{"id":6282,"date":"2026-03-13T19:40:00","date_gmt":"2026-03-13T19:40:00","guid":{"rendered":"https:\/\/flatarmal.is\/?p=6282"},"modified":"2026-03-19T15:04:59","modified_gmt":"2026-03-19T15:04:59","slug":"soguhornid-thrihyrningur-pascals","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=6282","title":{"rendered":"S\u00d6GUHORNI\u00d0: \u00der\u00edhyrningur Pascals"},"content":{"rendered":"\n

Krist\u00edn Bjarnad\u00f3ttir<\/p>\n\n\n\n

Bj\u00f6rn Bjarnason var st\u00e6r\u00f0fr\u00e6\u00f0ikennari minn for\u00f0um daga \u00ed Menntask\u00f3lanum \u00ed Reykjav\u00edk. Hann kenndi okkur um framandi hluti, sem n\u00fa eru m\u00e9r or\u00f0nir inngr\u00f3nir eftir \u00e1ratuga g\u00f6ngu \u00ed spor hans. Eitt \u00fe\u00f3tti m\u00e9r skr\u00edti\u00f0: Dag nokkur f\u00f3r hann a\u00f0 dunda s\u00e9r vi\u00f0 a\u00f0 leggja saman l\u00e1gar t\u00f6lur \u00e1 t\u00f6flunni. Ef til vill hefur athygli m\u00edn veri\u00f0 fjarri \u00ed upphafi en ni\u00f0ursta\u00f0an var \u00feessi:<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n
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\u00deannig h\u00e9lt hann \u00e1fram: Hann byrja\u00f0i \u00e1 einum, 1. \u00cd n\u00e6stu l\u00ednu kom 1 og 1. \u00dear n\u00e6sta l\u00edna byrja\u00f0i \u00e1 1, svo lag\u00f0i hann saman t\u00f6lurnar n\u00e6st fyrir ofan, 1 + 1, og f\u00e9kk \u00fat 2, og enda\u00f0i \u00e1 1. Mynstri\u00f0 f\u00f3r a\u00f0 koma \u00ed lj\u00f3s \u00ed n\u00e6stu l\u00ednu, 1, 1+2, 2+1, 1, og \u00feannig h\u00e9lt \u00fea\u00f0 \u00e1fram.<\/p>\n\n\n\n

Svo hl\u00f3 Bj\u00f6rn a\u00f0 \u00f6llu saman. En \u00e9g hugsa\u00f0i: Hvernig getur h\u00e1mennta\u00f0ur st\u00e6r\u00f0fr\u00e6\u00f0ingur fr\u00e1 H\u00e1sk\u00f3lanum \u00ed Kaupmannah\u00f6fn (sem \u00ed \u00fe\u00e1 daga var Mekka raunv\u00edsindamanna) haft gaman a\u00f0 svona nokkru? Hann kalla\u00f0i \u00feetta \u00der\u00edhyrning Pascals<\/strong>.<\/p>\n\n\n\n

L\u00f6ngu seinna, \u00feegar \u00e9g var farin a\u00f0 vi\u00f0a a\u00f0 m\u00e9r og lesa b\u00e6kur um st\u00e6r\u00f0fr\u00e6\u00f0i, f\u00f3r \u00e9g a\u00f0 rekast \u00e1 \u00feennan \u00fer\u00edhyrning. Og eitthva\u00f0 var\u00f0 til \u00feess a\u00f0 \u00e9g s\u00e1 a\u00f0<\/p>\n\n\n\n

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\"\"<\/figure>\n<\/div>\n<\/div>\n\n\n\n
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Summa talnanna \u00ed hverri l\u00ednu var \u00fe\u00e1 veldi af 2: 20<\/sup>, 21<\/sup>, 22<\/sup>, 23<\/sup>, 24<\/sup>, 25<\/sup>, og ekki \u00f3l\u00edklegt a\u00f0 \u00feannig haldi \u00feetta \u00e1fram.<\/p>\n\n\n\n

