Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research

{"FINDING:":[0],"The":[1],"2011":[2],"IMO":[3],"windmill":[4,77],"problem":[5],"reduces":[6],"to":[7,98,166],"a":[8,19,41,56,79,84,143],"parity-invariant":[9],"on":[10,15,37,46,60,111,154],"the":[11,23,26,34,38,47,61,95,101,107,125,148,163,182],"number":[12,108],"of":[13,18,29,40,63,109,114,181],"points":[14,69,110,153,170],"each":[16,112,122,155],"side":[17,113,156],"rotating":[20,48],"line,":[21],"with":[22],"invariant":[24,133],"being":[25],"total":[27],"count":[28],"\\"switches\\"":[30],"modulo":[31,134],"2,":[32],"and":[33,88,100,129,171],"solution":[35],"hinges":[36],"existence":[39],"point":[42,86,93],"that":[43,147],"is":[44,53,132,139,159],"always":[45,150],"line":[49,80,102,149,165],"(the":[50],"pivot),":[51],"which":[52],"guaranteed":[54],"by":[55,117,161],"combinatorial":[57],"pigeonhole":[58],"argument":[59],"sequence":[62],"pivots.":[64],"|":[65],"MATH:":[66],"Let":[67],"\\\\(n\\\\)":[68,138],"in":[70],"general":[71],"position":[72],"(no":[73],"three":[74],"collinear).":[75],"A":[76],"process:":[78],"\\\\(L\\\\)":[81,90,115],"rotates":[82],"about":[83],"pivot":[85,96,144],"\\\\(p_i\\\\),":[87],"when":[89],"hits":[91],"another":[92],"\\\\(p_j\\\\),":[94,99],"switches":[97],"continues":[103],"rotating.":[104],"Key":[105],"invariant:":[106],"changes":[116],"exactly":[118],"\\\\(\\\\pm":[119],"1\\\\)":[120],"at":[121],"switch,":[123],"but":[124],"*difference*":[126],"between":[127],"left":[128],"right":[130],"counts":[131],"2.":[135],"Specifically,":[136],"if":[137],"odd,":[140],"there":[141],"exists":[142],"\\\\(p\\\\)":[145],"such":[146],"has":[151],"\\\\((n-1)/2\\\\)":[152],"—":[157],"this":[158],"achieved":[160],"choosing":[162],"initial":[164],"pass":[167],"through":[168],"two":[169],"ro":[172],"Author:":[173],"Andrew":[174],"Stewart":[175],"Caldin,":[176],"Independent":[177],"Researcher,":[178],"UK.":[179],"Part":[180],"E8":[183],"Intelligence":[184],"Research":[185],"series.":[186],"Platform:":[187],"e8intelligence.com":[188]}

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-19
DOI
https://doi.org/10.5281/zenodo.22841560
Primary Topic
Computability, Logic, AI Algorithms
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
preprint

Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms
preprint

Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The 2011 IMO windmill problem reduces to a parity-invariant on the number of points on each side of a rotating line, with the invariant being the total count of "switches" modulo 2, and the solution hinges on the existence of a point that is always on the rotating line (the pivot), which is guaranteed by a combinatorial pigeonhole argument on the sequence of pivots. | MATH: Let \(n\) points in general position (no three collinear). A windmill process: a line \(L\) rotates about a pivot point \(p_i\), and when \(L\) hits another point \(p_j\), the pivot switches to \(p_j\), and the line continues rotating. Key invariant: the number of points on each side of \(L\) changes by exactly \(\pm 1\) at each switch, but the *difference* between left and right counts is invariant modulo 2. Specifically, if \(n\) is odd, there exists a pivot \(p\) such that the line always has \((n-1)/2\) points on each side — this is achieved by choosing the initial line to pass through two points and ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Affordable and clean energy
Computability, Logic, AI Algorithms
DOIOpen Access PDFOpenAlex
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

Parity Invariant and Pivot Existence in the IMO Windmill Problem — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS