How many not further subdivided regions are there?

Asked Jan 14 '20 at 13:06

Active Jan 14 '20 at 13:06

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I have no idea how to solve the following problem, but I found it interesting (maybe because it's very challenging to me):

$1 + 10^{1,234,567,890}$ triangles are drawn on a plane surface. What is the maximum number of areas, not further subdivided, that can be formed as these triangles intersect each other?

Source: https://web.archive.org/web/20091211190114/http://www.sigmasociety.com/sigma_teste/sigma_teste_eng.asp (question 23)

I've tried to change the problem to only three triangles, and made some drawings and manual countings, but no insight came from it. Please, could someone help me?

asked Jan 14 '20 at 13:06

Viktor K.

1

From the liinked page: "Please, don't discuss the solutions for these problems in open nor private forums. This would invalidate this test. " – Hagen von Eitzen Jan 14 '20 at 13:10
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@HagenvonEitzen: I am not taking the test, the original website is gone. This is an archived version – Viktor K. Jan 14 '20 at 13:11
Does this answer your question? Show that $n$ lines separate the plane into $\frac{(n^2+n+2)}{2}$ regions – almagest Jan 14 '20 at 14:56

How many not further subdivided regions are there?

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