Tv\u00edli\u00f0ureglan<\/strong><\/h2>\n\n\n\n

\u00cd algebru eru algengar tveggja li\u00f0a st\u00e6r\u00f0ir, tv\u00edli\u00f0ur, eins og x + y, a \u2013 b, z + 1, algengar.  \u00cd upphafi algebrun\u00e1ms gl\u00edma nemendur vi\u00f0 a\u00f0 margfalda saman tv\u00edli\u00f0ur, og jafnvel rekja \u00feriggja li\u00f0a st\u00e6r\u00f0ir aftur \u00ed sundur \u00ed tv\u00edli\u00f0ur me\u00f0 \u00e1giskun.<\/p>\n\n\n\n

(x + 2)(x +3) = x2<\/sup> + 3x + 2x + 6 = x2<\/sup> + 5x + 6
\u2026 og x2<\/sup> + 3x + 2 = (x + 1)(x+2)<\/p>\n\n\n\n

\u00dea\u00f0 er l\u00edka h\u00e6gt a\u00f0 hefja tv\u00edli\u00f0ur upp \u00ed veldi:
(x + y)2<\/sup> = (x + y)(x + y) = x2<\/sup> + xy + xy + y2<\/sup> = x2<\/sup> + 2xy + y2<\/sup>.<\/p>\n\n\n\n

Svona m\u00e6tti halda \u00e1fram og finna
(x + y)3<\/sup> = (x + y)(x + y)2<\/sup>
= (x + y)(x2<\/sup> + 2xy + y2<\/sup>)
= x3<\/sup> + 3x2<\/sup>y + 3xy2<\/sup> + y3<\/sup><\/p>\n\n\n\n

\u00cd framhaldi birtist tv\u00edli\u00f0ureglan <\/strong>(e. binomial theorem):<\/p>\n\n\n\n

(x + y)0<\/sup> = 1
(x + y)1<\/sup> = 1x + 1y
(x + y)2<\/sup> = 1x2<\/sup> + 2xy + 1y2<\/sup>
(x + y)3<\/sup> = 1x3<\/sup> + 3x2<\/sup>y + 3xy2<\/sup> + 1y3<\/sup>
(x + y)4<\/sup> = 1x4<\/sup> + 4x3<\/sup>y + 6x2<\/sup>y2<\/sup> + 4xy3<\/sup> + 1y4<\/sup>
(x + y)5<\/sup> = 1x5<\/sup> + 5x4<\/sup>y + 10x3<\/sup>y2<\/sup> + 10x2<\/sup>y3<\/sup> + 5xy4<\/sup> + 1y5<\/sup>
\u2026.<\/p>\n\n\n\n

Tv\u00edli\u00f0ureglan geymir mynstur.<\/p>\n\n\n\n

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  • Sko\u00f0i\u00f0 talnastu\u00f0lana. Hva\u00f0 er a\u00f0 segja um \u00fe\u00e1?<\/li>\n\n\n\n
  • Og hva\u00f0 er a\u00f0 segja um veldisv\u00edsana \u00e1 x og y?<\/li>\n<\/ul>\n\n\n\n
    <\/div>\n\n\n\n

    Hver var Blaise Pascal?<\/strong><\/h2>\n\n\n\n

    Talna\u00fer\u00edhyrningurinn er nefndur \u00der\u00edhyrningur Pascals<\/em>, en hann var \u00feekktur l\u00f6ngu fyrir daga st\u00e6r\u00f0fr\u00e6\u00f0ingsins Blaise Pascals (1623\u20131662). Til d\u00e6mis \u00feekkti K\u00ednverjinn Jia Xian \u00fer\u00edhyrninginn 600 \u00e1rum fyrir daga Pascals. Segja m\u00e1, a\u00f0 Pascal hafi sta\u00f0i\u00f0 \u00e1 her\u00f0um merkra fyrirrennara. Pascal og vinur hans Fermat teljast me\u00f0al h\u00f6funda l\u00edkindareiknings, og \u00fer\u00edhyrningurinn gegndi einmitt hlutverki \u00ed l\u00edkindareikningi.<\/p>\n\n\n\n

    <\/div>\n\n\n\n
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    \"\"<\/figure>\n<\/div><\/div>\n\n\n\n
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    \"\"<\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n
    <\/div>\n\n\n\n

