This document summarizes 12 unsolved problems in geometry. Some of the problems discussed include determining the largest number of cubes a cube can be cut into (the Hadwiger problem), whether every polygon can be fully illuminated by a light source within it, packing spheres together to minimize volume (the penny packing problem), and finding aperiodic tiles that can tile the plane without repeating. Many of the problems have some partial solutions or bounds proven but remain open overall.

1. A Dozen Unsolved Problems
in Geometry
Erich Friedman
Stetson University
9/17/03

2. 1. The Hadwiger problem
• In d-dimensions, define L(d) to be the
largest integer n for which a cube can not
be cut into n cubes. What is L(d)?
• L(2)=5, as shown below.

3. 1. The Hadwiger problem
• D. Hickerson
showed L(3) = 47.
• The last partition to be
found was the division
into 54 smaller cubes,
as shown to the right.
• Partitions into 49 and 51
cubes are also challenging.

4. 1. The Hadwiger problem
• L(4)≤853 and L(5)≤1890.
• The best bound known (due to Erdös) is L(d)<(e-1)
(2d)d.
• Smart $: L(d) is probably o(dd).

5. 2. The Polygonal
Illumination Problem
• Given a polygon S constructed with
mirrors as sides, and given a point P in
the interior of S,
we can ask
if the inside of S
will be completely
illuminated by a
light source at P?

6. 2. The Polygonal
Illumination Problem
• It is conjectured that for every S and P, the
answer is yes.
• No counterexample is known, but no one
has a proof.
• Even this easier problem is open: Does
every polygon S have any point P where a
light source would illuminate the interior?

7. 2. The Polygonal
Illumination Problem
• For non-polygonal regions, the conjecture
is false, as shown by the example below.
• The top and bottom
are elliptical arcs
with foci shown,
connected with
some circular arcs.

8. 2. The Polygonal
Illumination Problem
• There are continuously differentiable
regions where an arbitrarily large number
of light sources are necessary.
• To get a region requiring an infinite
number of light sources, you need one
non-differentiable point (J. Rauch).
• Smart $: The conjecture is true.

9. 3. The Penny Packing Problem
• How can n non-overlapping d-dimensional
spheres be arranged to minimize the
volume of their convex hull?
• (The convex hull is the set of all points on
a line segment between points in two
different spheres.)

10. 3. The Penny Packing Problem
• In 2 dimensions, the answers are clusters,
or “hexagonal as possible”.

11. 3. The Penny Packing Problem
• In 3 dimensions, the answers for n≤56 are
sausages, with the centers in a straight line.
• For d=3 and n≥57, the answers are clusters.
• For d=4, the answers are sausages for n up to
somewhere between 50,000 and 100,000!

12. 3. The Penny Packing Problem
• The Sausage Conjecture: (F. Tóth)
In dimensions 5 and higher, the optimal
configuration is always a sausage.
• U. Betke, M. Henk, and J. Wills proved
the sausage conjecture for d≥42 in 1998.
• Smart $: The conjecture is true.

13. 4. The Chromatic Number
of the Plane
• What is the smallest number of colors χ
with which we can color the plane so
that no two points of the same color are
distance 1 apart?
• This is just the chromatic number of the
graph whose vertices are in the plane
and two vertices are connected if they
are unit distance from each other.

14. 4. The Chromatic Number
of the Plane
• The chromatic
number of this
unit distance
graph (which
is called the
Moser spindle)
is 4, so χ≥4.

15. 4. The Chromatic Number
of the Plane
• The plane can
be colored
with 7 colors
to avoid unit
pairs having
the same
color, so χ≤7.

16. 4. The Chromatic Number
of the Plane
• If the sets of points of a given color have
to be measurable, χ≥5.
• If the sets have to be closed, χ≥6.
• Smart $: χ=7.

17. 5. Kissing Numbers
• In d dimensions, the kissing number K(d)
is the maximum number of disjoint unit
spheres that can touch a given sphere.
• K(2)= 6
• K(3)=12.

18. 5. Kissing Numbers
• J. Conway and N. Sloane proved
K(5)=40, K(6)=72, and K(7)=126 in 1992.
• A. Odlyzko and N. Sloane proved
K(8)=240, and K(24)=196,560 in 1979.
• All other dimensions are still unsolved.
• Smart $: K(9)=306.

19. 6. Perfect Cuboids
• A perfect cuboid is a rectangular box
whose sides, face diagonals, and space
diagonals are all integers.

20. 6. Perfect Cuboids
• It is not known whether a perfect cuboid exists.
• Several near misses are known:
a=240 b=117 c=44 dab=267 dac=244 dbc=125
a=672 b=153 c=104 dac=680 dbc=185 dabc=697
a = 18720 b=√211773121 c = 7800
dab=23711 dac=20280 dbc=16511 dabc=24961

21. 6. Perfect Cuboids
• If there is a perfect cuboid, it has been
shown that the smallest side must be at
least 232 = 4,294,967,296.
• Smart $: There is no perfect cuboid.

22. 7. Cutting Rectangles into
Congruent Non-Rectangular Parts
• For which values of n is it possible to cut a
rectangle into n equal non-rectangular parts?
• Using triangles, we can do this for all even n.

