A346265 Aug 24 2021 Bernard Schott A346265(4) = 5. Drawings of the five squares 30 X 30 that are tiled with squares of two different sizes so that the numbers of large or small squares are equal. Notation: s = side of the tiled square, a = side of small squares, b = side of large squares, z = number of small squares = number of large squares. 1) Three squares 30 X 30 with distinct tilings of Type 1 (see A344331) ______ ______ ______ _ ______ ______ ______ _ ______ ______ ______ _ | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|_____ |______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| | | | |_| | | |_| | | |_| | | | |_| | | |_| | | |_| |______|______|______|_|______|______|______|_|______|______|______|_| Square 30 X 30, type 1 a=1, b=3, s=30, z=90 Primitive square 30 X 30 --------------------------------------------------------------------------------------- ____ ____ _ ____ ____ _ ____ ____ _ ____ ____ _ ____ ____ _ ____ ____ _ | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| | | |_| | |_| | |_| | |_| | |_| | |_| |____|____|_|____|____|_|____|____|_|____|____|_|____|____|_|____|____|_| Square 30 X 30, type 1 a=1, b=2, s=30, z=180 9 copies of primitive square 10 X 10 with z=20 -------------------------------------------------------------------------------------------------- _____________ _____________ ______ _____________ _____________ ______ | | | | | | | | | | | | | | | | |______| | |______| | | | | | | | | | | | | | | |_____________|_____________|______|_____________|_____________|______| | | | | | | | | | | | | | | | | |______| | |______| | | | | | | | | | | | | | | |_____________|_____________|______|_____________|_____________|______| | | | | | | | | | | | | | | | | |______| | |______| | | | | | | | | | | | | | | |_____________|_____________|______|_____________|_____________|______| | | | | | | | | | | | | | | | | |______| | |______| | | | | | | | | | | | | | | |_____________|_____________|______|_____________|_____________|______| | | | | | | | | | | | | | | | | |______| | |______| | | | | | | | | | | | | | | |_____________|_____________|______|_____________|_____________|______| Square 30 X 30, type 1 a=3, b=6, s=30, z=20 Elementary square 30 X 30 with z=20 --------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------- 2) Two squares 30 X 30 with distinct tilings of Type 2 (see A344332) ________ ________ ________ ______ ________ ________ ________ ______ | | | | | | | | | | | | | | | | | | | | | |______| | | |______| |________|________|________| |________|________|________| | | | | | | | | | | | | | |______| | | |______| | | | | | | | | | |________|________|________| |________|________|________| | | | | |______| | | |______| | | | | | | | | | | | | | | | | | | |______ _|___ ____|_ ______|______|______ _|___ ____|_ ______|______| | | | | | | | | | | | | | | | | | | | | | | |______|_ ___|____ _|______|______|______|_ ___|____ _|______|______| | | | | | | | | | | | | | | | | | | | | | |______| | | |______| |________|________|________| |________|________|________| | | | | | | | | | | | | | |______| | | |______| | | | | | | | | | |________|________|________| |________|________|________| | | | | |______| | | |______| | | | | | | | | | | | | | | | | | | |______ _|___ ____|_ ______|______|______ _|___ ____|_ ______|______| | | | | | | | | | | | | | | | | | | | | | | |______|_ ___|____ _|______|______|______|_ ___|____ _|______|______| Square 30 X 30, type 2 a=3, b=4, s=30, z=36 Equals 4 copies of primitive square 15 X 15 with z=9 -------------------------------------------------------------------------------------------------------- ________________ _________________ _________________ ____________ | | | | | | | | | | | | | | | | | | | | | | | | | | | | |____________| | | | | | |________________|_________________|_________________| | | | | | | | | | | | | | | | | | | | |____________| | | | | | | | | | | | | | | | |________________|_________________|_________________| | | | | | | | | | |____________| | | | | | | | | | | | | | | | | | | | | | | | | | |____________ ___|_________ _______|____ ____________|____________| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |____________|_____________|____________|____________|____________| Square 30 X 30, type 2 a=6, b=8, s=30, z=9 Elementary square 30 X 30 with z=9