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First Written: May 22, 2018 ; This version: Aug. 5, 2018.
[Appeared in Mathematical Intelligencer v. 41 issue 1 (March 2019), pp. 16.]
One of the crowning achievements of human ingenuity is Lars Onsager's 1944 exact solution of
the celebrated twodimensional Ising model in zero magnetic field. This proof is notoriously complicated
and adhoc, as are all the subsequent `simpler' proofs. Here we present a new derivation that is embarrassingly simple,
using (very basic) symbolic computation, and `experimental mathematics' (aka guessing). While nonrigorous
it is absolutely certain, and should be totally acceptable to physicists who do not share
mathematicians' fanatical (and often misplaced) insistence on rigor.
Added Aug. 30, 2018: Read a
fascinating message from Tony Guttmann.
Maple Package
Lars.txt (version of May 2, 2018),
Doing Onsager by pure guessing
[One of the procedures, vSeriesCh(K), requires the data file:
Ons48PC.txt
]
C program for fast computation of the trace of a power of a matrix
Sample Input and Output for Maple package Lars.txt

If you want to see an ab initio derivation of Onsager's formula, using experimental mathematics,
the input leads to the
output.

If you want to see the vSeries up to 11 terms, Directly in terms of the definition (but by "cheating", using the precomputed weightenumerators
for the n by n torodial rectangle for n=24)
input leads to the
output.
The Ising Polnomials
In the following list of length 16, the ith entry is the number of socalled lattice polygons
with 2i sides, that can be placed in a torodial rectangle with N lattice points sidelenghts are both larger than 2i+2.
L:=[ 0,N,2*N,(N*(9 + N))/2,2*N*(6 + N),(N*(7 + N)*(32 + N))/6,N*(130 + 21*N + N^2),N*(11766 + 1715*N + 102*N^2 + N^3)/24,
N*(5876 + 776*N + 49*N^2 + N^3)/3,(N*(980904 + 118830*N + 7415*N^2 + 210*N^3 + N^4))/120,
N*(423624 + 47666*N + 2855*N^2 + 94*N^3 + N^4)/12,N*(112852800 + 11919274*N + 678945*N^2 + 23725*N^3 + 375*N^4 + N^5)/720,
N*(42723120 + 4272044*N + 231260*N^2 + 8175*N^3 + 160*N^4 + N^5)/60,
N*(16620978240 + 1584498216*N + 81728374*N^2 + 2851695*N^3 + 62545*N^4+ 609*N^5 + N^6)/5040,
N*(5589930384 + 510961484*N + 25204804*N^2 + 857825*N^3 + 19891*N^4 +251*N^5 + N^6)/360,
N*(2990184306480 + 263323487916*N + 12469823436*N^2 + 411847009*N^3 +9754920*N^4 + 143794*N^5 + 924*N^6 + N^7)/40320
]:
Acknowledgement: We were greatly helped by