A New Rose : The First Simple Symmetric 11-Venn Diagram
6 ᖺ D ⤌ ከ㈡┾ᣦᑟᩍဨ ᕝཱྀ ៅ஧㸯㸬せ⣙ ⚾ࡣከᩘࡢ㞟ྜ࠿ࡽᡂࡿVennᅗ࡟ࡘ࠸࡚◊✲ࡋ࡚࠸ࡿࠋ௒ᅇࡣKhalegh Mamakani ࡜Frank Ruskey ࡟ࡼࡿ”A New Rose : The First Simple Symmetric 11-Venn Diagram ([1])”ࢆࡶ࡜࡟11ಶࡢ㞟ྜࡢVennᅗ࡟ࡘ࠸࡚⪃ᐹࡋࡓࠋ࣮࣮࢟࣡ࢻ  n -Vennᅗࠊࢡࣟࢫ࢝ࢵࢺࠊࢡࣛࢫࢱࠊࢩ࣮ࢣࣥࢫ
㸰㸬◊✲ࡢ⫼ᬒ࡜┠ⓗMore than Four Classesenn Diagrams for⚾ࡣ4ᖺ⏕ࡢ࡜ࡁ࠿ࡽከᩘࡢ㞟ྜࡢVennᅗࡢ୍⯡໬࡟㛵ࡋ࡚◊✲ࡋࠊࡇࢀࡲ࡛ࡢ◊More than Four Classes ([2])”࡛ド᫂ࡉࢀ࡚࠸ࡿࠋᅇ㌿ᑐ⛠ n -Vennᅗ࡟ᑐࡋ࡚ࠊn ಶࡢ᭤✲࡛ࠊฝᆺࡢ᭤⥺࡛ᵓᡂࡉࢀࡓVennᅗࡣ5ಶࡢ㞟ྜࡢࡶࡢࡲ࡛⪃᱌ࡍࡿࡇ࡜ࡀ࡛ࡁࡓࠋ⥺ࢆ᭱ࡶእഃࡢ㡿ᇦ࡟᥋ࡍࡿ㒊ศ࡟㛵ࡋࡋ࠿ࡋ6ಶ௨ୖࡢ㞟ྜࡢሙྜࠊᅗࡀ」㞧࡟࡞࡚᫬ィᅇࡾ࡟ C  C    Cn ࡜ྡ௜ࡅࡿࠋࡾVennᅗࢆ⮬సࡍࡿࡇ࡜ࡀ࡛ࡁ࡞࠿ࡗࡓࡢ࡛ࠊ௒ᅇࡣඛ⾜◊✲࡛࠶ࡿ”A New Rose :㸱㸫㸰㸬ࢡࣟࢫ࢝ࢵࢺ

The First Simple Symmetric 11-Venn

㸱㸫㸰㸫㸯㸬ᐃ⩏Diagram ([1])”ࢆㅮㄞࡋ⪃ᐹࡍࡿࡇ࡜࡟ࡋ‫ࢺࢵ࢝ࢫࣟࢡڦ‬ࡓࠋ ࠶ࡿ᭤⥺࡟࠾࠸࡚ࠊ཯᚟ࡍࡿࡇ࡜࡞ࡃ࡯࠿ࡢ᭤⥺࡜㐃⥆ⓗ࡟஺ᕪࡍࡿ᭤⥺ࡢษ∦ࡢ㸱㸬◊✲ෆᐜࡇ࡜ࢆࢡࢡࣟࢫ࢝ࢵࢺ࡜ᐃ⩏ࡍࡿࠋ ౛࠼ࡤᅇ㸱㸫㸯㸬ᇶᮏ஦㡯㌿ᑐ⛠7-Vennᅗ࡛ࡣᅗ㸯ࡢ㯤Ⰽ࡛ሬࡽࢀ

