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Magic squares and water reten0on
In a magic square, all rows, columns and diagonals add to the same value, and all numbers 1, 2, …, L2 occur just once.
Magic squares have been studied since at least 650 BC in China. (The above magic square is the Lo Shu Square), and con0nue to be objects of intense study in recrea0onal mathema0cs today.
Chris Knecht proposed studying water reten0on on magic squares, and it became a problem in an interna0onal computa0on compe00on, with Walter Trump as one of the winners.
Magic Square work drove the discovery of water reten0on algorithms. This photo by Walter Trump shows the order 5 magic square that retains the most water.
Magic Square work drove the discovery of water reten0on algorithms. This photo by Walter Trump shows the order 5 magic square that retains the most water.
• hNp://www.knechtmagicsquare.paulscompu0ng.com/
Zimmermann’s contest in March 2010 was to find the maximum water reten8on for magic squares orders 4 – 28. The contest demonstrates that magic squares of any order can be produced and manipulated to explore their physical proper8es. hEp://www.knechtmagicsquare.paulscompu8ng.com/
Water reten0on on Ben Franklin’s magic square
In a Franklin magic square, diagonals do not add to the value of the columns, but all bent diagonals (black sites here) add to the value of the columns!
Albrecht Dürer’s Magic Square
Dürer 1471 – 1528
• hNp://kk.haum-‐bs.de/?id=a-‐duerer-‐wb3-‐0122
It has been said that the first scien0fic paper wriNen in the new world was a paper on magic squares by Ben Franklin. He was bored out of his mind siang in the commiNees wri0ng out the US cons0tu0on and doodled out three magic squares that are named acer him to this day. • hNp://www.magic-‐squares.net/square-‐update.htm
First US postage stamp!
Dürer’s Melancholia I
1514
17
17
Centrally symmetric pairs add to 17 (“associa0ve magic square”)
17
17
34 34
34 34
In addi0on, here all corner squares add to 34
34
34
Corners of 2 x 4 rectangles
34 34
34 34
Corners of 3x3 squares:
34 34
Also:
34 34
Also:
1514 ”D” ”A”
Finally,
Interchanging two rows gives four addi0onal cruciform sums… “We want to call out to the angel, ‘Don’t worry! Be happy! Interchange the second and third rows!’ ” hNp://www.nr.com/whp/notes/DurerSquare.pdf
2012
Water reten0on on Dürer magic square:
Order
semi-magic (A)
normal (B)
associative (C)
pandiagonal (D)
ultramagic (E)
3 9 1 1 0 0 4 68 688 880 48 48 0 5 579 043 051 200 275 305 224 48 544 3 600 16
6 9.4597 (13) !1022 1.775399 (42) !1019 0 0 0
7 4.2848 (17) !1038 3.79809 (50) !1034 1.125151 (51) !1018
1.21 (12) !1017 20 190 684
8 1.0806 (12) !1059 5.2225 (18) !1054 2.5228 (14) !1027 C8 + ? 4.677 (17) !1015
9 2.9008 (22) !1084 7.8448 (38) !1079 7.28 (15) !1040 81!E9 + ? 1.363 (21) !1024
10 1.4626 (16) !10115 2.4149 (12) !10110 0 0 0
Variants of a square by means of rotations and reflections are not counted. Statistical notation: 1.2345 (25) !109 means that the number is not known precisely but is in the interval (1.2345 ±0.0025) !109 with a probability of 99%. Ultramagic squares are associative (centrally symmetrical) and pandiagonal. !
Source: Walter Trump, hNp://www.trump.de/magic-‐squares/howmany.html
How many magic square are there? Results of historical and computer enumera0ons and es0mates.
Water-‐filled magic square created for the 500th anniversary of Dürer’s Melancholia
Durer’s birth/death: 1471-‐1528
Blue = sites retaining water.
Total reten0on = 2014 units
This is a 196x196 magic square made by convolving the smaller 14x14 magic squares, showing the water reten0on! Harry White, Sept. 5, 2012
Reten0on of first 14 x 14 square
• Diagrams created by Harry White and Craig Knecht.
• A constraint-‐based solver for water reten0on was also wriNen by Johan Öfverstedt, Uppsala University (Bachelor’s thesis under Pierre Flener).
• hNp://www.ted.com/profiles/963652
