Absolute-difference triangles:
order 5:
Fill the numbers from 1 to 15 (= the 5th triangular number)
in a triangular table with side length 5 such that each entry
is the absolute difference between the two above neighbours.
order 6:
Fill the numbers from 1 to 21 (= the 6th triangular number)
in a triangular table with side length 6 such that each entry
is the absolute difference between the two above neighbours.
Show that this task is impossible.
Solution (order 5):
unique solution (up to mirroring)
6 14 15 3 13
8 1 12 10
7 11 2
4 9
5
Solution (order 6):
This turns out to be a parity problem.
Addition and subtraction are the same modulo 2.
Therefore (modulo 2) we can examine the addition triangle instead.
a1 a2 a3 a4 a5 a6
-- -- -- -- --
-- -- -- --
-- -- --
-- --
--
Starting with six numbers a1 to a6, what the sum of all entries
in this triangle?
sum = 6 a1 + 10 a2 + 12 a3 + 12 a4 + 10 a5 + 6 a6
As all factors are even the sum is even.
If there were an absolute-difference triangle with numbers 1 to 21
we would have an modulo 2 addition triangle with 10 even and 11 odd
numbers. But there total sum is odd. Which is impossible.
References:
Rainer Bodendiek, Gustav Burosch;
Streifz"uge durch die Kombinatorik,
Aufgaben und L"osungen aus dem Schatz der Mathematik-Olympiaden,
Spektrum Akademischer Verlag, Heidelberg, 1995,
ISBN 3-86025-393-X
Kapitel: Aufgaben zu Invarianten, Aufgabe 5.19
There is no absolute-difference triangles of order 6.
Martin Gardner;
MG13SA: Penrose Tiles to Trapdoor Chiphers ... and the return of Dr. Matrix
MG13SA: Freeman (1989) New York
MG13SA.9.1 Pool-Ball Tiangles
MG13SA.9.1. absolute-difference triangles of consecutive numbers must have 1
MG13SA.9.1. as its lowest number (C. Trigg). Only order 1..5 are possible.
MG13SA.9.1. No. of solutions: 1..5; 1, 2, 4, 4, 1.
MG13SA.9.1. A triangular array (even) has always an even-odd sum pattern with
MG13SA.9.1. an equal number of even and odd ones (H. Harborth).
MG13SA.9.1. modulo-m-sum triangle of 0..m-1 (order 4 ok, order 5, 6 no)