‫ ڦ‬n -Vennᅗ࣭ᅇ㌿ᑐ⛠ n -Vennᅗࡓ᭤⥺ࡀࢡࣟࢫ࢝ࢵࢺ࡛࠶ࡿࠋ  ᖹ㠃ୖ࡟ n ಶࡢ㛢ࡌࡓ᭤⥺ࢆ㞟ࡵࡓࡶࡢ

ࢆ n -Vennᅗ࡜ࡼࡪࠋࡇࡢ࠺ࡕࠊྛ஺Ⅼ࡛2



S
ᅇ
n



㌿ࡉࡏࡓ࡜ࡁ࡟ࡶ࡜ࡢVennᅗ࡟୍⮴ࡍࡿ

Vennᅗࢆᅇᅇ㌿ᑐ⛠ n -Vennᅗ࡜ࡼࡪࠋ ᅇ㌿ᑐ⛠ n -Venn ᅗࡣ n ࡀ⣲ᩘ࡛࡞࠸࡜

ᥥࡅ࡞࠸ࡇ࡜ࡣ1963 ᖺ࡟David W.

ᮏࡢ᭤⥺ࡢࡳࡀ஺ᕪࡋ࡚࠸࡚ࠊࡲࡓ     Hendersonࡢㄽᩥ ”Venn Diagrams for                    ᅗ㸯 㸱㸫㸰㸫㸰㸬⿵㢟ɂ⩌ Vennᅗࡢ᭤⥺ࡢษ∦࡟ࡼࡗ࡚ᅖࡲࢀࡓ⿵㢟㸯㡿ᇦࡢ㞟ࡲࡾࡢࡇ࡜ࢆ⩌⩌࡜࠸࠺ࠋࡇࡢ࡜ࡁࠊ

 n !  ࡢሙྜࠊᅇ㌿ᑐ⛠ n -Venn ᅗࡣ㸯⩌ࡢ኱ࡁࡉࡣࠊࡑࢀ࡟ྵࡲࢀࡿ㡿ᇦࡢᩘ࡛ࡘࡢ᭤⥺࠶ࡓࡾ᭱኱1ࡘࡢࢡࣟࢫ࢝ࢵࢺ

⾲ࡍࠋࢆࡶࡘࠋ‫ࢱࢫࣛࢡڦ‬㸦ド᫂㸧 ᭱ࡶእഃࡢ㡿ᇦ࡜᭱ࡶෆഃࡢ㡿ᇦࢆ㝖࠸ Vennᅗࢆᵓᡂࡍࡿ௵ពࡢ᭤⥺࡟࠾࠸࡚ࠊࢡࣟࢫ࢝ࢵࢺࡢጞⅬࢆ᭱ࡶእഃࡢ㡿ᇦ࡟᥋࡚ᅇ㌿ᑐ⛠ n -Vennᅗࡀ኱ࡁࡉ

n  
ࡢnಶnࡍࡿ㒊ศࠊ⤊Ⅼࡣ᭱ࡶෆഃࡢ㡿ᇦ࡟᥋ࡍࡿ

ࡢ⩌࡟ศࡅࡽࢀࡿࠋࡇࡢ࡜ࡁࠊࡇࡢ⩌ࢆࠕࢡࢡ㒊ศ࡜⪃࠼ࡿࠋࡇࡢ࡜ࡁࠊ௵ពࡢ᭤⥺ࡣ᭱ࣛࢫࢱࠖ࡜ࡼࡪࡇ࡜࡟ࡍࡿࠋࢡࣛࢫࢱࡣᅇࡶእഃࡢ㡿ᇦ࡜᭱ࡶෆഃࡢ㡿ᇦ࡟1ᅇࡎࡘ㌿ࡢᇶᮏ㡿ᇦ࡟࡞ࡿࠋ㸦ᅗ㸰ࡢⰍ௜ࡅࡋࡓ

