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Poisson Games and Sudden-Death Overtime

Representing a Number as the Sum of Three Squares using a Probabilistic Shotgun

See this post on MathOverflowEfficient Computation of Integer Representation as Sum of Three Squares.

One example:

88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888883 =                                                                                                              
3728708309692149319069486105836547569635831800611344866231809626605743446543140741338906572157111906908174220385938813888215444725007055559361034118288392106711441267490894902067351394374510189603371757744263341018713166151526298683346214125342893805012291639475722227078443761154317206380555332240855575850923487133276807555304260120129179410975007599951176226386425345496311175560711401472026399392161325224552285910886215686341134733538354919800609823980774575909850703888338271526201573341101297625937194645663741887567408937166745004054837902992947738719498311260341062479838525863315246124587329025994737831896962572251605303542008917816688105857525263899973501824564669305499^2 + 8659423954866836497222055442609569479198212006663142432779025107564215236268212089964838333998727328589210056339682895508921922562067554420365202477992005631619765951799790747911029105975651521423183103854427982379701565857188948798827049644230457335425566175730313559096014470591244755124818823620481973835142141854050641587503177488728535676941288815996010222969186688888101284376165512662389139600159011876940028243578093146223369745603888487795213374055272652379453140354950015534128847565124896267173708689765986573722289960008854542751323743735684696670546547009812267118488250157564430050257057333477696216049998331821788940995913966994169767925303505778930077956125196943941^2 + 799^2


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Standard notation:

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