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How many triangles? 
Mathematics  
How many triangles are there in this picture? 94.7% of Americans miss the answer. Try it first, then see the answer below.
Answer: 6 small red triangles 3 small white triangles 3 medium red triangles with a small white triangle center 1 large triangle with 3 small triangles in the center  total = 13
Triangles within a TriangleDate: 11/10/96 at 07:38:57 From: Cato H. Jensen Subject: Geometry, Math Project (GCSE) 1996 This is an English project called "Triangles in Triangles" for I and H level. We are doing this in 10th grade. The question is: Investigate equilateral triangles of other sizes in order to examine the relationship between the size of a triangle and the total number of triangles it contains. The numbers (where n = total number of triangles and b = number of triangles in the base) given are : b n 1 1 2 5 3 13 4 27 5 48 6 78 7 118 8 170 I need to find a formula for the total number of triangles when there are X base triangles. I have done the rest of the project, but this part has gotten me really stuck. Date: 01/04/97 at 15:58:04 From: Doctor Chaos Subject: Re: Geometry, Math Project (GCSE) 1996 I think the best way to approach your problem is to examine the pattern you were given in order to calculate the total number of triangles. Look at these pictures: This triangle has 3 little triangles in the base so b=3. /\ In this triangle, there are nine (9) little /__\ triangles or b^2. Then there are three (3) /\ /\ triangles which are larger (containing 4 little ones) /__\/__\ and one (1) largest (containing all 9). /\ /\ /\ /__\/__\/__\ 9+3+1 = 13 triangles. This triangle has 4 little triangles in the base so b = 4. /\ The total of little triangles is b^2 or 16. /__\ The number of bigger triangles (containing 4 little /\ /\ ones) is 7. The number of still bigger triangles /__\/__\ (containing 9 little ones) is 3. The number of /\ /\ /\ largest triangles contains all 16 and is 1. /__\/__\/__\ /\ /\ /\ /\ 16+7+3+1 = 27 triangles /__\/__\/__\/__\ If we examine the numbers of triangles at every level, we may see a pattern which may then suggest a formula. b = Number of triangles in the base l = Number of little triangles f = Next largest triangle (contains 4) n = Next largest triangle (contains 9) s = Next largest triangle (contains 16) T = Total number of triangles b l f n s T 1 1 0 0 0 1 2 4 1 0 0 5 3 9 3 1 0 13 4 16 7 3 1 27 5 25 ? ? ? 48 6 36 ? ? ? 78 7 49 ? ? ? 118 8 64 ? ? ? 170 Organize the data you have already collected and see if there is a formula that will help you add up the various sizes of triangles to arrive at the total. You might even need 2 different formulas; one for an odd number on the base and one for even numbers. You never know until you try. Good luck. Doctor Chaos, The Math Forum Check out our web site! http://mathforum.org/dr.math/

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Herb O Buckand has an enormous interest in the structure of threes and has been collecting and examining these concepts for years. He has an exciting website for his Threseology Research Journal located at http://www.threesology.org. Here he delves into concepts in threes in many different areas. He is a true generalist. 