Switchback Swagger II
Courtesy of a commenter at God Plays Dice we have this nice link about the fact that there are no solutions to the sum of squares problem for $n > 24$. This was proved in 1918 by G. N. Watson, in the paper The problem of the square pyramid. In fact, the only solutions are $n = 1$ and $n = 24$. The equation
$\frac{1}{6} n (n+1) (2n + 1) = k^{2}$
originally described a pile of cannonballs, built from a base layer of $k \times k$ balls into a square pyramid of height $n$. So it's really a sphere packing problem.
$\frac{1}{6} n (n+1) (2n + 1) = k^{2}$
originally described a pile of cannonballs, built from a base layer of $k \times k$ balls into a square pyramid of height $n$. So it's really a sphere packing problem.
posted by Kea | 8:26 AM
4 Comments:
Very interesting novelty maths, but your decision to reverse the definitions of n and k, relative to their definitions in the linked article http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html confused me a bit. Also, the statement that a solution is n = 1 i.e. one ball, is hardly much of a solution when the problem is about building pyramids or squares. Since when has a single sphere been a pyramid or a square? They might as well try to find solutions to that equation using zero or negative numbers or invoking imaginary pyramids on the complex plane. You have an square of n*n cannon balls laid out on the ground (with no piling). If n = 70, n*n = 4900.
Then you take those n*n or 4900 cannon balls and re-arrange them in a square-based pyramid which is k balls high, where k is defined by 1/6 k(1 + k)(1 + 2k) = n^2.
The only solution is k = 24, which is a pyramid 24 layers high formed from 4900 balls, which if arranged into a flat square would have side 70.
"... what is the smallest number of balls that can first be laid out on the ground as an n*n square, then piled into a square pyramid k balls high?" - http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html
The first part sounds as if it is merely and for no real purpose stipulating that the square root of the total number of balls must be an integer ("what is the smallest number of balls that can first be laid out on the ground as an n*n square"), and the second part is requiring adherence to the formula 1/6 k(1 + k)(1 + 2k) = n^2 ("then piled into a square pyramid k balls high").
The main difference between this "problem" and the ancient efforts to find the size of a square whose area equalled that of a circle of known radius, is that this "problem" has a solution...
Hi Nigel. I suspect that the original cannonball problem was a genuine military problem involving actual cannonballs! I am trying to track down the Watson paper at the library here, because it is so much fun, and I don't view this as mere novelty. In the context of M theory, it is amazing to see that the pyramidal numbers are related to the 6 sides of a hexagon.
Sorry Kea, I only meant "novelty" in the cannonball stacking context which was something I could attempt to grasp. In the context of far more abstract maths, it's clearly not something my limited mind can understand without years of study.
I hope you great weather and a fun time receiving your PhD later this week in New Zealand. The 21st December is the shortest day of the year here (it gets dark now at about 3pm), so it must be the longest day in your hemisphere.
Hi Nigel. I finally fixed the typos in the post. Sorry for swapping ns and ks. Our summer solstice (sun reaching its southernmost point) will be at 07:35am local time on Saturday, December 22, which is the longest day of the year.
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