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Stairway

This task was found somewhere in internet.
Trying to get "analytical" solution I have got the following...

Task

Build a stairway of maximum length (in horizontal direction) using N homogeneous blocks of the same mass and size.

The blocks are not "glued" to each other in any way, but simply lie on top of each other. Thus, the projection of the center of gravity of the upper blocks onto the horizontal must be within the support area (within the lowermost block).

Let's try ?

1. Kids solution

The center of gravity of each block will be in its "geometric" center. If L is the length of the block, then its center of gravity is at a distance L/2 from the edge of the block.

Thus, the upper block can be shifted relative to the lower one by no more than S = L / 2

Adding one more block to the top (with an offset in the same direction to get a stairs) is impossible, since in this case the center of gravity of the two upper blocks will go beyond the limits of the lower block (which is their support).

The max length of the stairway is 1.5 * L

2. And if to think a little?
Why to use the maximum offset right away?

What happens if the shift S is less then L/2? It looks like the number of steps could be increased.
Let's try to formalize the problem.

Let X-coordinate of the center of gravity of the lower block is equal to Xo. Each subsequent block is shifted (relative to the previous one) in the same direction (for example, to the right) by S (to make a stairs). The coordinates of the blocks (centers) are:

Then X-coordinate of the center of gravity of a set of (upper) n blocks is:

(1)

Since the lowest block ( k = 0 ) is located at Xo , the stability condition of the stairway is:

(2)

Using (1) we rewrite (2) as:

(3)

Recall that the sum of the first n integers is:

Then, for the shift S from (3) we get:

And the length of the stairway
along X (horizontal projection)

does not exceed 2L.

By the way, the height of the stairs grows as n, increasing its steepness.

It looks like a parallelepiped, the center of which is projected onto the base.

3. And now let's think a little more.
Why should all stairs steps be with same shift?

If the shifts of the blocks are not the same, then the x-coordinates of their centers (of gravity) are:

(4)

The x-coordinate of the center of gravity of a set of n blocks as in (1) is:

Taking into account (4), the center of gravity is:

And the stability condition of the stairway when the lowest block ( k = 0 ) is located at Xo - see (2) - now is:

For convenience, let's "normalize" the shifts by L/2 :

Now the stair stability condition is:

That is, the sum of n numbers is less than or equal to n . Obvious solution: each term is less than or equal to 1 . I.e:

(5)

The sign of equity corresponds to maximum possible shift values. Smallest displacement is at the bottom (at k=1):
k = 1, f1 = 1 / n ;
k = 2, f2 = 1 / (n-1) ;
k = 3, f3 = 1 / (n-2) ;
...
k = n-2, fn-2 = 1/3 ;
k = n-1, fn-1 = 1/2 ;
k = n, fn = 1
the largest one is for the top block (k=n).

Note that if you change the block numbering (start counting from the top), then (5) will look simpler:

So, we got stairway of non-const steepness. The length of the stairway (horizontally) grows with n as

So, it is possible to build a stairway of any length!

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