[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84851-en":3,"doc-seo-84851-105":29,"detail-sidebar-cat-0-en-105":82},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84851,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Sudoku Grids That Require Many Clues","Motivated by worst-case time bounds for solving Sudoku, this work proves that most filled-in \u003Cm^2 by m^2> (i.e., \u003Cm×m blocks) Sudoku grids need nearly all information: all but a logarithmic fraction of cells must be provided as clues. For 9×9 and 16×16 variants, explicit constructions require 18 and 80 clues respectively. The study frames the minimum-clue problem, develops an algorithmic motivation via exact-cover counting, and proves bounds on how many grids can be specified by a limited clue budget.","Sudoku Grids That Require Many Clues  \nDavid Eppstein∗ Xinyu (Cindy) Zhang  \nComputer Science Department, University of California, Irvine  \nAbstract  \nMotivated by worst-case algorithmic time bounds for solving sudoku, we prove that a majority of filled-in 􀀼2 × 􀀼2 sudoku grids require all but a logarithmic fraction of cells to be filled by clues. For 9 × 9 and 16 × 16 sudoku, we construct grids that require 18 clues and 80 clues.  \n1 Introduction  \nIn sudoku puzzles, a 9 × 9 grid of square cells, partitioned into nine 3 × 3 blocks, is to be filled with the digits from 1 to 9 so each row, column, and block contains each digit. To make the solution unique, some cells are initially filled as clues. Often 30%–35% of cells are clues, but many fewer can be needed: the minimum number of clues in a uniquely-solvable puzzle is 17. Sudoku can be generalized to 􀀼2 × 􀀼2 grids partitioned into 􀀼2 blocks of size 􀀼×􀀼, and 16×16 sudoku is conjectured to require 56 clues [8] . For 􀀼2 × 􀀼2 sudoku the minimum number of clues is conjecturally Ω (􀀼4 ) ; only Ω (􀀼2 ) has been proven, but special clue sets that allow a solution using very restricted deduction rules require Ω (􀀼4 ) clues. Further, some sudoku puzzles can be uniquely specified using ⌊􀀼4 /4⌋ clues [7] .  \nWe study a variant of this minimum-clue problem. Instead of the fewest clues among all sudoku puzzles, we seek the fewest clues for a given filled-in grid, and a grid that maximizes this minimum number of clues. We prove that most 􀀼2 × 􀀼2 filled grids require 􀀼4 − 􀀤(􀀼4 /log 􀀼) clues, leaving only a logarithmic fraction of cells unfilled. We construct many 9 × 9 sudoku grids requiring 18 clues, matching a report of McGuire et al. [8] that 17 clues do not always suffice, and a 16 × 16 grid requiring 80 clues, well above the conjectured 56-clue minimum. Our results are motivated by the worst-case algorithmic complexity of sudoku, and by a result that an 􀀼2 ×􀀼2 sudoku puzzle with 􀀻 clues can be solved in time 􀀤(􀀼4 2􀀼4 −􀀻 ) . For this solution method, grids with more clues can be solved more quickly.  \nA related problem, adding the fewest clues to a partial set of clues to produce a unique solution, is hard for the second level of the polynomial hierarchy [2] . Our problem differs in seeking the fewest clues for the solution itself. It would be of interest to find its computational complexity; we do not.  \n2 Algorithmic motivation  \nPublished sudoku puzzles often can be solved by deduction rules of polynomial complexity [3], but sudoku is NP-hard and hence unlikely to have polynomial worst-case time [ 10] . Backtracking often works well but this has not been validated theo-  \n∗Research supported in part by NSF grant CCF-2212129 .  \nretically. To close the gap between theory and practice, we seek worst-case times that are exponential but as fast as possible. A brute force search that checks all ways of filling empty cells with digits would take time 􀀤(􀀼4 􀀼2 (􀀼2 −􀀻)) . Instead, we adapt an algorithm by Björklund, Husfeldt, and Koivisto that uses dynamic programming and inclusion–exclusion on the lattice of subsets of a given set to find exact covers [ 1] .  \nDefine a valid placement of a digit to be a set of cells that, together with the clues for that digit, include one cell in each row, column, and block of the puzzle. A sudoku solution partitions all cells into valid placements, one per digit. This is not quite an exact cover because each digit must contribute exactly one valid placement to the cover. Instead we use the sum weighted partitions problem defined by Björklund et al. [1] . It takes as input a family of 􀀹 functions 􀀵􀀷 from subsets of a given set to numbers; the output is the sum of products ∑􀀥 ∏ 􀀷􀀵􀀷 (􀀥 􀀷) where the sum is over partitions 􀀥 into 􀀹 sets 􀀥 􀀷 . For sudoku, let 􀀹 = 􀀼2 and let 􀀵􀀷 (􀀨) be one when 􀀨 is a valid placement of digit 􀀷 and zero otherwise; then the sum weighted partitions value is the number of puzzle solutions. (Conventionally this should equal one but we ","cbCaivnIbwzV1v02","https://ap.wps.com/l/cbCaivnIbwzV1v02","pdf",89850,1,2,"English","en",105,"# Introduction\n# Algorithmic motivation\n# Existence","[{\"question\":\"Why is the minimum-clue problem relevant to solving Sudoku computationally?\",\"answer\":\"The results are motivated by worst-case algorithmic complexity: grids with more clues can be solved faster, so understanding how many clues are necessary helps bound and interpret solving time in the hardest cases.\"}]",1784198808,5,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":77,"head_meta":79,"extra_data":81,"updated_unix":27},"sudoku-grids-that-require-many-clues","",{"@graph":35,"@context":76},[36,52,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,46,49],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":21},"https://docshare.wps.com/document/","Document",{"item":47,"name":12,"@type":42,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":42,"position":51},"https://docshare.wps.com/document/sudoku-grids-that-require-many-clues/84851/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70],{"name":71,"@type":72,"acceptedAnswer":73},"Why is the minimum-clue problem relevant to solving Sudoku computationally?","Question",{"text":74,"@type":75},"The results are motivated by worst-case algorithmic complexity: grids with more clues can be solved faster, so understanding how many clues are necessary helps bound and interpret solving time in the hardest cases.","Answer","https://schema.org",{"og:url":50,"og:type":78,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":80,"canonical":50},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":83},[84,88,92,96,100,105,110,113,118,121,125],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":111,"slug":112},30,"research-report",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},9,"Religion & Spirituality",20,"religion-spirituality",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":119,"show_sort_weight":116,"slug":120},"World Cup","world-cup",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":122,"slug":124},10,"Lifestyle","lifestyle",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":28,"slug":128},19,"General","general"]