{"id":806804,"date":"2021-09-15T09:00:00","date_gmt":"2021-09-15T07:00:00","guid":{"rendered":"https:\/\/www.gamesfanatic.pl\/?p=806804"},"modified":"2021-09-15T09:28:26","modified_gmt":"2021-09-15T07:28:26","slug":"scrabble","status":"publish","type":"post","link":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/2021\/09\/15\/scrabble\/","title":{"rendered":"Scrabble"},"content":{"rendered":"

\"\"<\/a>Na pierwszy rzut oka Scrabble to gra humanistyczna, w kt\u00f3rej decyduj\u0105c\u0105 rol\u0119 odgrywa zas\u00f3b s\u0142ownictwa, a matematyka sprowadza si\u0119 do prostych dzia\u0142a\u0144 \u2013 sumowania punkt\u00f3w za litery i ewentualnie mno\u017cenia, gdy uk\u0142adany wyraz przechodzi przez pole premii literowej lub s\u0142ownej. W tym tek\u015bcie postaram si\u0119 wykaza\u0107, \u017ce do dok\u0142adniejszej analizy gry matematyka jest przydatna, a nawet niezb\u0119dna.<\/p>\n

<\/p>\n

Podstawowym zagadnieniem \u201ematematycznym\u201d jest ustalenie, ile powinno by\u0107 p\u0142ytek z dan\u0105 liter\u0105 i jaka powinna by\u0107 warto\u015b\u0107 punktowa tej litery. Wiadomo, \u017ce niekt\u00f3re litery wyst\u0119puj\u0105 w polskim s\u0142ownictwie cz\u0119\u015bciej, a inne rzadziej. Tych cz\u0119\u015bciej wyst\u0119puj\u0105cych powinno by\u0107 w zestawie wi\u0119cej, bo wyst\u0119puj\u0105 w wi\u0119kszej liczbie wyraz\u00f3w i s\u0105 przez to bardziej potrzebne. Ale z drugiej strony \u0142atwiej takich liter mo\u017cna u\u017cy\u0107, wi\u0119c powinny by\u0107 ni\u017cej punktowane. Oczywi\u015bcie \u0142atwo zauwa\u017cy\u0107, \u017ce litera A wyst\u0119puje w j\u0119zyku polskim cz\u0119\u015bciej ni\u017c \u0179 ale por\u00f3wnanie, czy wi\u0119cej jest wyraz\u00f3w z liter\u0105 B czy D ju\u017c takie \u0142atwe nie jest. A sama hierarchia cz\u0119sto\u015bci wyst\u0119powania nie wystarczy, bo trzeba jeszcze dobra\u0107 odpowiednie proporcje.<\/p>\n

Gdy w 1979 roku Marek Penszko opublikowa\u0142 w Kalendarzu Szaradzisty przepisy wzorowanej na Scrabble \u201eGry w krzy\u017c\u00f3wk\u0119\u201d, przy doborze liczb poszczeg\u00f3lnych liter i punkt\u00f3w za ka\u017cd\u0105 liter\u0119 wykorzysta\u0142 dane, zawarte w artykule Barbary Kowalskiej \u201eCz\u0119sto\u015b\u0107 wyst\u0119powania liter w j\u0119zyku polskim\u201d, opublikowanym w 1967 roku w bran\u017cowym pi\u015bmie poligraf\u00f3w \u201eLitera\u201d. Tym samym \u017ar\u00f3d\u0142em pos\u0142u\u017cy\u0142 si\u0119 Jacek Ciesielski, przedstawiaj\u0105c najpierw w tygodniku \u201eRazem\u201d (1986) zasady gry wraz z plansz\u0105 i zestawem \u017ceton\u00f3w do samodzielnego wyci\u0119cia, a nast\u0119pnie przygotowuj\u0105c pierwsze polskie wydanie Scrabble w 1993 roku.<\/p>\n

