Leaning tower

[Genady]

Imagine that you got planks 1m long each, as many of them as you wish. The planks have a width and a thickness, but those are not important here. What is important, they have weight.

The only thing you can do with the planks is to put them on top of each other, like this:

Your goal is to make the tower with the largest possible "shadow", x. IOW, the largest horizontal distance between the leftmost and the rightmost edges of the planks.

The question is: what is the largest x so that the tower and its parts don't fall?

[Genady]

Let's start from the top of the tower. The longest the top plank can stick out without falling is 0.5m:

In this case, x = 1.5m. Obviously, it can be made longer using more planks ...

[Genady]

Let's go to the next plank under the top two. To stick out without falling these two can have their center of gravity on the edge of the plank under them:

This position adds another 0.25m making x = 1 + 1/2 + 1/4 m.

How far can we go if we continue like this?

[Genady]

It might seem that if we continue this way, we get a series 1 +1/2 + 1/4 + 1/8 + ..., which converges to 2. That would mean that by building the tower this way we can make it maximum 2m "wide". However,

Spoiler

calculating the next offset gives us 1/6 rather than 1/8. The series is 1 + 1/2 + 1/4 + 1/6 + 1/8 + 1/10 + ...

Thus, what is the maximal possible x?

 

[TheVat]

Spoiler

It appears to be a harmonic series that is divergent.  There is no limit to x.  

 

[Genady]

39 minutes ago, TheVat said:

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It appears to be a harmonic series that is divergent.  There is no limit to x.  

 

Right. +1. Counterintuitive, isn't it?