ゐࢀࡿࡢ࡛ࠊVennᅗࡢ᭤⥺ࡣ3ࡘ௨ୖࡢࢡ

㒊ศࡣ7-Vennᅗࡢࢡࣛࢫࢱࢆ♧ࡋࠊᅗ㸱࡛ࠊࣟࢫ࢝ࢵࢺࢆࡶࡘࡇ࡜ࡣ࡛ࡁ࡞࠸ࠋ

ࢡࣛࢫࢱࢆ෌ᥥ෗ࡋ࡚࠸ࡿࠋᅗ㸱࡛ࢡࣛࢫࡑࡇ࡛ n -Vennᅗࡢ࠶ࡿ᭤⥺ C ࡟2ࡘࡢࢱࡣࢡࣟࢫ࢝ࢵࢺᑐ⛠࡟࡞ࡗ࡚࠸ࡿࡇ࡜

ࢡࣟࢫ࢝ࢵࢺࡀ࠶ࡿ࡜௬ᐃࡍࡿࠋ᭤⥺ C ࡢ2ࡀࢃ࠿ࡿࠋ㸧ᮏࡢࢡࣟࢫ࢝ࢵࢺࡣ᭱ࡶእഃࡢ㡿ᇦ࡟᥋ࡍࡿ㒊ศ࠿ࡽጞࡲࡾࠊ᭱ࡶෆഃࡢ㡿ᇦ࡟᥋ࡍࡿ㒊ศ࡛⤊ࢃࡿࡢ࡛᭤⥺ C ࡣ࡯࠿ࡢ᭤⥺࡜ n   ಶࡢ஺ⅬࢆࡶࡘࠋࡇࡢVennᅗ࡟ࡣ

n ಶࡢ᭤⥺ࡀ࠶ࡿࡢ࡛ࠊVennᅗࡀࡶࡘ஺Ⅼࡣ඲㒊࡛ n n   ಶࡢ஺Ⅼࢆࡶࡘࠋ͐ձ
୍᪉࡛ࠊVennᅗࡀᅇ㌿ᑐ⛠࡞ࡽࡤࠊࡇࡢnVennᅗࡣ඲㒊࡛ࠊ    ಶ͐ղࡢ஺Ⅼࢆࡶᅗ㸰ࡘࡣࡎ࡛࠶ࡿࠋ
ձ࡜ղࡢᘧࡼࡾࠊ

n n   n  ࢆゎࡃ࡜ࠊ n    ࡟ࡘ࠸࡚ࡣᡂ❧ࡍࡿࡀࠊnӍ ࡛ࡣᡂࡾ❧ࡓ࡞࠸ࠋࡺ࠼࡟ࠊn ! ࡢ࡜ࡁࠊ n -Vennᅗࡀ2ࡘࡢࢡࣟࢫ࢝ࢵࢺࢆ

ᅗ㸱ࡶࡘ࡜࠸࠺௬ᐃࡣ㛫㐪ࡗ࡚࠸ࡓࠋ ࡼࡗ࡚㢟ពࡣ♧ࡉࢀࡓࠋ (Q. E. D.)ࡲࡓࠊ᭤⥺ Ck ࡢࢡࣟࢫ࢝ࢵࢺࢆࡶࡘࢡࣛ
㸱㸫㸱㸬ࢡࣛࢫࢱ

ࢫࢱࢆࢡࣛࢫࢱ S k ࡜ࡼࡪࡇ࡜࡟ࡍࡿࠋ㸱㸫㸱㸫㸯㸬ᐃ⩏ 㸱㸫㸲㸬ࢩ࣮ࢣࣥࢫࡢ࠺ࡕࠊ㸱㸫㸲㸫㸯㸬ᐃ⩏

Li  k ࢆࢡࣛࢫࢱ S k ࡟࠾࠸࡚ࠊ᭤⥺ Ci ࢆ᫬n  ࡢせ⣲ࡔࡅ࡛ࠊVennᅗ඲యnࢆ⾲ࡍࡇ࡜ࡀ࡛ࡁࡿࠋ᭱ึࡢ኱ࡁࡉ

n  
ࡢnィᅇࡾ࡟ࡓ࡝ࡗ࡚࠸ࡗࡓ࡜ࡁ࡟㐼㐝ࡍࡿ᭤ࢩ࣮ࢣࣥࢫࢆᇶᇶᮏࢩ࣮ࢣࣥࢫ࡜ࡼࡪࠋ⥺ࡢࣜࢫࢺ࡜ࡍࡿࠋࢡࣛࢫࢱࡣࢡࣟࢫ࢝ࢵ  ౛࠼ࡤࠊᅇ㌿ᑐ⛠7-Vennᅗ࡟࠾࠸࡚ࠊ

