Mike Paterson
Uri Zwick
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).
No friction
Length parallel to table
“Real-life” 3D version
Idealized 2D version
The classical solution
Using n blocks we can
get an overhang of
Harmonic Stacks
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
F4
F3
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
F5
Force equation
Moment equation
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
F2
F1
F3
F4
F5
F11
F7
F8
F9
F10
F13
F14
F15
F16
F6
F12
Equivalent to the feasibility
of a set of linear inequalities:
F17
F18
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
Blocks = 4
Overhang = 1.16789
Blocks = 5
Overhang = 1.30455
Blocks = 6
Overhang = 1.4367
Blocks = 7
Overhang = 1.53005
Small optimal stacks
Blocks = 16
Overhang = 2.14384
Blocks = 17
Overhang = 2.1909
Blocks = 18
Overhang = 2.23457
Blocks = 19
Overhang = 2.27713
Support and balancing blocks
Balancing
set
Principal
block
Support
set
Support and balancing blocks
Balancing
set
Principal
block
Support
set
Loaded stacks
Stacks with
downward external
forces acting on them
Size
=
Principal
block
number of
blocks
+
sum of external
forces.
Support
set
Spinal stacks
Stacks in which the
support set contains
only one block
at each level
Principal
block
Support
set
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:
Conjecture:
A factor of 2 improvement
over harmonic stacks!
Optimal 100-block spinal stack
Towers
Shield
Spine
Optimal weight 100
loaded spinal stack
Loaded spinal stack + shield
spinal stack + shield + towers
Are spinal stacks optimal?
No!
Support set
is not spinal!
Blocks = 20
Overhang = 2.32014
Tiny gap
Optimal 30-block stack
Blocks = 30
Overhang = 2.70909
Optimal (?) weight 100 construction
Weight = 100
Blocks = 49
Overhang = 4.2390
Brick-wall constructions
Brick-wall constructions
“Parabolic” constructions
6-stack
Number of blocks:
Balanced!
Overhang:
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
Yes!
1/3
(n )
the final answer?
Mike Paterson
Yuval Peres
Mikkel Thorup
Peter Winkler
Uri Zwick
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
1
-3
-2
-1
0
1
2
3
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?