Mike PatersonUri Zwick

Overhang

The overhang problem

How far off the edge of the table can we reach by stacking n identical blocks of length 1?

J.B. Phear – Elementary Mechanics (1850)J.G. Coffin – Problem 3009, American Mathematical Monthly (1923).

“Real-life” 3D version Idealized 2D version

No frictionLength parallel to table

The classical solution

Harmonic Stacks

Using n blocks we can get an overhang of

Is the classical solution optimal?

Obviously not!

Inverted triangles?

Balanced?

???

Inverted triangles?

Balanced?

Inverted triangles?

Unbalanced!

Diamonds?

The 4-diamond is balanced

Diamonds?

The 5-diamond is …

Diamonds?

The 5-diamond is Unbalanced!

What really happens?

What really happens!

Why is this unbalanced?

… and this balanced?

Equilibrium

F1 + F2 + F3 = F4 + F5

x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5

Force equation

Moment equation

F1

F5F4

F3

F2

Forces between blocks

Assumption: No friction.All forces are vertical.

Equivalent sets of forces

Balance

Definition: A stack of blocks is balanced iff there is an admissible set of forces under which each block is in equilibrium.

1 1

3

Checking balance

Checking balance

F1F2 F3 F4 F5 F6

F7F8 F9 F10

F11 F12

F13F14 F15 F16

F17 F18

Equivalent to the feasibilityof a set of linear inequalities:

Static indeterminacy:balancing forces, if they exist, are usually not unique!

Balance, Stability and Collapse

Most of the stacks considered are precariously balanced, i.e.,

they are in an unstable equilibrium.

In most cases the stacks can be made stable by small modifications.

The way unbalanced stacks collapse can be determined in polynomial time

Small optimal stacks

Overhang = 1.16789Blocks = 4

Overhang = 1.30455Blocks = 5

Overhang = 1.4367Blocks = 6

Overhang = 1.53005Blocks = 7

Small optimal stacks

Overhang = 2.14384Blocks = 16

Overhang = 2.1909Blocks = 17

Overhang = 2.23457Blocks = 18

Overhang = 2.27713Blocks = 19

Support and balancing blocks

Principalblock

Support set

Balancing

set

Support and balancing blocks

Principalblock

Support set

Balancing

set

Principalblock

Support set

Stacks with downward external

forces acting on them

Loaded stacks

Size =

number of blocks

+ sum of external

forces.

Principalblock

Support set

Stacks in which the support set contains

only one block at each level

Spinal stacks

Assumed to be optimal in:

J.F. Hall, Fun with stacking Blocks, American Journal of Physics 73(12), 1107-1116, 2005.

Loaded vs. standard stacks

1

1

Loaded stacks are slightly more powerful.

Conjecture: The difference is bounded by a constant.

Optimal spinal stacks

…

Optimality condition:

Spinal overhang

Let S (n) be the maximal overhang achievable using a spinal stack with n blocks.

Let S*(n) be the maximal overhang achievable using a loaded spinal stack on total weight n.

Theorem:

A factor of 2 improvement over harmonic stacks!

Conjecture:

Optimal 100-block spinal stack

Spine

Shield

Towers

Optimal weight 100 loaded spinal stack

Loaded spinal stack + shield

spinal stack + shield + towers

Are spinal stacks optimal?

No!

Support set is not spinal!

Overhang = 2.32014Blocks = 20

Tiny gap

Optimal 30-block stack

Overhang = 2.70909Blocks = 30

Optimal (?) weight 100 construction

Overhang = 4.2390Blocks = 49

Weight = 100

Brick-wall constructions

Brick-wall constructions

“Parabolic” constructions

6-stack

Number of blocks: Overhang:

Balanced!

Using n blocks we can get an overhang of (n1/3) !!!

An exponential improvement over the O(log n) overhang of

spinal stacks !!!

“Parabolic” constructions

6-slab

5-slab

4-slab

r-slab

r-slab

r-slab within a (r+1)-slab

“Vases”

Weight = 1151.76

Blocks = 1043

Overhang = 10

“Vases”

Weight = 115467.

Blocks = 112421

Overhang = 50

Forces within “vases”

Unloaded “vases”

“Oil lamps”

Weight = 1112.84

Blocks = 921

Overhang = 10

Forces within “oil lamps”

Brick-by-brick constructions

Is the (n1/3) the final answer?

Mike PatersonYuval Peres

Mikkel ThorupPeter Winkler

Uri Zwick

MaximumOverhangYes!

1

0 1 2 3-3 -2 -1

Splitting game Start with 1 at the origin

How many splits are needed to get, say, a quarter of the mass to

distance n?

At each step, split the mass in a given

position between the two adjacent

positions

Open problems

● What is the asymptotic shape of “vases”?● What is the asymptotic shape of “oil lamps”?● What is the gap between brick-wall stacks

and general stacks?● What is the gap between loaded stacks

and standard stacks?