23. 7. Cutting Rectangles into
Congruent Non-Rectangular Parts
• This is harder to do for odd n.
• Here are solutions for n=11 and n=15.

24. 7. Cutting Rectangles into
Congruent Non-Rectangular Parts
• Trivially, there is no solution for n=1.
• Solutions are known for all other n except
n=3, 5, 7, and 9, which remain open.
• What is true in higher dimensions?
• Smart $: There are no solutions for these n.

25. 8. Overlapping Congruent Shapes
• Let A and B be congruent overlapping
rectangles with perimeters AP and BP .
• What are the best possible bounds for
length(A∩BP )
R = ------------------ ?
length(AP ∩B)

26. 8. Overlapping Congruent Shapes
• It is fairly easy to prove 1/4 ≤ R ≤ 4.
• It is conjectured that 1/3 ≤ R ≤ 3.
• Same ratio defined for triangles?
• It is conjectured that the best bounds for a
triangle with smallest angle θ are
sin(θ/2) ≤ R∆ ≤ csc(θ/2).

27. 8. Overlapping Congruent Shapes
• In d dimensions, is the best upper bound
on the ratio of (d-1)-dimensional surface
area equal to 2d-1?
• Of course, for circles, RΟ = 1.
• Smart $: 1/3 ≤ R ≤ 3.

28. 9. Distances Between Points
• If we have n points in the plane, they
determine 1+2+3+…+(n-1) distances.
• Can we arrange n points in general
position so that one distance occurs once,
one distance occurs twice, … and one
distance occurs n-1 times?
• (General position means no 3 points on a

29. 9. Distances Between Points
• This is easy to accomplish for small n.
• An example for n=4 is shown below.

30. 9. Distances Between Points
• Solutions are
only known
for n≤8.
• A solution (by
I. Pilásti) for
n=8 is shown
to the right.

31. 9. Distances Between Points
• Is there a solution for n=9?
• Is there a solution for all integers n?
• Erdös offered $500 for a proof of “yes” and
$50 for a proof of “no”.
• Very little has been done on the same
problem in higher dimensions.
• Smart $: There is a solution for n=9, but not
for large n.

32. 10. The Kabon Triangle Problem
• How many disjoint triangles can be
created with n lines in the plane?
• The sequence K(n) starts
0, 0, 1, 2, 5, 7, 11, 15, 21, .…
• The optimal arrangements for n≤9 are
shown on the next slide.

33. 10. The Kabon Triangle Problem
• How many disjoint triangles can be
created with n lines in the plane?

34. 10. The Kabon Triangle Problem
• What is K(10)?
• How fast does K(n) grow?
• S. Tamura proved that K(n) ≤ n(n-2)/3.
• Smart $: This bound can be improved.

35. 11. Aperiodic Tiles
• A tiling of the plane is called periodic if
it can be translated onto itself with two
non-parallel translations.

36. 11. Aperiodic Tiles
• A set of tiles is called aperiodic if they
tile the plane, but not in a periodic way.
• Even though a
square can tile the
plane in a non-
periodic way, it is
not aperiodic.

37. 11. Aperiodic Tiles
• In 1966, Berger produced the first set of
20,426 aperiodic tiles, which he soon
lowered to 104 tiles.
• In 1968, D. Knuth discovered 92 tiles.
• Shortly thereafter, R. Robinson reduced
this to 35 tiles, R. Penrose found a set of
34 tiles, and R. Ammann lowered to 16

38. 11. Aperiodic Tiles
• In 1971, R. Robinson found this set of 6
aperiodic tiles based on notched squares.

39. 11. Aperiodic Tiles
• In 1974, R. Penrose found this set of 2 colored
aperiodic tiles, now called Penrose Tiles.

40. 11. Aperiodic Tiles
• The coloring can be dispensed with if we
notch these pieces.

41. 11. Aperiodic Tiles
• This is part of a tiling using Penrose Tiles.

42. 11. Aperiodic Tiles
• Is there a single tile which is aperiodic?
• There is a set of 3 convex (meaning no
notches) aperiodic tiles. Are there 2? 1?
• In 3 dimensions, R. Ammann has found 2
aperiodic polyhedra, and L. Danzer has
found 4 aperiodic tetrahedra.

43. 12. Heesch’s Problem
• The Heesch number of a planar shape is
the number of times it can be completely
surrounded by copies of itself.
• For example, the
shape to the right has
Heesch number 1.
• What’s the largest
finite Heesch number?

44. 12. Heesch’s Problem
• A hexagon
with two
external
notches and
3 internal
notches has
Heesch
number 4!

45. 12. Heesch’s Problem
• The
highest
known
Heesch
number
is 5.
• Smart $:
There are
higher ones.

46. References
• V. Klee, Some Unsolved Problems in Plane
Geometry, Math Mag. 52 (1979) 131-145.
• H. Croft, K. Falconer, and R. Guy, Unsolved
Problems in Geometry, Springer Verlag, New
York, 1991.
• Eric Weisstein’s World of Mathematics,
http://mathworld.wolfram.com/.
• The Geometry Junkyard,
http://www.ics.uci.edu/~eppstein/junkyard/.