56 Triangles Arrangement

Probability Level 4

What is the maximum number of regions that 56 triangles can divide a plane into?

For example, if we have 2 triangles arranged in the Star of David, we get 8 regions.


The answer is 9242.

Aayush Mani by Aayush Mani , 21, India

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3 solutions

Shravan Ashok
May 23, 2014

The formula for solving this is 3n^2 - 3n + 2. For a more detailed article on how the recurrence relation works check out USAMTS Problem 4/4/14 in this link: http://stuff.mit.edu/~agustya/Math/4-14.html

2 Helpful 0 Interesting 0 Brilliant 0 Confused

The formula is also in 20 MATHCOUNTS Skills by Jane Chen.

Kaan Dokmeci - 7 years ago
Ang Yu Jian
Jun 6, 2014

It's much easier to see if one constructs the triangles by placing the vertices equal distances apart on a circle. Then we see that the spaces between a triangle and the next is divided into n-1 smaller regions. With 3 such regions per triangle and n triangles as well as the inner and outer spaces we arrive at 2+3(n)(n-1) as the final answer

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Rajiv Poduval
Jun 3, 2014

N^3 - (N-1)^3 +1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you explain where you obtained this formula from?

Calvin Lin Staff - 7 years ago

I must confess that my solution is not elegant and would be happy to see one. There are couple of ways I got to this answer. I looked at N=1 and N=2 (David's triangle) and constructed a N=3 around it. One can see that the pattern (leaving the outside region) is 0 + 1, 6 + 1, 9 + 9 + 1.. (Level 1 traingales, level 2 triangles, and so on.... ) This is nothing but 3N * (N-1) + 1.. Add 1 to this figure to include the open region and one can get to the solution. The second approach: One can see that this is 1,7,19 and so on.. Which is nothing between the difference of adjacent cubes and hence the formula shared above

Rajiv Poduval - 7 years ago

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