Christian Eggermont Radboud University Nijmegen Metagrobologist and Mathematician Webdesign and Internet specialist Puzz

1. Christian Eggermont Radboud University Nijmegen Metagrobologist and Mathematician Webdesign and Internet specialist Puzzledesigner

2. Magic Squares Multimagic Squares Numbers 1, 2, …, n2(order n>1) Rows have same sum S Columns have same sum S m a g i c Diagonals have same sum S Constructionmethods for all orders > 2 Lo Shu ± 2200 B.C.

3. p-Multimagic Squares M is a p-multimagic square if for each 1 ≤ i ≤ p the matrix obtained by raising each element of M to the i-th power is a magic square. m u l t i Numbers 1, 2, … , n2 Rows have same sumColumns have same sum Diagonals have same sum 2-multimagic squareH.E.Dudeney(<1917) m a g i c

4. p-Multimagic Squares Properties Some ‘simple’ facts The Questions How to make them? }=> p-multimagic order n p-multimagic order m p-multimagic order n*m How many are there (for a given p and order n)? • There are no 3-multimagic squares of order 4*k+2 If you drop the condition of different numbers then diagonal Latin Squares are ‘∞-multimagic squares’:

5. 2-Multimagic Squares 1890: Order 8 Pfefffermann (Ray?) Order 9 Pfeffermann 2004: Order 10 Jansson Order 11 Jansson

6. 3-Multimagic Squares 1905: Order 128 Tarry 1933: Order 64 Cazalas 1949: Order 32 Benson 2002: Order 12 Trump

7. 4-,5-,6-Multimagic Squares 1984?: 4-multimagic Order 256 Charles Devimeux? 2001: 4-multimagic Order 512 Boyer & Viricel 5-multimagic Order 1024 Boyer & Viricel 2003: 4-multimagic Order 256 Boyer 4-multimagic Order 256 Gao Zhiyuan & Wu Shuoxin 5-multimagic Order 729 Li Wen 6-multimagic Order 4096 Pan Fengchu

8. p-Multimagic Squares of order n

9. Result There is a p-multimagic square for all p Harm Derksen, Christian Eggermont, Arno van den Essen To appear More information (later) on http://www.puzzled.nl Explicit p-multimagic squares of order qp where q is the smallest prime ≥ 2*p-1 e.g. the first 7-multimagic of order 137 1cm for each number then the square would cover the benelux