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The following question was asked in a competitive exam

Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor.What is the maximum number of tiles that can be accommodated on the floor ?

Solution obtained by trial and error. enter image description here answer from above diagram is 6 .

but is there a equation or inequality based approach (because I cannot rely on trial and error) like $$130*110 - 70*30*n < 70*30 $$ $$ n >5.8 $$ so n is the next integer so 6 , so we might get the answer right , but is the approach right ? for example if it is some other shape which is also of area $130\times110 = 14300$ but not a rectangle ,there is no guaranty that we can place inside it 6 smaller this approach is not applicable at all places .

so guide me in knowing the right approach . please help . I am stuck .

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There are a lot of heuristics and also results about both packing and tiling problems. But no general formula AFAIK. For these small cases imagination is your friend. Counting the area won't give you everything. For an easy to grasp counterexample: How many 60x60 tiles can you place on a 110x110 floor? – Jyrki Lahtonen Aug 31 '13 at 12:24
Please be more clear about what your real question is. – Marc van Leeuwen Aug 31 '13 at 12:45
approach to solve this problem without trial and error . That is I am searching for a solution using equations . – ILoveStackExchange Aug 31 '13 at 12:52

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