ࢺ࡛ᑐ⛠࡟࡞ࡿࡢ࡛ࠊ nӍ ࡟࠾࠸࡚ࣜࢫࢺn  nࡶᑐ⛠࡟࡞ࡗ࡚࠸ࡿࠋ ࡼࡾࠊᇶᮏࢩ࣮ࢣࣥࢫࡢ኱ࡁࡉࡣ18࡜࡞ࡿࠋᅗ㸰ࡢᅇ㌿ᑐ⛠7-Vennᅗࡢᇶᮏ⾲㸯 7-Vennᅗࡢࣜࢫࢺ

ࢩ࣮ࢣࣥࢫࡣࠊ௨ୗࡢࡼ࠺࡟࡞ࡿࠋ㸱㸫㸳㸬ࢡࣟࢫ࢝ࢵࢺᑐ⛠ࡢᐃ⌮ ᐃ⌮㸯ᅇ㌿ᑐ⛠n-Vennᅗ࡟࠾࠸࡚ࠊᇶᮏࢩ࣮ࢣ
ࣥࢫࡀࢩ࣮ࢣࣥࢫࠕȨȘțȘȚ㸩࡛ࠖ⾲ɂࢩ࣮ࢣࣥࢫ᭱ࡶෆഃࡢ㡿ᇦࡢ࠶ࡿⅬX࠿ࡽࠊ᭱ࡶእഃࡏࡿሙྜࡢࡳࠊࢡࣟࢫ࢝ࢵࢺᑐ⛠࡛࠶ࡿ࡜ࡢ㡿ᇦ࡟ྥ࠿ࡗ࡚┤⥺LࢆࡦࡁࠊXࢆ୰ᚰ࡟࠸࠼ࡿࠋࡓࡔࡋࠊȨȘțȘȚ㸩ࡣ௨ୗࡢ┤⥺ࢆ᫬ィᅇࡾ࡟ᅇ㌿ࡉࡏࡓ࡜ࡁ࡟஺ᕪࡍࡼ࠺࡟ᐃ⩏ࡍࡿࠋࡿVennᅗࡢ஺Ⅼࡢ␒ྕࢆ㡰࡟୪࡭ࡓࡶࡢ࣭Ȩ㸦ࢡࣛࢫࢱᕥቃ⏺ࡢ஺Ⅼ㸧ࢆ       n   n   ࢆࢩࢩ࣮ࢣࣥࢫ࡜࠸࠺ࠋࡓࡔࡋࠊ஺Ⅼࡢ␒ྕࡣ௨ୗࡢἲ๎࡟ᚑࡗ࡚ࡘࡅࡿࠋࢩࢩ࣮ࢣࣥࢫ࣭ț㸦ࢡࣟࢫ࢝ࢵࢺୖࡢ஺Ⅼ㸧ࢆࡢ኱ࡁࡉ࡜ࡣࠊࢩ࣮ࢣࣥࢫ࡟ྵࡲࢀࡿせ⣲n   n       ࡢಶᩘࢆ⾲ࡍࠋ࣭Ș㸦Ȩࡼࡾྑഃ࡛ࢡࣟࢫ࢝ࢵࢺࡲ࡛ࡢ஺



Ⅼ㸧࡜ȘȚ㸩㸦ࢡࣟࢫ࢝ࢵࢺࡼࡾྑഃ࡛ࢡ

㸺஺Ⅼ␒ྕࡢἲ๎㸼ࣛࢫࢱࡢྑቃ⏺ࡼࡾᕥഃ㸧ࢆࠊȘȚ㸩ࡣȘ࣭᭱ࡶእഃࡢ㡿ᇦ࡟᥋ࡍࡿ஺Ⅼń1

ࢆ㏫㌿ࡉࡏ࡚1㊊ࡋ࡚ᚓࡽࢀࡿࡼ࠺࡞኱࣭እഃ࠿ࡽ2␒┠ࡢ㡿ᇦ࡟᥋ࡍࡿ஺Ⅼń2ࡁࡉ࣭እഃ࠿ࡽ n   ␒┠ࡢ㡿ᇦ㸦ෆഃ࠿ࡽ2␒┠㸧࡟᥋ࡍࡿ஺Ⅼń n   ^n `  n  nࡢࢩ࣮ࢣࣥࢫ࡜ࡋ࡚ᐃ⩏ࡍࡿࠋࡇࡢ࡜ࡁࠊȘȚ㸩> i @ Ș>_Ș_ i  @࡜࠿ࡅᅇ㌿ᑐ⛠ࡢVennᅗࡣ඲యࡢࢩ࣮ࢣࣥࢫ

