[math-fun] A powerful Zero Torque Square
Consider one of the 880 order 4 magic squares. .6, 12, .9, .7, 15, .1, .4, 14, .3, 13, 16, .2, 10, .8, .5, 11 If these weights are placed on the odd-odd vertices from (-3,-3) to (3,3), then this will balance perfectly on the center point (0,0). Square each number, and it will still remain perfectly balanced. There are 48 4x4 magic squares with this property. .36, 144, .81, .49, 225, ..1, .16, 196, ..9, 169, 256, ..4, 100, .64, .25, 121 Cube each number, and this magic square is still perfectly balanced. And the same when the numbers are raised to the 4th powers. (But not the 5th or 6th powers) This is the only magic square that remains balanced at powers 3 and 4. .216, 1728, .729, .343, 3375, ...1, ..64, 2744, ..27, 2197, 4096, ...8, 1000, .512, .125, 1331 --Ed Pegg Jr
Very pretty! (What do the repeated decimal points mean?) —Dan
On Sep 15, 2016, at 9:53 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
Consider one of the 880 order 4 magic squares.
.6, 12, .9, .7, 15, .1, .4, 14, .3, 13, 16, .2, 10, .8, .5, 11
If these weights are placed on the odd-odd vertices from (-3,-3) to (3,3), then this will balance perfectly on the center point (0,0). Square each number, and it will still remain perfectly balanced. There are 48 4x4 magic squares with this property.
.36, 144, .81, .49, 225, ..1, .16, 196, ..9, 169, 256, ..4, 100, .64, .25, 121
Cube each number, and this magic square is still perfectly balanced. And the same when the numbers are raised to the 4th powers. (But not the 5th or 6th powers) This is the only magic square that remains balanced at powers 3 and 4.
.216, 1728, .729, .343, 3375, ...1, ..64, 2744, ..27, 2197, 4096, ...8, 1000, .512, .125, 1331
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Excellent! @Dan : each decimal point means space or zero. Useful for a correct display of columns, using fixed fonts. -----Message d'origine----- De : math-fun [mailto:math-fun-bounces@mailman.xmission.com] De la part de Dan Asimov Envoyé : jeudi 15 septembre 2016 19:24 À : math-fun <math-fun@mailman.xmission.com> Objet : Re: [math-fun] A powerful Zero Torque Square Very pretty! (What do the repeated decimal points mean?) —Dan
On Sep 15, 2016, at 9:53 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
Consider one of the 880 order 4 magic squares.
.6, 12, .9, .7, 15, .1, .4, 14, .3, 13, 16, .2, 10, .8, .5, 11
If these weights are placed on the odd-odd vertices from (-3,-3) to (3,3), then this will balance perfectly on the center point (0,0). Square each number, and it will still remain perfectly balanced. There are 48 4x4 magic squares with this property.
.36, 144, .81, .49, 225, ..1, .16, 196, ..9, 169, 256, ..4, 100, .64, .25, 121
Cube each number, and this magic square is still perfectly balanced. And the same when the numbers are raised to the 4th powers. (But not the 5th or 6th powers) This is the only magic square that remains balanced at powers 3 and 4.
.216, 1728, .729, .343, 3375, ...1, ..64, 2744, ..27, 2197, 4096, ...8, 1000, .512, .125, 1331
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Christian Boyer -
Dan Asimov -
Ed Pegg Jr