 1 and Square 2. As Square 2 is now covered, the only adjacent square to Square 3 is Square 5(below it), so a domino must cover these two. This means that the only adjacent square to Square 6 is Square 9 (below). After we put a domino on Squares 6 and 9, the only adjacent square to Square 10 is Square 14. And so it goes on. However when we reach the last square of the bottom row [ if there are n squares, this square will be n(n+1)/2]. As this is the bottom row, there is no square below it, so Bill cannot put a domino on this square. Therefore it is impossible to fit an exact number of dominos on any squarecase.
2) Imagine Bill shades some of the squares black to create a chequerboard pattern (i.e. 1,3,4,6,8,10,11,13,15 etc), a domino will fit over one white square and one black square. However large the squarecase is, it will always contain more black squares than white squares, as the main diagonal is black. Therefore, again, it is impossible to fit an exact number of dominos on any squarecase.