ࡿࠋ 㸦ド᫂㸧࡟ྑ࠿ࡽࢡࣟࢫ࢝ࢵࢺ㸦ᅗ㸲࡛ࡣ C ࡟ྥ Vࢆࠊࢡࣟࢫ࢝ࢵࢺࢆ᭷ࡍࡿᅇ㌿ᑐ⛠࠿ࡗ࡚ᅇ㌿ࡉࡏ࡚࠸ࡃࠋᅇ㌿㛤ጞ┤ᚋࡣࠊVennᅗ࡛࠶ࡿ࡜ࡋࠊ᭤⥺ C ࡢษ∦ࡀࢡࣟࢫ࢝ࢵࢺࢆᙧᡂࡍࡿVࡢࢡࣛࢫࢱࢆ⪃࠼ࡿࠋ

M >Cn   Cn  Cn   Cn      C  C  C @ N >C  Cn   Cn  Cn     C  C  C @ ࢡࣟࢫ࢝ࢵࢺᑐ⛠ࡼࡾࠊ Ln  >Cn    Cn  @࡜࡞ࡾࠊ n   ಶࡢ஺Ⅼ࡜஺ᕪࡋࡓ┤ᚋࡣࠊ࡛࠶ࡿ࠿ࡽࠊ Cn ࡣ C ࡜ࡢ஺Ⅼࡢࡍࡄୗࡢ
ࢡࣛࢫࢱࡢྑ➃ࡢ࠶ࡿⅬ v ࡛ Cn  ࡜஺ᕪࡋM >Cn  Cn    Cn   Cn      C  C  C @ N >C  Cn  Cn    Cn     C  C  C @ ࡞ࡅࢀࡤ࠸ࡅ࡞࠸ࠋVࡀᅇ㌿ᑐ⛠࡛࠶ࡿ࠿ࡽࠊ࡜࡞ࡿࠋVࡀᅇ㌿ᑐ⛠࡛࠶ࡿࡢ࡛ࠊNࡣᚲࡎSᅇ㌿ࡉࡏࡿࡇ࡜࡛ࠊᕥቃ⏺ୖࡢnⅬ s ࢆ♧ࡍࡇ࡜ࡀ࡛ࡁࡿࠋⅬ s ࡛ࡣࠊCn  ࡜Mࡢせ⣲ࢆ1ࡘྑ࡟ࡎࡽࡋࡓࡶࡢ࡛࠶ࡿࠋࡘCn   ࡀ஺ᕪࡍࡿࠋࡢ㛫࡛㉳ࡇࡿࠋࡼࡗ࡚ࠊM( i )࡜M( i   )ࡢⅬvࢆྠᵝ࡟⪃࠼࡚ࠊࢡࣟࢫ࢝ࢵࢺᑐ⛠ࡼࡾࠊLn  >Cn  Cn      Cn    C n @ ࡞ࡢ࡛ࠊࡲࡾࠊM( i )࡜M( i   )࡜ࡢ㛫࡛஺ᕪࡀ㉳ࡇࡿ࡜ࡁࠊࡘࡂࡢ஺ᕪࡣN( i   )࡜N( i   )࡜
஺ⅬࡣN( i   )࡜N( i   )ࡢ஺Ⅼ࡜ᑐᛂࡋ࡚࠸ࡿ࡜࠸࠼ࡿࠋCn  ࡣࢡࣛࢫࢱࡢྑቃ⏺࡛ Cn   ࡜஺ᕪࡋ ࡇࢀࡼࡾࠊȘȚ㸩ࡣȘࡢせ⣲ࢆ㏫㌿ࡉࡏ࡚ࠊSᅇ㌿ࡉࡏࡿࡇ࡜࡛ࠊᕥቃ⏺ୖnࡢⅬ w ࢆ♧ࡍࠋⅬ w ࡛ࡣࠊCn   ࡜ Cn  ࡀ஺

