# Combinatorics and Graph Theory

## 1906 Submissions

[3] **viXra:1906.0144 [pdf]**
*replaced on 2019-06-10 17:12:23*

### The Case N_1=n_2>n_3 of the N_1 X N_2 X N_3 Dots Puzzle: Improved Upper Bound

**Authors:** Valerio Bencini

**Comments:** 12 Pages.

In this paper, I show an improved upper bound for the case n_1=n_2>n_3 of the n_1 X n_2 X n_3 Dots Puzzle, and I extend all the upper bounds I found to the k-dimensional case, with k>=4.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1906.0110 [pdf]**
*submitted on 2019-06-07 13:21:19*

### n_1 X n_2 X n_3 Dots_Puzzle: A Method to Improve the Current Upper Bound

**Authors:** Valerio Bencini

**Comments:** 8 Pages.

The aim of this paper is to lower down the current upper bound for the Ripà's n_1 X n_2 X n_3 Dots Problem, with n_1>n_2>=n_3, using the same method Ripà and I used for the case n_1=n_2=n_3:=n. The new value is now 1/2*floor(1/(3*n_1-3*n_2-3*n_3+5))*((n_1-n_2-n_3)*(n_1-n_2-n_3+1)-2*floor(1/2*(-n_1+n_2+n_3)))+2*n_2*n_3-1.
At the end of the article, I also extend this result, and that I previously found with Ripà for n_1=n_2=n_3=:n, to the k-dimensional problem n_1 X n_2 X n_3 X n_4 X ... X n_k, using the equation found by Ripà in 2013.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1906.0031 [pdf]**
*submitted on 2019-06-03 12:10:14*

### Die Würfelschlange

**Authors:** Henning Thielemann

**Comments:** 5 Pages. Language: German

The dice sequence
The dice sequence is an adaption of Kruskal's card trick to dice.
We compute precise and approximated probabilities that the trick works.
The adaption to dice simplifies the problem considerably
because the probabilities of the dice rolls are independent.
Initially I wrote the text for Wikipedia but in order to meet Wikipedia's exclusion of original research I wrote up this paper.

**Category:** Combinatorics and Graph Theory