X Tiles Challenge: Patterns and Matching
In this X Tiles Challenge, Holly arranges square tiles with colored triangles to create all-matched squares and rectangles. The task involves creating specific patterns, identifying unique X Tiles, and constructing various configurations using the tiles. Explore the creative possibilities and matching strategies in this engaging math challenge.
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AMT CLASSROOM PROBLEMS MATHS CHALLENGE EDITION X Tiles Intermediate | Years 9 10 Topic: Patterns Source: Maths Challenge 2013
X TILES CHALLENGE PROBLEM T1 T3 T2 An X Tile is a square tile with its two diagonals marked on the front to create four triangles. Each triangle is coloured black, white, or grey. The back of each X Tile is red, so they are never flipped over. Two patterns are the same if they are just rotations of one another, for example, T1 and T2. However reflections may be different, for example, T2 and T3.
X TILES CHALLENGE PROBLEM Holly had the following set of 20 X Tiles.
X TILES CHALLENGE PROBLEM Holly took four of her X Tiles and arranged them into a square so that the colours matched where the tiles met and there was the same colour around the edge of her square. Because the square consisted of two rows of two tiles each, she called it a 2 2 all-matched square. Here is an example.
X TILES CHALLENGE PROBLEM All-matched rectangles can be constructed in a similar way. Two all- matched squares or rectangles are the same if they are just rotations of one another, however reflections may be different. a. Draw another 2 2 all-matched square using four of Holly s X Tiles. b. Draw an all-matched rectangle using six of Holly s X Tiles. c. There are in fact 24 different X Tiles. Draw the four that are not in Holly s set. d. Draw two different 2 2 all-matched squares that each contain at least one X Tile that is not in Holly s set.
X TILES MARKING SCHEME a. A correct 2 2 all-matched square: 1 mark b. A correct 1 6 or 2 3 all-matched rectangle: 1 mark c. Four correct X Tiles: 1 mark d. Two correct 2 2 all-matched squares: 1 mark
X TILES SOLUTIONS a. Here are two 2 2 all-matched squares. There are several others. 1 mark
X TILES SOLUTIONS b. Here are two all-matched rectangles using six of Holly s X Tiles. There are several others. 1 mark
X TILES SOLUTIONS c. Here are four correct X Tiles. 1 mark
X TILES SOLUTIONS d. Here are two 2 x 2 all-matched squares. There are several others. 1 mark
X TILES EXTENSIONS 1. How many different 2 2 all-matched squares could be constructed if any four X Tiles could be used for each square. 2. Explain why a 2 10 all-matched rectangle is not possible, even if all 24 X Tiles are available. 3. Construct a 4 6 all-matched rectangle from the full set of 24 X Tiles.
X TILES SOLUTIONS TO EXTENSIONS 1. First suppose all triangles on the perimeter of a 2 2 all-matched square are white. There are three cases. Case 1: no pair of white triangles meet at an edge. There are only four suitable X Tiles and they can form only one square.
X TILES SOLUTIONS TO EXTENSIONS Case 2: exactly one pair of white triangles meet at an edge. There are only two suitable X Tiles that can form this pair of white triangles. Hence there are only four squares.
X TILES SOLUTIONS TO EXTENSIONS Case 3: exactly two pairs of white triangles meet at an edge. In this case we have only two squares. Thus there are seven 2 2 all-matched squares with all triangles on the perimeter white. Similarly, there are seven 2 2 all-matched squares with all triangles on the perimeter black and seven with all triangles on the perimeter grey. Hence there are 21 different 2 2 all-matched squares.
X TILES SOLUTIONS TO EXTENSIONS 2. Suppose there was a 2 10 all-matched rectangle and all the triangles on the perimeter were white. This would require 20 X Tiles with at least one white triangle. There are only 18 X Tiles with at least one white triangle. So there is no 2 10 all-matched rectangle with all perimeter triangles white. Similarly there is no 2 10 all-matched rectangle with all perimeter triangles black and no 2 10 all-matched rectangle with all perimeter triangles grey.
X TILES SOLUTIONS TO EXTENSIONS 3. There are many thousands of 4 6 all- matched rectangles. Here is one:
X TILES ACKNOWLEDGEMENT The 24 X Tiles in this problem are called MacMahon squares after Major Percy Alexander MacMahon R.A., D.Sc, Sc.D., LL.D., F.R.S. He mentions them on Page 23 of his book New Mathematical Pastimes, which was published in 1921 by Cambridge University Press.
