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The question is as follows:

We know that in order to create 1, 2, 3 and 4 congruent unit equilateral triangles on a flat plane, we need 3, 5, 7 and 9 matchsticks respectively.

What is the minimum number of matchsticks on a flat plane needed to create 7 congruent unit equilateral triangles?

 13, 14, 15, or 16?

I am thinking that the answer is 14 because I have drawn out 12 matchsticks that would create six equilateral triangles, and by an addition of two more matchsticks, I was able to get seven equilateral triangles. My diagram may be wrong, therefore, I am asking about what the correct answer could be.

Ethan Bolker
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geo_freak
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2 Answers2

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We need 3 matchsticks to make a triangle, but a matchstick in the interior of our figure can contribute to two triangles.

If we want 7 triangles, those have 21 sides; to do this with 13 matchsticks, we need 8 of the matchsticks to be inside the figure (contributing to 16 sides), and 5 on the outside (contributing to 5 more sides, 21 in total).

But a shape with perimeter 5 has area at most $\frac{25}{4\pi} \approx 1.99$: the area of a circle with circumference 5. This is not enough to fit 7 equilateral triangles, which have total area $\frac{7\sqrt3}{4} \approx 3.03$.

So there cannot be a 13-matchstick construction, and at least 14 matchsticks are required. As you've noticed, there is a 14-matchstick construction, so that is the best value possible.

Misha Lavrov
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following on ..... the matchstick analysis looks at the size=1 triangles in Triangle formations, Star formations and Hexagon formations.

Adding 2 matches will always add 1 triangle; sometimes adding 3 will develop 2; sometimes adding 1 will be enough.

The sequence would run 0, 3, 5, 7, 9 (Triangle), 11, 12 (Hexagon), 14, 16, 18 [18 matches in a T3 triangle for 9 T1 triangles) then 20m -> 10T; 22m -> 11T; then 24 in a Star for 12 triangles [simply by +6m => +3T) or a T4 Triangle. 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, ... This sequence is NOT in OEIS.org.

Onward with Stars and Triangles : 45 matches in a T5 triangle delivers 25 T1 triangles BUT 42 matches in a Hexagon delivers 24 triangles (and 2 more gives 25T from only 44m).

Around the 84-90 match level, the Triangle has 84m -> 49T; the Star 84m -> 48T while the Hexagon has 90m -> 54T and clearly a 'best' figure of 84m -> 50T.

It is unclear which is the most efficient formation for delivering T1 triangles. JK

J King
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