Enigma 454: Stepped rectangles
29 June 2018
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From New Scientist #1605, 24th March 1988 [link]
Oölith the Rational, on conquering the town of Major-Minor, the two parts of which are separated by a river, decreed that two piazzas be built, one (the larger of the two) in Major and one in Minor. They were to be stepped rectangles (see diagram) of square slabs, each measuring 1 groddly by 1 groddly. Each piazza was to contain a number of slabs equal to the perimeter in groddlies multiplied by the king’s age (an exact number of years).
The vizier explained to the grand mason appointed to this task: “A stepped rectangle is a plane array of square slabs laid edge to edge with no overlap. The sides of the figure are zig-zag: if you imagine walking around its perimeter clockwise, you must turn alternately left and right except at the four extreme corners, at each of which you must make exactly two consecutive right turns.”
The mason knew the king’s age (he was in his twenties) and realised that exactly and only two different stepped rectangles were possible which would fit the conditions.
How many slabs did he require to build:
(a) the Major piazza;
(b) the Minor piazza?