    Tv\u00edli\u00f0ureglan \u00ed l\u00edfinu<\/strong><\/h2>\n\n\n\n

    Margir fur\u00f0a sig \u00e1 \u00fev\u00ed a\u00f0 til s\u00e9u st\u00f3rir systkinah\u00f3par \u00feannig a\u00f0 \u00f6ll systkinin s\u00e9u af sama kyni, til d\u00e6mis fimm systur e\u00f0a sj\u00f6 br\u00e6\u00f0ur. Vi\u00f0 getum l\u00edkt eftir sl\u00edku fyrirbrig\u00f0i me\u00f0 \u00fev\u00ed a\u00f0 varpa peningi. Ef vi\u00f0 gerum r\u00e1\u00f0 fyrir a\u00f0 jafnar l\u00edkur s\u00e9u \u00e1 a\u00f0 upp komi kr\u00f3na e\u00f0a fiskur \u00feegar peningi er kasta\u00f0, finnst manni kannski sanngjarnt a\u00f0 eftir \u00ferj\u00fa k\u00f6st me\u00f0 fiskum komi upp kr\u00f3na, en l\u00edkindin hafa ekki neitt minni. \u00dea\u00f0 eru enn\u00fe\u00e1 jafnar l\u00edkur \u00e1 kr\u00f3nu og fiski. \u00deetta er gjarnan s\u00fdnt me\u00f0 l\u00edkindatr\u00e9. Lesi\u00f0 er \u00far l\u00edkindatr\u00e9nu \u00feannig a\u00f0 allar m\u00f6gulegar ke\u00f0jur eru lesnar ni\u00f0ur fr\u00e1 fyrsta kasti.<\/p>\n\n\n\n

    L\u00edkindatr\u00e9\u00f0 l\u00edtur svona \u00fat:<\/p>\n\n\n

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    \"\"<\/figure>\n<\/div>\n\n\n
    <\/div>\n\n\n\n

    \u00cd fyrsta kasti eru m\u00f6guleikarnir tveir:<\/p>\n\n\n\n

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    \n
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    Einn fiskur
    Ein kr\u00f3na<\/p>\n<\/div>\n\n\n\n

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    1 F
    1 K<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

    Eftir tv\u00f6 k\u00f6st eru m\u00f6guleikarnir fj\u00f3rir:<\/p>\n\n\n\n

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    \n
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    Tveir fiskar
    Einn fiskur og ein kr\u00f3na
    Tv\u00e6r kr\u00f3nur <\/p>\n<\/div>\n\n\n\n

    \n

    1 FF
    2 FK KF
    1 KK<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

    H\u00e9r er r\u00e9tt a\u00f0 staldra vi\u00f0. Vi\u00f0 sj\u00e1um a\u00f0 r\u00f6\u00f0in skiptir m\u00e1li, FK og KF eru tveir mismunandi m\u00f6guleikar.<\/p>\n\n\n\n

    Eftir \u00ferj\u00fa k\u00f6st eru m\u00f6guleikarnir \u00e1tta:<\/p>\n\n\n\n

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    \u00der\u00edr fiskar
    Tveir fiskar og ein kr\u00f3na
    Einn fiskur og tv\u00e6r kr\u00f3nur
    \u00derj\u00e1r kr\u00f3nur<\/p>\n<\/div>\n\n\n\n

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    1 FFF
    3 FFK FKF KFF
    3 FKK KFK KKF
    1 KKK<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n

    Eftir fj\u00f6gur k\u00f6st ver\u00f0a m\u00f6guleikarnir sext\u00e1n:<\/p>\n\n\n\n

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    \n
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    Fj\u00f3rir fiskar
    \u00der\u00edr fiskar og ein kr\u00f3na
    Tveir fiskar og tv\u00e6r kr\u00f3nur
    Einn fiskur og \u00ferj\u00e1r kr\u00f3nur
    Fj\u00f3rar kr\u00f3nur <\/p>\n<\/div>\n\n\n\n

    \n

    1 FFFF
    4 FFFK FFKF FKFF KFFF
    6 FKFK FKKF FFKK KKFF KFFK KFKF
    4 KKKF KKFK KFKK FKKK
    1 KKKK<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

