A Logistic Transition Model for Perceived Difficulty in Sequential Problem Solving

A Logistic Transition Model for Perceived Difficulty in Sequential Problem Solving | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Logistic Transition Model for Perceived Difficulty in Sequential Problem Solving Nnaemeka Kingsley Ugwumba, Peter Sunday Jaja This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9237812/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper introduces a mathematical framework for modeling the temporal evolution of perceived difficulty in sequential problem-solving processes. The study is motivated by the observation that sufficiently complex tasks exhibit a characteristic progression in which difficulty is initially high, remains elevated for a period, and then decreases rapidly following a critical transition point. To formalize this behavior, a continuous model based on logistic decay is proposed to represent perceived difficulty as a function of time or progress. The model defines a critical transition parameter that captures the onset of rapid reduction in difficulty, corresponding to a breakthrough phase in problem solving. A complementary function is derived to represent perceived ease, demonstrating a symmetric but inverse relationship with difficulty. The analytical properties of the model are examined, including boundary behavior, monotonicity, and convergence. The proposed framework provides a generalized theoretical structure that can be applied across domains where problem solving processes exhibit nonlinear progression. The model contributes to the mathematical understanding of cognitive dynamics and offers a foundation for further analytical and applied investigations. Applied Mathematics Perceived difficulty logistic model nonlinear dynamics problem solving theory mathematical modeling transition point analysis sigmoid functions cognitive modeling convergence behavior theoretical mathematics Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9237812","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":612885958,"identity":"9595495b-b9db-4512-9936-7b6de2ee06a4","order_by":0,"name":"Nnaemeka Kingsley Ugwumba","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0000-2493-9846","institution":"Laskenta Technologies Limited","correspondingAuthor":true,"prefix":"","firstName":"Nnaemeka","middleName":"Kingsley","lastName":"Ugwumba","suffix":""},{"id":612885959,"identity":"621f9df6-ff09-45d7-aa83-5f312adc1d4d","order_by":1,"name":"Peter Sunday Jaja","email":"data:image/png;base64,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","orcid":"","institution":"Laskenta Technologies Limited","correspondingAuthor":true,"prefix":"","firstName":"Peter","middleName":"Sunday","lastName":"Jaja","suffix":""}],"badges":[],"createdAt":"2026-03-26 20:13:58","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9237812/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9237812/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105703387,"identity":"b56c858c-8a3c-4192-bb80-9aaf4916906b","added_by":"auto","created_at":"2026-03-30 06:37:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":420818,"visible":true,"origin":"","legend":"","description":"","filename":"PSEP.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9237812/v1_covered_052b5207-4df9-4e3b-975c-cf801177630f.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eA Logistic Transition Model for Perceived Difficulty in Sequential Problem Solving\u003c/strong\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Laskenta Technologies Limited","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Perceived difficulty, logistic model, nonlinear dynamics, problem solving theory, mathematical modeling, transition point analysis, sigmoid functions, cognitive modeling, convergence behavior, theoretical mathematics","lastPublishedDoi":"10.21203/rs.3.rs-9237812/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9237812/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper introduces a mathematical framework for modeling the temporal evolution of perceived difficulty in sequential problem-solving processes. The study is motivated by the observation that sufficiently complex tasks exhibit a characteristic progression in which difficulty is initially high, remains elevated for a period, and then decreases rapidly following a critical transition point. To formalize this behavior, a continuous model based on logistic decay is proposed to represent perceived difficulty as a function of time or progress. The model defines a critical transition parameter that captures the onset of rapid reduction in difficulty, corresponding to a breakthrough phase in problem solving. A complementary function is derived to represent perceived ease, demonstrating a symmetric but inverse relationship with difficulty. The analytical properties of the model are examined, including boundary behavior, monotonicity, and convergence. The proposed framework provides a generalized theoretical structure that can be applied across domains where problem solving processes exhibit nonlinear progression. The model contributes to the mathematical understanding of cognitive dynamics and offers a foundation for further analytical and applied investigations.\u003c/p\u003e","manuscriptTitle":"A Logistic Transition Model for Perceived Difficulty in Sequential Problem Solving","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-30 06:37:00","doi":"10.21203/rs.3.rs-9237812/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"2ebaabe4-3c99-4688-8619-ac5d173b08b4","owner":[],"postedDate":"March 30th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":65217559,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2026-03-30T06:37:00+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-30 06:37:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9237812","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9237812","identity":"rs-9237812","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}