Base 53: Maximum Iteration Sequences: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ

Color Key
OrangeFixed Pointnumber that becomes itself after one iteration
GreenFinal Numbersall numbers eventually reached, from all cycles
PurpleCycle Startiterations include entry into cycle, assuming it repeats
RedFull Cycleiterations include entire cycle, proving it repeats
BlueStart Numberssubset (optimization) of all numbers; minimal for full coverage
GrayNo Resultsany calculation that has no results

Digits Max Steps Max Step Example
Full Cycle
12 max steps100
212 max steps10 → 0q → p1 → n3j7bFLV9hXJDdPR1pn3
329 max steps100 → 0qq → pq1 → oq2 → nq3 → mq4 → lq5 → kq6 → jq7 → iq8 → hq9 → gqA → fqB → eqC → dqD → cqE → bqF → aqG → ZqH → YqI → XqJ → WqK → VqL → UqM → TqN → SqO → RqP → QqQPqRQqQ
430 max stepsR000 → QqqQ → QPQR → 1qqp → p0p2pm32ni74jaF8bKVGLAfWVAfMV8hMZ8hIZGZIJGZYJEbYNEbUN6jUd6jEdOREP2nSl2n6leB6fSNCT4lOh4lAhWJAXCdKRCdQR0pQp0p2
527 max stepsP0000 → OqqqS → SNqSP → T2qnO → oOqR3 → nPqQ4 → mNqS5 → lOqR6 → kMqT7 → jNqS8 → iLqU9 → hMqTA → gKqVB → fLqUC → eJqWD → dKqVE → cIqXF → bJqWG → aHqYH → ZIqXI → YGqZJ → aFqaH → bIqXG → aIqXHZHqYIZGqZIaGqZHaIqXH
6206 max stepsK000nn → nnJW34 → kjCd77 → dcQPEE → PO0pSS → p42nm2 → nlh854 → jgYHA8 → bWGZKG → LJBeXW → TEAfcO → VO4lSM → h93mhA → jYWJI8 → bGCdaG → RLJWVQ → DA0pge → pWQPK2 → nC0pe4 → pjRO72 → nc2nE4 → ljNS76 → fc4lEC → hTNSNA → X64lkK → heCdCA → XSQPOK → D40pme → piQP82 → na0pG4 → pjJW72 → ncCdE4 → jRNSP8 → b51olG → ngKVA4 → jWAfK8 → bVBeLG → TL9gVO → XA4lgK → hWCdKA → XRBePK → TD1odO → nQ4lQ4 → jgqq98 → ih2n99 → lZXIH6 → fIEbYC → TNFaTO → L64lkW → heAfCA → XVROLK → DA2nge → lWQPK6 → fC0peC → pTRON2 → n62nk4 → ljdC76 → fcQPEC → TO0pSO → p53ml2 → njfA74 → jcULE8 → bO8hSG → ZL3mVI → jH9gZ8 → bXHYJG → LHDcZW → PIAfYS → VG2naM → lK8hW6 → fZBeHC → USHYON → H73mja → jcIXE8 → bOEbSG → NL3mVU → jA6jg8 → dbVKFE → PMAfUS → V82niM → la8hG6 → fZJWHC → TICdYO → RG4laQ → hK0pWA → pXBeJ2 → nTDcN4 → jP5kR8 → fb1oFC → nTLUN4 → j95kh8 → fbXIFC → TMEbUO → N84liU → ha6jGA → dXJWJE → PECdcS → RO2nSQ → l40pm6 → pieB82 → naSNG4 → jK4lW8 → hbBeFA → XTLUNK → D95khe → fYQPIC → TG0paO → pK4lW2 → nhBe94 → jYSNI8 → bG4laG → hLJWVA → XD9gdK → XQCdQK → RCqqdQ → eQBeQD → TQqqPO → SR2nPP → l31on6 → nkeB64 → jeSNC8 → bS4lOG → hL3mVA → jX9gJ8 → bXDcJG → PLDcVS → PA2ngS → lW2nK6 → lfBeB6 → fUSNMC → T84liO → ha4lGA → hXJWJA → XECdcK → ROCdSQ → R40pmQ → pi0p82 → pnZG32 → nkIX64 → jeEbC8 → bSMTOG → L73mjW → jcAfE8 → bVNSLG → LA4lgW → hWAfKA → XVBeLK → TD9gdO → XQ4lQK → hCqqdA → ge3mCB → jVROL8 → bA2ngG → lWKVK6 → fCAfeC → VTRONM → 962nki → leYHC6 → fSGZOC → TJ3mXO → jE4lc8 → hbNSFA → XM4lUK → hD7idA → bXPQJG → LE0pcW → pOAfS2 → nV3mL4 → ki9g87 → daWJGE → PKCdWS → RC2neQ → lS0pO6 → pf3mB2 → njTM74 → jc6jE8 → dbNSFE → PM4lUS → h82niA → laWJG6 → fKCdWC → TRBePO → T51olO → ng4lA4 → jhVK98 → bYAfIG → VLFaVMLA8hgWZWAfKIVHBeZMTI8hYOZG4laIhKGZWAXJBeXKTECdcORO4lSQh40pmApiWJ82naCdG4jRJWP8bD1odGnQKVQ4jAqqf8ig3mA9jZVKH8bIAfYGVLFaVM
