The decoration at the right is associated with the Buddhist Lent. Assuming that the decorative object is two-dimensional, how many triangles can be found? Shown below is a black and white diagram without the extra decorations.
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Answer to the Problem of the Week for the week of October 17, 2005 |
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| Buddhist Lent Puzzle from the Mathematics Teacher |
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The decoration at the right is associated with the Buddhist Lent. Assuming that the decorative object is two-dimensional, how many triangles can be found? Shown below is a black and white diagram without the extra decorations. |
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Solution to the Problem: |
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There are 104 triangles in the figure.
To organize our approach, we create a two-dimensional representation of
the decorative object, labeling the triangles 1 to 24, noticing that the interior of
the figure is a regular hexagon.
Next we look for triangles that are created from three single triangles. An example of this type of triangle is triangles 1 and 13 and 24. Finding mirror images, we count a total of 12 such triangles, and we find no triangles that are exclusively in the regular hexagon of the figure. Now we examine the figure for triangles made up of four smaller triangles, such as triangles 1 & 13 & 24 & 23. There are 12 of this type of triangle.There are no large triangles made up of five smaller triangles. We find 6 large triangles containing six smaller triangles, one of which is triangles 1 & 13 & 24 & 23 & 22 & 10.No large triangles are formed by seven smaller triangles. Large triangles containing eight smaller triangles can be created in two ways. We found 6 similar to triangles 1 & 13 & 24 & 23 & 22 & 21 & 20 & 8,and 6 similar to triangles 1 & 2 & 13 & 14 & 24 & 15 & 23 & 16. We found 12 large triangles that contain nine smaller triangles, one of which is triangle 1 & 13 & 24 & 23 & 22 & 10 & 9 & 21 & 20. No large triangles can be made from ten, eleven, or even twelve smaller triangles. Indeed, the final large triangle that can be created in this figure is made of eighteen smaller triangles, and 2 of this type can be found in the figure. We have found a total of 104 triangles in the figure. |
| 1. Sagar Patel | Columbus, Georgia |
| 2. Tristan Collins | Virginia Tech Blacksburg, Virginia |
| 3. David & Judy Dixon | Bennettsville, South Carolina |
| 4. Praveen Nandamuru | Columbus, Georgia |
| 5. Heather Brown | Winchester, Virginia |
| 6. Stephen O'Hara | Winchester, Virginia |
| 7. Kelley Kolar | Winchester, Virginia |
| 8. Elizabeth Carlson | Winchester, Virginia |
| 9. Jessie Biggs | Winchester, Virginia |
| 10. Charles Washington | Winchester, Virginia |
| 11. Andrew Montoya | Winchester, Virginia |
First, we count 24 single triangles. Then we count triangles that are constructed
from two single triangles, such as the triangle created by combining triangles 1 and 13
and that triangle's mirror image, triangles 2 and 14. We find a total of
12 triangles of this type. Two more triangles created by combining two
single triangles are triangles 1 and 2 and its mirror image, triangles 13 and 14. We find 12 more
triangles of this type. Therefore, we count a total of 24 triangles constructed from two single triangles.