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)