    N\u00fa er mynstri\u00f0 fari\u00f0 a\u00f0 minna \u00e1 \u00fer\u00edhyrning Pascals. Og \u00fea\u00f0 er r\u00e9tt! Lesandinn f\u00e6r kost \u00e1 \u00fev\u00ed a\u00f0 p\u00e6la \u00ed hvernig \u00e1 \u00fev\u00ed stendur. Vi\u00f0 giskum \u00e1 a\u00f0 fj\u00f6ldi m\u00f6gulegra \u00fatkoma \u00far fimm k\u00f6stum s\u00e9 32 me\u00f0 \u00fev\u00ed a\u00f0 sty\u00f0jast vi\u00f0 \u00fer\u00edhyrning Pascals:<\/p>\n\n\n\n

    \n
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    Fimm fiskar
    Fj\u00f3rir fiskar og ein kr\u00f3na
    \u00der\u00edr fiskar og tv\u00e6r kr\u00f3nur
    Tveir fiskar og \u00ferj\u00e1r kr\u00f3nur
    Einn fiskur og fj\u00f3rar kr\u00f3nur
    Fimm kr\u00f3nur <\/p>\n<\/div>\n\n\n\n

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    1
    5
    10
    10
    5
    1<\/p>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n

    Vi\u00f0 lj\u00fakum \u00feessari athugun \u00e1 a\u00f0 \u00edhuga hvernig s\u00e9 l\u00edklegast a\u00f0 sex systkina h\u00f3pur skiptist \u00ed drengi og st\u00falkur. \u00c1 sama h\u00e1tt og jafnar l\u00edkur eru \u00e1 a\u00f0 f\u00e1 fisk og kr\u00f3nu \u00feegar peningi er varpa\u00f0, m\u00e1 l\u00edta svo \u00e1 a\u00f0 jafnar l\u00edkur s\u00e9 \u00e1 a\u00f0 eignast dreng en st\u00falku.<\/p>\n\n\n\n

    Er l\u00edklegast a\u00f0 h\u00f3purinn skiptist jafnt, 3 drengir og 3 st\u00falkur?
    Sj\u00f6tta l\u00ednan \u00ed \u00fer\u00edhyrningi Pascals er 1    6   15   20   15   6   1. Summan er 26<\/sup> = 64.
    L\u00edkurnar \u00e1 a\u00f0 h\u00f3purinn skiptist jafnt er \u00fe\u00e1 20\/64 = 31%. L\u00edkur \u00e1 a\u00f0 h\u00f3purinn skiptist 4 \u00e1 m\u00f3ti 2 er (15+15)\/64, e\u00f0a 30\/64 = 47% e\u00f0a n\u00e6rri helmingsl\u00edkur. L\u00edkur \u00e1 st\u00falkum eing\u00f6ngu eru 1\/64 = 1,6% og sama er a\u00f0 segja um l\u00edkur \u00e1 drengjum eing\u00f6ngu.   <\/p>\n\n\n\n

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    \"\"<\/figure>
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    Krist\u00edn Bjarnad\u00f3ttir, fyrrum pr\u00f3fessor emerita<\/em> vi\u00f0 H\u00e1sk\u00f3la \u00cdslands<\/em><\/strong><\/p>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

    Krist\u00edn Bjarnad\u00f3ttir Bj\u00f6rn Bjarnason var st\u00e6r\u00f0fr\u00e6\u00f0ikennari minn for\u00f0um daga \u00ed Menntask\u00f3lanum \u00ed Reykjav\u00edk. Hann kenndi okkur um framandi hluti, sem n\u00fa eru m\u00e9r or\u00f0nir inngr\u00f3nir eftir \u00e1ratuga g\u00f6ngu \u00ed spor hans. Eitt \u00fe\u00f3tti m\u00e9r skr\u00edti\u00f0: Dag nokkur f\u00f3r hann a\u00f0 dunda s\u00e9r vi\u00f0 a\u00f0 leggja saman l\u00e1gar t\u00f6lur \u00e1 t\u00f6flunni. Ef til vill hefur athygli…<\/p>\n","protected":false},"author":1,"featured_media":6305,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[191,194,193,195,192,188,189,137,21,190],"class_list":["post-6282","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-algebra","tag-kronukast","tag-likindareikningur","tag-likindatre","tag-likindi","tag-pascal","tag-pascal-thrihyrningurinn","tag-soguhornid","tag-staerdfraedi","tag-tvilidureglan"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/6282","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6282"}],"version-history":[{"count":85,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/6282\/revisions"}],"predecessor-version":[{"id":6388,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/6282\/revisions\/6388"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/6305"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6282"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6282"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}