782 /www/nrich/html/content/01/06/six3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>You can move the 4 pieces of this jigsaw around and fit them into the outlines shown below. Try for yourself with <a href="/content/01/06/six3/triangle-illusion.swf">this interactivity</a>. One square unit is missing in the lower shape.<mdo:flash height="260" id="/content/01/06/six3/triangle-illusion.swf" width="320"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/01/06/six3/triangle-illusion.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="260"></param> <param name="width" value="320"></param> </mdo:flash></p> <p>Explain what has happened to the missing area? (Of course muggles magic is not magic at all!)</p> <div class="framework">Puzzles such as these first appeared in Zeitschrift für Mathematik und Physik Leipzig, 1868, vol XIII, page 162, although Martin Gardner states that Sam Lloyd the Elder, (there were two, father and son, who were both great puzzle producers and publishers in England) also talked about them around 1865. This particular one seems to have been invented by Roy Nauw of Kloetinge, from whom permission is being sought.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Chua Zhi Yu, from River Valley High School, Singapore and Andrei Lazanu, from 12, School No. 205, Bucharest, Romania both saw through the deceptiveness of these diagrams and sent explanations of why one square unit of area seems to disappear when the pieces are rearranged.</p> <p>Look at the triangle with arms of lengths 5 and 13 units. We shall call this triangle T. Although the figure at the top looks like this triangle it is not a triangle at all. What appears to be the longest side is not a straight line but actually two lines along the hypotenuses of the red and blue triangles. The gradient or slope of the hypotenuse of the red triangle is 3/8, while the gradient of the hypotenuse of the blue triangle is 2/5. We know 3/8 &lt; 2/5 so the figure at the top has two edges that 'dip inwards' making it into a concave quadrilateral. This shows that the four pieces of this jigsaw fit inside the right angled triangle T leaving a small space uncovered. Exchanging the positions of the red and blue triangles makes these two hypotenuses project outwards enclosing extra area in the shape of a long thin parallelogram.</p> <p>Adding up the areas, the red triangle has area 12 square units, the blue triangle 5 square units and the other two pieces together 15 square units making a total of 32 square units. The right angled triangle T with arms of length 5 and 13 units has area 32.5 square units, an extra half a square unit. By rearranging the pieces the extra area included along the hypotenuse of triangle T is twice this, namely one square unit, and this accounts for the indentation on the bottom of the lower figure.</p> <mdo:image width="320" height="260" border="0" alt="" src="triangle-illusion.gif"></mdo:image> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You might like to explore this problem with a Logo program. <br></br> See the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=4863&amp;part=index"> Muggles, Logo and Gradients</a> . <br></br> <br></br> For a similar problem on this topic see <a href="http://nrich.maths.org/public/viewer.php?obj_id=677&amp;part=index"> Lying and Cheating</a><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Think about areas. Think about gradients.<br></br></mdoxml> 2 3 0 0 1 0 0 You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area. Measures and Mensuration Area Trigonometry Tangent Sequences, Functions and Graphs Gradients Using, Applying and Reasoning about Mathematics Visualising Information and Communications Technology Interactivities