737 max steps200000Q → Q1qqqoR → pQOqRQ2 → oR0qpP3 → qmOqR41 → pmNqS42 → olOqR53 → njMqT74 → mgMqTA5 → lcJqWE6 → kXIqXJ7 → jSDqcO8 → iWDqcK9 → hVHqYLA → gQCqdQB → fUCqdMC → eTGqZND → dOBqeSE → fQEqbQC → eRAqfPD → gSEqbOB → fSCqdOC → eTEqbND → dQDqcQE → dPBqeRE → fQDqcQC → eSBqeOD → fRFqaPC → eQAqfQD → gSDqcOB → fTDqcNC → eSEqbOD → dQCqdQE → ePCqdRD → eRDqcPD → dRCqdPEePDqcRDdRCqdPE
8189 max steps50000Ioo → ooI4lY23 → mlhFa955 → hgYKVIAA → XWGAfaKK → VKDBedWM → TQC8heQO → ZS4qqlOI → mYN6jTI5 → hdG5kaDA → fXQJWQJC → TECqqdcO → ecF8hbED → ZROLUSPI → H941omha → niYIXI84 → jaGEbaG8 → bNKJWWTG → LDC5kedW → fSQAfQOC → VT3qqmNM → nUP1oRM4 → nj81oi74 → njcZGE74 → jcOIXSE8 → bOF3mbSG → jML3mVU8 → jbA7igF8 → caWLUKGF → NKC8heWU → ZSC6jeOI → dSH3mZOE → jPI3mYR8 → jbG1oaF8 → nbMJWUF4 → jMD7idU8 → caQ7iQGF → bNJqqWTG → aXE2ncJH → lOJDcXS6 → fPE3mcRC → jTO1oSN8 → nb63mkF4 → kieLUC87 → daS8hOGE → ZPK3mWRI → jHC1oeZ8 → nbSHYOF4 → jMH3mZU8 → jbI7iYF8 → caMFaUGF → NLK7iWVU → bCA6jgeG → dWSKVOKE → PCB3mfeS → jUS2nOM8 → lb83miF6 → jfaLUGB8 → bUK8hWMG → ZLC7ieVI → bSH9gZOG → XLI3mYVK → jGD9gda8 → bXQJWQJG → LECqqdcW → ecI5kYED → fROFaSPC → TL41omVO → niA4lg84 → jhaVKG98 → bYKAfWIG → VLGBeaVM → TKA8hgWO → ZWC4leKI → hSHBeZOA → XTI3mYNK → jGD5kda8 → fbQJWQFC → TMCqqdUO → eUF0pbMD → pRM7iUP2 → nb81oiF4 → njaLUG74 → jcK8hWE8 → bZOBeSHG → TLI3mYVO → jGA4lga8 → hbWJWKFA → XMDBedUK → TQD7idQO → bQ4qqlQG → maLAfVG5 → hVK9gWLA → YWC9geKJ → XSFBebOK → TMD3mdUO → jQ84liQ8 → hbZqqGFA → gbR1oPFB → nVM1oUL4 → njA7ig74 → jcbVKFE8 → bOMAfUSG → VL83miVM → jaA8hgG8 → bZWJWKHG → LIDBedYW → TQGAfaQO → VK4qqlWM → mWP0pRK5 → phC1oe92 → omYROI43 → liG2na86 → lfaJWGB6 → fUKCdWMC → TRC7iePO → bS51olOG → ngL3mVA4 → kiW9gK87 → daXBeJGE → TPKDcWRO → PC51oleS → ngS2nOA4 → ljW3mK76 → jfcBeEB8 → bUTNSNMG → L864lkiW → heaAfGCA → XVSJWOLK → ECA3mged → jWSOROK8 → bC42nmeG → liSKVO86 → faB3mfGC → jUTJWNM8 → bD85kidG → faQKVQGC → TKAqqfWO → gWH2nZKB → lVIBeYL6 → fTG9gaNC → XTK5kWNK → fDC5kedC → fTSPQONC → T640pmkO → pie4lC82 → nhaROG94 → jYK2nWI8 → lbGBeaF6 → fTMJWUNC → TD85kidO → faQ4lQGC → hTJqqWNA → gXK2nWJB → lVEBecL6 → fTO9gSNC → XT63mkNK → jeD5kdC8 → fbSPQOFCTM40pmUOpi84li82nhaZGG94jYKIXWI8bGFBebaGTMLJWVUODA84ligehaWQPKGAXKC0peWKpSDBedO2nTQ3mQN4ki5qqk87ljc1oE76nfcNSEB4jUO4lSM8hb83miFAjaXLUJG8bKE8hcWGZOLBeVSITHA3mgZOjWI4lYK8hbGBeaFAXTMJWUNKEC85kiedfaSOROGCTK42nmWOliC4le86hfaROGBAXUK2nWMKlDC7ied6fbSPQOFC