Need explantion

Question

Toothpicks are used to design the following figures. Figure number 1 has an area of 1 square unit. What is the area of Figure number 10? Since I can't post a picture on here, figure number 1 shows 1 square, Figure number 2 shows 3 squares, and Figure number 3 shows 6 squares.

Answer 1

So this is the squares to stairs problem based on your explanation.

Given that, the easiest answer should be the visual one.

First think about it first in the abstract.

You have a polygon that is hard to work with, so lets make it easier. If we put a second one conjoined with the original after being rotated 180 degrees, we can get a rectangle by matching up the zig zags.

By considering the original case, we can say that any given figure is going to have 2 sides (the straight sides of the greatest magnitude) that have the same number of toothpicks, t, as the figure number, n. Thus we can say that n determines t.

Then, consider our new rectangle, what length are each of its sides. one side should be t=n and the orthogonal sides should be t=n+1. We know what the area of a rectangle is so it's safe to assume that the original polygon will have an area corresponding to half of the area from the resultant rectangle.

Now, you should feel comfortable in writing an equation for the area based on the figure, n. Let the area be equal to half of the area of the rectangle you get from pushing two of the figures together.

Let the following represent the provided algorithm in the case of fig 3.

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fig. 3

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Flipped fig 3.

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Rectangle from combining them.

Consider then the areas of each and how they relate to one another.