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from María Victoria Canullo, Jorge Dignani, Claudia Didoné y Robin Willson (Sociedad Magic Penny Patagonia)
for children everywhere, and especially those in Provincial School 168, Puerto Madryn and in Provincial School, 15 Paso de los Indios, Patagonia , Argentina; and in Wylam First School, Northumbria, UK.
2000 June 21       LEONARDO HOME           en ESPAŅOL           update 2000-07-05

Patagonian Golden Hexagon Magic Penny Packing Theorem

Escuela 168 Puerto Madryn

For any objects of circular cross section, the same number can be arranged in a close packed regular hexagon of side x objects, or in a square grid of side y objects, provided the following equation is obeyed, and x and y are integers.

y2 =3x2 - 3x + 1

The smallest number of objects that can be packed in both of these ways is 169.  
Of all the numbers from 1 to 1000000000000000000 (1018) there are only 7 that are magical in this way. They are as follows:

number                         (side of square, hexagon)
169                               (13, 8)
32761                           (181, 105)
6355441                       (2521, 1458)
1232922769                 (35113, 20273)
239180661721             (489061, 282360)
46399815451081         (6811741, 3932761)
9001325016847969     (894875313, 54776288)

The same numbers of objects, less a lower number in the series or 1, can be arranged in regular hexagons or squares with regular hexagonal, or square holes respectively, in their middle: for example, 168 or 8954925201396888.

These and related magical properties of 168 - 169 and the associated numbers of the Golden Hexagon of Patagonia series are being used in the development of various games and learning aids. For these the Magic Penny Trust holds the copyright. -more

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