Visual puzzles are some of my favorites, and while they take a little effort on your part to explain the answers, I think they're worth it. If you try drawing some pictures, make sure they look okay in a monospaced font, and that you use only spaces, not tabs. I've included a hexagon (below) you can play around with. This problem also has a lot of parts to it - make sure you answer them all!Split a regular hexagon into three identical parts. What shape is each part? Now split it into six identical parts in at least two different ways. What shapes are your pieces? Can you figure out how to split it into six kites?
Annie says: Solutions
This is a hard problem, both for those submitting answers and sometimes for the person reading them! How do I decide what is precise enough? It is tough to explain this accurately without pictures, and as many of you know, it's also hard to draw pictures using only text characters. It takes a lot of patience and a little skill. Mostly patience.
I've highlighted four solutions this week. Dave Peterson, a homeschooling parent, provided a pretty thorough explanation of a zillion ways to solve this. Eric Faden from Georgetown Day school drew some really nice pictures (I'm a sucker for nice pictures!). Amy Mousaw of Cheshire High School wrote a very nice solution detailing how she approached the problem and came up with her answer. And Katie Quinn-Kerins from Germantown Academy explained how to split a hexagon into six congruent parts in a lot of different ways. I encourage you to read these solutions carefully!
A list of all the people who got this problem right and most of the solutions are also available.
Dave Peterson ABB Industrial Systems, Inc. Rochester, New York This problem intrigued me, because at first it seemed trivial, but then I found some unexpected solutions. I wonder if any students will find them all. For three pieces, there's an obvious answer consisting of rhombuses, and a less regular one consisting of irregular (but symmetrical) pentagons: ______ _______ /\ \ / | \ / \ \ / | \ / \_____\ / | \ \ / / \ / \ / \ / / \/ \/ \/_____/ \_______/ X1 X2 In fact, if you draw any ray from the center, and rotate it twice by 120 degrees, you will get three identical irregular pentagons, so there are an infinite number of answers, of which only the two above produce symmetrical pieces. _______ / \ \ / \ \ / \ \ \ / - _/ \ / / \_______/ X (includes cases X1, X2) Now for six pieces, we can obviously take the two symmetrical cases above and divide each piece in half; because the rhombuses have two lines of symmetry, there are two ways (A1, B1) to divide them in identical halves: ______ ______ _______ /\ /\ /|\ - \ / | \ / \ / \ / | \ - \ /\ | /\ /____\/____\ / | \____\ / \ | / \ \ /\ / \ | / -/ \ / | \ / \ / \ / \ | / - / \/ | \/ \/____\/ \|/____/ \___|___/ A1 B1 A2 The first two consist of triangles; the last consists of kites, as requested. But there's yet another way to cut the rhombuses in half! _______ / /\ /------/ \ /______/ / \ \ \ \ / / \ \ \ / \___\__\/ B2 This gives six parallelograms. Finally, there are two more infinite sets of solutions. First, take the solution with equilateral triangles (A1), and rotate the radial lines by any angle; or to say it more carefully, draw any ray from the center and rotate it five times by 60 degrees. You will get six irregular quadrilaterals, which will change, as you rotate the radial lines, from the equilateral triangles (A1) through the kites (A2) and back. _______ / \ \ /_ /\ / - \/ \ \ / \ - _/ \/ / \_____\_/ A (includes cases A1, A2) Then take the three rhombuses, and cut one of them in half using any line (not just the lines through pairs of vertices), producing a pair of identical trapezoids; then rotate the cut rhombus around the center to find the angle at which to cut the next rhombus; or to describe it differently, take the three lines added to create the equilateral triangles (A1) from the rhombuses and rotate them all by the same angle. This family of solutions includes the other triangle solution (B1) and the parallograms (B2) as well. ______ /\_ \ /\ \ _\ / \_____\ \ / \ / \ \/ / \/___\_/ B (includes cases A1, B1, B2) These aren't very clearly drawn, but I hope they give the idea. What I've done is to describe two infinite sets of solutions, which include the four special cases, and overlap by both including case A1. (A1 and B1, of course, deserve special mention because the quadrilaterals turn into triangles in those cases.)
Eric Faden School: Georgetown Day School, Washington, DC Teacher: Paul Nass
Amy Mousaw Grade: 10 School: Cheshire High School, Cheshire, Connecticut Teacher: Mrs. Miller We were given visual puzzles to "play around with" in order to find answers to three questions. These questions were all about how to split a rectangle into: 1) three identical parts, 2) two different sets of six identical parts, and 3) six identical "kites." In order to do this I had to give reasons, include sketches, and discuss the geometric properties that each division had. When I saw the first question, I thought I would try to put a dot in the middle of the hexagon and divide it from there. I saw that there were six vertices, so I divided that number by three to get two. I then drew a line at every second vertex. These divisions caused the hexagon to become divided into three rhombuses. It ends up looking like this: ________ / /\ / 2 / \ /_______/ 1 \ \ \ / \ 3 \ / \_______\/ Since a rhombus is an equilateral quadrilateral, it has completely equal lengths on all sides. If you take one of these three rhombuses and cut one out, you can place it perfectly on top of another. This is because they are equal. The second problem had two parts, to split the hexagon into six parts in two different ways. For my first way, I took the information I obtained from the first problem and used it to answer this question. I used the same shape as the last one (with the rhombuses) and connected the line segments at the vertices. This made six triangles. Two of the sides of the triangle are equal, and two of the angles are equal. The triangles are all equal to each other and can be placed on top of each other perfectly. The end result will look like this. _____ / /\ / / / |\ / /_ / | \ \\ \ | / \ \ \ |/ \_ \ \/ The second way I divided the regular hexagon is again by placing a dot in the center and drawing a line to each vertex. Since there are six vertices it worked out making six equal triangle. All the sides of these triangles are equal and so are the angles. It is an equilateral triangle, and the six are all equal to each other. The way I divided the regular hexagon to make "six kites" is also another answer to number two. I figured out this answer by trial and error. After a few attempts, I placed a line down the center of the hexagon from one line segment to the one opposite it. I then drew lines from the center of the hexagons' line segments through the center of the hexagon and to the center of the line segment opposite it. This made kites that have four sides. Of these four sides two pairs equal each other. _______ / | \ /\ | / \ / \ / \ \ / \ / \/ | \/ \___|___/ By dividing up the regular hexagon, I have made four different geometric shapes including: a rhombus, two different kinds of triangles, and a kite. All these shapes have been equal to the others that make up the hexagon. The illustrations help to prove this point.
Katie Quinn-Kerins Grade: 10 School: Germantown High School, Fort Washington, Pennsylvania Teacher: Mrs. Carver A regular hexagon has 6 sides and 6 intersecting points. If lines were drawn from 3 of the points to the middle of the hexagon, the hexagon would have 3 parts. In order for those parts to be identical, every other point would have to be used - no two points next to one another could be used. These 3 parts would be identical parallelograms. To split a regular hexagon into 6 parts, lines could be drawn from each of the points of the hexagon; all connecting in the middle. This would result in 6 equal triangles. Another way to divide the hexagon into 6 equal parts is to draw a line from the center of each of the lines that make up the hexagon to the middle. Each of these identical figures would be shaped like kites. There are many other ways to split a hexagon into 6 equal parts. Each one involves drawing a line from each side of the hexagon to the middle. Each of the lines to the middle leaves an individual side of the hexagon and each leaves from the same point on that side. For example, if each side of the hexagon is 5 units long and each of the 6 lines is drawn to the middle from one unit to the right of each point (when looking at the side from the middle's perspective), then 6 equal lop-sided, kite-looking, shapes would be formed.
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