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Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\/{\\sqrt{2}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:msqrt>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msqrt>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also consider a similar problem about 3-regular unit distance graphs in\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00454-023-00503-2","type":"journal-article","created":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T13:02:19Z","timestamp":1689339739000},"page":"764-782","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Some Non-Rigid Unit Distance Patterns"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4939-4835","authenticated-orcid":false,"given":"N\u00f3ra","family":"Frankl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dora","family":"Woodruff","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,14]]},"reference":[{"issue":"2","key":"503_CR1","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s00454-002-0727-x","volume":"28","author":"PK Agarwal","year":"2002","unstructured":"Agarwal, P.K., Sharir, M.: The number of congruent simplices in a point set. 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