Weekly Problem 24

2834 /www/nrich/html/content/id/2834/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> What is the maximum number of pieces with the shape T which can be placed within the $5 \times 5$ grid shown, without overlapping, and with their edges along the lines of the grid? <div><mdo:image alt="t shape and 5x5 grid" height="162" src="wp24.JPG" width="265"></mdo:image></div> If you liked this problem, <a href="http://nrich.maths.org/4975">here is an NRICH task</a> which challenges you to use similar mathematical ideas. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <mdo:image height="162" width="161" src="wp24s.JPG" alt="a solution"></mdo:image> The diagram shows that it is possible to fit five T shapes in the square. In order to fit six T shapes into the square, exactly one of the $25$ squares would be left uncovered; hence at least $3$ corner squares must be covered. We now label a corner square $H$ or $V$ if it is covered by a T shape which has the top part of the T horizontal or vertical respectively. If all four corner squares are covered then there must be at least two cases of an $H$ corner with an adjacent $V$ corner. Each such combination produces a non-corner square which cannot be covered e.g. the second square from the right on the top row of the diagram. If only $3$ corner squares are covered, there must again be at least one $H$ corner with an adjacent $V$ corner and therefore a non-corner square uncovered, as well as the uncovered fourth corner. In both cases, at least $2$ squares are uncovered, which means that it is impossible to fit six T shapes into the square. </mdoxml> 2 3 0 1 1 0 0 Weekly Problem 24 - 2013 Weekly Problem 24-2013 Measures and Mensuration Area