Golden Hexagon of Patagonia (10 Centavos Argentina)
in PatagoniaMath Frames
Encouraging:
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Golden Hexagon of Patagonia (10 Centavos Argentina) |
Suitability: 14 + years
Equipment:
pencil paper / compasses / squares
/ calculator / computer / PatagoniaMath frames
Theorem
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.
y
2 =3x2 - 3x + 1The 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 similar in this way. They are as follows:
| number | (side of square) | (side of 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 9001325016847969 - 46399815451081 ie 8954925201396888.
Background and Derivation
The above equation was derived from first principles whilst playing with coins in Patagonia in 2000. The numbers greater than 169 were obtained with the help of Excel on a desktop computer . On 2000 June 28 these first seven numbers were submitted to NJA (Neil) Sloane for possible inclusion in his renowned Encyclopedia of Sequences hosted on the ATT server. An email was also sent to Steven Finch of Mathsoft asking if he had any knowledge of the series being published previously. As a result of the ensuing correspondence, in which it was learnt that Martin Gardner had referred to this series of numbers in his book "Time Travel and other Mathematical Bewilderments (Freeman NY 1988), the series has now been expanded and several probably hitherto unpublished numbers have been posted on the internet (2000-07-04).
By 0800 hrs the following day (2000-07-05) a photocopy of Chapter Two of Gardnerīs book had been obtained from the library of Brunel University, London. The chapter entitled "Hexes and Stars" was found to be beautifully written and illustrated.
The book should be essential reading for all interested in mathematical patterns. The mathematics underlying the series is all there, together with the above equation. It is gratifying that playing with coins can lead to similar conclusions and diagrams. The contibutions of Martin Gardner and Neil and Steven and Brunel were much appreciated and it was was intended to post more information here once the copyright had been checked with the publishers. These and related "magical" properties of 168 - 169 and the associated numbers of the "Golden Hexagon of Patagonia" Square- Hex Series are being used in the development of various games and learning aids.see www.research.att.com/~njas/sequences (A 001570, A001922, A006051)
www.mathsoft.com
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