Talk:Fujimura's problem

Let [math]\displaystyle{ \overline{\overline{c}}^\mu_n }[/math] be the largest subset of the tetrahedral grid:

[math]\displaystyle{ \{ (a,b,c,d) \in {\Bbb Z}_+^4: a+b+c+d=n \} }[/math]

which contains no tetrahedrons [math]\displaystyle{ (a+r,b,c,d), (a,b+r,c,d), (a,b,c+r,d), (a,b,c,d+r) }[/math] with [math]\displaystyle{ r \gt 0 }[/math]; call such sets tetrahedron-free.

These are the currently known values of the sequence:

n	0	1	2
[math]\displaystyle{ \overline{\overline{c}}^\mu_n }[/math]	1	3	7

n=0

[math]\displaystyle{ \overline{\overline{c}}^\mu_0 = 1 }[/math]:

There are no trapezoids, so no removals are needed.

n=1

[math]\displaystyle{ \overline{\overline{c}}^\mu_1 = 3 }[/math]:

Removing any one point on the grid will leave the set tetrahedron-free.

n=2

[math]\displaystyle{ \overline{\overline{c}}^\mu_2 = 7 }[/math]:

Suppose the set can be tetrahedron-free in two removals. One of 2000, 0200, 0020, and 0002 must be removed. Removing any one of the four leaves three tetrahedrons to remove. However, no point coincides with all three tetrahedrons, therefore there must be more than two removals.

Three removals (for example 0002, 1100 and 0020) leaves the set tetrahedron-free with a set size of 7.

General n

For a given n the tetrahedral lattice has [math]\displaystyle{ \frac{1}{24}n(n+1)(n+2)(n+3) }[/math] tetrahedrons.