1ࢆ㊊ࡋ࡚ᚓࡽࢀࡿࢩ࣮ࢣࣥࢫ࡟࡞ࡿ࡜࠸ᕪࡍࡿࠋț࡟ࡘ࠸࡚ࠊࡣࢡࣟࢫ࢝ࢵࢺୖࡢ஺Ⅼࡔ࠿ ࡇࢀࢆ⥆ࡅ࡚࠸ࡃ࡜ࠊࢡࣛࢫࢱࡢᕥྑࡢ

ࡽ᫂ࡽ࠿࡟ț㸻 n   n       ࡜࡞ቃ⏺⥺ࡣᅗ㸲࡟♧ࡍࡼ࠺࡞ࢪࢢࢨࢢࡢቃ⏺

ࡿࠋࡇࢀࡼࡾࠊᇶᮏࢩ࣮ࢣࣥࢫࡣࢆࡶࡘࡇ࡜ࡀࢃ࠿ࡿࠋ>       n   n   D 

㸦Ⅼ t 㸧ࠊ࠼ࡿࠋࡲࡓࠊࢡࣛࢫࢱࡢᕥቃ⏺࠿ࡽࠊȨ㸻        n   n   ࡜ࢃ࠿ࡿࠋn   n       D J  @࡜࡞ࡿࡇ࡜ࡀド࡛᫂ࡁࡓࠋ
᭱ᚋ࡟㏫ࢆド᫂ࡍࡿࠋ᮲௳ࢆ‶ࡓࡍȨ㸪Ș㸪ț㸪ȘȚ㸩ࢆ୚࠼ࡿ࡜ࠊᇶᮏࢩ࣮ࢣࣥࢫ
ࡣࠊ>       n   n   D n   n       D J  @࡜࡞ࡿࠋț࠿ࡽࠊࢡࣟࢫ࢝ࢵࢺࡀ࠶ࡿࡇ࡜ᅗ㸲ࡀࢃ࠿ࡾࠊȨ࠿ࡽࢡࣛࢫࢱࡀࢪࢢࢨࢢቃ⏺

ࢆࡶࡘࡇ࡜ࡀࢃ࠿ࡿࠋࢪࢢࢨࢢቃ⏺ࢆࡶࡘࡇࡇ࡛ࠊᕥྑࡢቃ⏺⥺࡟ὀ┠ࡍࡿࠋVennᅗࡢ᭱ࡶෆഃࡢ㡿ᇦࡢ࠶ࡿⅬX࠿ࡽእഃ࡟ࡇ࡜࡜Ș㸪ȘȚ㸩࠿ࡽࠊࢡࣟࢫ࢝ࢵࢺᑐ⛠ᛶࡀ♧ࡉࢀࡿࠋࡼࡗ࡚㢟ពࡣ♧ࡉࢀࡓࠋྥ࠿ࡗ࡚2ᮏࡢ┤⥺M, NࢆᘬࡁࠊMࢆᕥቃ
⏺࠿ࡽࢃࡎ࠿࡟ᕥࠊNࢆྑቃ⏺࠿ࡽࢃࡎ࠿

(Q. E. D.) 

