Polyiamonds are figures made by fitting together two or more equilateral triangles. The simplest polyiamond, a 'diamond', consists of two equilateral triangles.There are twelve different polyiamonds that contain six equilateral triangles each. They are shown below. Can you figure out how to fit eight of them together to form a six-pointed star? It may be necessary to turn some of the pieces over.
1. __ __ 2. __ __ __ 3. /\ 4. __ /\ /\ / \ /\ /\ /\ /__\ /\ /\ /__\/__\/ \/__\/__\/__\ \ /\ /__\/__\ \ /\ \/__\ __ \ /\ / \/__\ \ /\ / \/__\/ \/__\/ /\ __ __ 5. __ /__\ 6. /\ /\ 7. /\ /\ 8. /\ /\ \ /\ /\ /__\/__\ __ /__\/__\ /__\/__\ \/__\/__\ \ /\ / \ /\ / \ /\ /\ \ / \/__\/ \/__\/ \/ \/__\ \/ \ / \/ __ /\ /\ / /\ /\ 9. __ /__\ __ 10. __ /__\/ 11. /__\ 12. /__\ __ \ /\ /\ / /\ /\ / /\ /\ /\ /\ /\ \/__\/__\/ /__\/__\/ /__\/__\ /__\/__\/__\ /\ / /__\/
Annie says: Solutions
This is a tough problem - it can take a long time to figure out, because there isn't really a "best" way to go about it. Many people put a lot of time into pictures, including Nicole Forostoski, Mike Johnston, and Roger Mong, but I think that Justin Lam outdid everyone - Justin has mastered the art of presenting answers with only text characters.
Erin Jacobs of Cheshire High School gave a really good description of first figuring out which pieces could fill the points, sort of like finding the corners and edge pieces when you start a jigsaw puzzle. I've also included pictures from Aleksandra Cuprys, Aurelia Thompson, and Leslie Roberts, Tetsuya Matsuguchi, Tana Kaplan and Marnie Hanel, and Gena Kerr and Katie Fredlund since they're pretty good, and we need some color on these pages!
A list of all the people who got this problem right and most of the solutions are also available.
Justin Lam Grade: 8 School: Sequoia Middle School /\ /__\ 12(R) (R) = Reversed /\ /\ /__\/__\ ________ ____________ \ /\ \ /\ / \ /\ /\ /\ \/__\_\/__\/ 11 2 \/__\/__\/__\ \ /\ / ________ ____\/__\/ \ /\ / /\ /\ 5 \/__\/ /__\/__\ /\ /\ \ /\ /\ ____ /__\/__\ \/8 \/__\ \ /\ ________ ___\/__\ 7 /\ /\ /\ /\ \ /\ /\ /__\/__\/__\ /__\ \/__\/__\ \ / \ /\ 3 (R) \/ \/__\ /\ / /__\/ 11 \ / \/ I hope you understand my drawing. The numbers correspond to the numbers for your drawing of the figures and (R) means you have to take the reverse of the figure. [Justin also submitted the following alternative solution] If we are allowed to use the same piece more than once, then the task would be easy - just use Number 7 for each of the six points and use two pieces of Number 9 (base against base) to fill in the middle. That should do the trick. I guess that is not what you are looking for.I will stick with my first solution.
Erin Jacobs School: Cheshire High School, Cheshire, Connecticut In order to find a solution to the six-pointed star problem, I first looked at which pieces would fit in the points of the star. There were only six points that would work. Therefore, I moved these around until they fit together and added in two other pieces which would fill in the gaps. The star consisted of figures 12,11,7,10,3, and 2 in that order around the star in each of the points. Some of the points required more than one figure to create. I then placed figure 5 next to the points formed by figures 2 and 3. The center was then filled in with figure 8. Some of the figures had to be flipped in order to fit in the star.
Aleksandra Cuprys, Aurelia Thompson, and Leslie Robertson Grades: 9, 10, 10 School: Lakeside School, Seattle, Washington We have decided that the shapes that make up our solution are: Nos. 2, 3, 5, 7, 8, 10, 11, and 12. We tried various different methods but this was the only conclusion we could make. It works and we used eight of the twelve shapes offered to complete the star. We are almost positive that there are different solutions but we could not find them.
Tetsuya Matsuguchi Grade: 11 School: Centennial High School, Boise, IdahoTet
Tana Kaplan and Marnie Hanel Grade: 10 School: Lakeside School, Seattle, Washington We found this by cutting out the pieces and by trial and error.Thank you, Tana Kaplan and Marnie Hanel
Gena Kerr and Katie Fredlund Grade: 10 School: Lakeside School, Seattle, Washington This is Gena Kerr's and Katie Fredlund's problem of the week. We are from Lakeside School in the tenth grade. Thanks for the fun!
[Privacy Policy] [Terms of Use]
Home || The Math Library || Quick Reference || Search || Help
The Math Forum is a research and educational enterprise of the Drexel School of Education.