㸱㸫㸴㸬ᅇ㌿ᑐ⛠9HQQᅗ࡟ࡘ࠸࡚㸲㸬௒ᚋࡢᒎᮃࠊࡲ࡜ࡵ ࢡࣟࢫ࢝ࢵࢺᑐ⛠ࡢᐃ⌮࠿ࡽࠊࢩ࣮ࢣࣥ ௒ᅇࡣᅇ㌿ᑐ⛠11-Vennᅗࡢసᡂ᪉ἲࢆ

ࢫȘࢆ᥈ࡍࡇ࡜࡛ᅇ㌿ᑐ⛠ n -Vennᅗࢆస⪃ᐹࡋࡓࡔࡅ࡛⤊ࢃࡗ࡚ࡋࡲࡗࡓࡢ࡛ࠊ௒

ᡂࡍࡿࡇ࡜ࡀ࡛ࡁࡿ࡜ࢃ࠿ࡗࡓࠋࡑࡇ࡛ࠊ

ᚋࡣసᡂࡋࡓVennᅗࢆࡶ࡜࡟⪃ᐹࡋࡓ࠸ࠋࢫ࣮ࣃ࣮ࢥࣥࣆ࣮ࣗࢱ࣮ࢆ⏝࠸࡚ᩘิࢆ⥙ ࡲࡓࠊ௒ᅇࠊࢩ࣮ࢣࣥࢫȘ࡟ࡘ࠸࡚ࡢල

⨶ⓗ࡟᥈⣴ࡋࠊᅇ㌿ᑐ⛠11-Vennᅗࢆᙧᡂయⓗ࡞ไ⣙ࢆㄪ࡭ࡿࡇ࡜ࡀ࡛ࡁࡎࠊࢩ࣮ࢣ

ࡋ࠺ࡿࢩ࣮ࢣࣥࢫȘࢆㄪ࡭ࡓ⤖ᯝࠊ200,000ࣥࢫࡀᐇ㝿࡟ᅇ㌿ᑐ⛠ n -Vennᅗࢆᙧᡂ࡛

✀㢮௨ୖࡢᅇ㌿ᑐ⛠11-Vennᅗࡀぢࡘ࠿ࡗࡁࡿ࠿ࡣࢫ࣮ࣃ࣮ࢥࣥࣆ࣮ࣗࢱ࣮ࢆ㢗ࡽࡊ

ࡓ>1@ࠋ௒ᅇࡣࠊࢩ࣮ࢣࣥࢫȘࡀ௨ୗࡢࡼ࠺

ࡿࢆᚓ࡞࠿ࡗࡓࡢ࡛ࠊ௒ᚋࡣࠊࢩ࣮ࢣࣥࢫ࡞ᅇ㌿ᑐ⛠11-Vennᅗࢆ♧ࡍ㸦ᅗ㸳㸧ࠋȘࡢ᮲௳࡟ࡘ࠸࡚ࡶ⪃࠼࡚࠸ࡁࡓ࠸ࠋ

ࢩ࣮ࢣࣥࢫȘ!

 ຍ࠼࡚ࠊ⚾ࡀ⪃᱌ࢆ┠ᣦࡋ࡚࠸ࡿ୕ḟඖ
ࡢVennᅗ࡟ᑐࡋ࡚ࡶࠊ௒ᅇド᫂ࡍࡿࡇ࡜ࡀ
࡛ࡁࡓࢡࣟࢫ࢝ࢵࢺᑐ⛠ࡢᐃ⌮ࢆᛂ⏝࡛ࡁ࡞࠸࠿⪃࠼࡚࠸ࡁࡓ࠸ࠋ
㸳㸬ཧ⪃ᩥ⊩㹙1㹛Khalegh Mamakani and Frank
Ruskey , “A New Rose : The First

Simple Symmetric 11-Venn
Diagram”, Math arXiv㸸1207.6452v1[cs.CG]㹙2㹛David W. Henderson, “Vennᅗ㸳

Diagrams for More than FourClasses”, Amer. Math. Monthly,
70(4)㸸424-426,(1963).
㸴㸬ㅰ㎡ᅗ㸴11-Vennᅗࡢ୍㒊௒ᅇࡢ◊✲࡟࠶ࡓࡾࠊࡈᣦᑟࡃࡔࡉࡗࡓ㢳ၥࡢᕝཱྀඛ⏕ࠊ࠶ࡾࡀ࡜࠺ࡈࡊ࠸ࡲࡋࡓࠋ
ࡲࡓࠊᰯෆⓎ⾲఍࡛ࡢ࣏ࢫࢱ࣮ࢭࢵࢩࣙࣥ࠾ࡼࡧࠕ㞟ࡲࢀ㸟⌮⣔ዪᏊ ዪᏊ⏕ᚐ࡟ࡼࡿ⛉Ꮫ◊✲Ⓨ⾲఍ 㛵す኱఍ࠖ࡟࠾࠸࡚ࡶࡓࡃࡉࢇࡢຓゝࢆ࠸ࡓࡔࡁࡲࡋࡓࠋࡇࡢሙࢆ೉ࡾ࡚࠾♩⏦ࡋୖࡆࡲࡍࠋᅗ㸵 11-Vennᅗࡢࢡࣛࢫࢱ 