Autorka artyku\u0142u w \u201eLiterze\u201d zebra\u0142a dost\u0119pne w owym czasie statystyki wyst\u0119powania liter i po prostu wyci\u0105gn\u0119\u0142a z nich \u015bredni\u0105 arytmetyczn\u0105. Na pierwszy rzut oka wydaje si\u0119, \u017ce u\u015brednienie wynik\u00f3w z siedmiu r\u00f3\u017cnych \u017ar\u00f3de\u0142 powinno przybli\u017ca\u0107 do \u201eprawdy obiektywnej\u201d ale w tym przypadku mo\u017cna mie\u0107 co do tego w\u0105tpliwo\u015bci. Przede wszystkim dlatego, \u017ce zbiory tekstowe, na podstawie kt\u00f3rych obliczano frekwencje liter, by\u0142y r\u00f3\u017cnej wielko\u015bci (od 10 tysi\u0119cy do 200 tysi\u0119cy znak\u00f3w) i w r\u00f3\u017cny spos\u00f3b powstawa\u0142y. Trzy z nich by\u0142y opracowaniami Polskiego Komitetu Normalizacji i zosta\u0142y przygotowane podczas prac nad polsk\u0105 wersj\u0105 klawiatury do maszyn do pisania, jeden powsta\u0142 na potrzeby artyku\u0142u naukowego, a dwa by\u0142y dzie\u0142em t\u0142umaczy, kt\u00f3rzy musieli zast\u0105pi\u0107 dane z oryginalnych ksi\u0105\u017cek, odnosz\u0105ce si\u0119 do frekwencji liter w j\u0119zyku rosyjskim. Ale by\u0142o jedno \u017ar\u00f3d\u0142o bardzo dziwne, bo oparte na cz\u0119sto\u015bci wyst\u0119powania nie liter, tylko g\u0142osek. Dane z tego \u017ar\u00f3d\u0142a znacznie odbiega\u0142y od pozosta\u0142ych, bo g\u0142oski rejestrowane \u201eze s\u0142uchu\u201d zani\u017cy\u0142y cz\u0119sto\u015b\u0107 wyst\u0119powania jednych liter, a zawy\u017cy\u0142y innych (m.in. dlatego, \u017ce np. w s\u0142owie \u201ep\u0105czek\u201d nie zosta\u0142a zarejestrowana litera \u201e\u0105\u201d, a zamiast niej litery \u201eo\u201d i \u201en\u201d). Pami\u0119ta\u0107 trzeba te\u017c o tym, \u017ce wszystkie opracowania \u017ar\u00f3d\u0142owe powsta\u0142y w latach 1951-61, a wi\u0119c z pewno\u015bci\u0105 by\u0142y obarczone b\u0142\u0119dami, wynikaj\u0105cymi z \u201er\u0119cznego\u201d liczenia liter.<\/p>\n

\"\"<\/a><\/p>\n

Warto zauwa\u017cy\u0107, \u017ce aktualna wersja gry Scrabble r\u00f3\u017cni si\u0119 od pierwszego wydania w kilku szczeg\u00f3\u0142ach: liter A jest 9 a nie 8, litera F jest na jednej p\u0142ytce, a nie na dw\u00f3ch i ma warto\u015b\u0107 5, a nie 4, natomiast litera \u0179 zdro\u017ca\u0142a z 7 do 9 punkt\u00f3w. Dlaczego litera F zosta\u0142a pocz\u0105tkowo uznana za cz\u0119\u015bciej wyst\u0119puj\u0105c\u0105 ni\u017c np. \u0104, cho\u0107 nawet \u015brednia z artyku\u0142u w \u201eLiterze\u201d na to nie wskazuje, tego nie wiadomo. By\u0107 mo\u017ce zdecydowa\u0142o to, \u017ce po jednej p\u0142ytce dosta\u0142y tylko typowo polskie litery (z wyj\u0105tkiem \u0141), a mo\u017ce to dlatego, \u017ce w jednym \u017ar\u00f3dle (akurat tym \u201efonetycznym\u201d) udzia\u0142 litery F wyni\u00f3s\u0142 a\u017c 1,5% ?<\/p>\n

Poni\u017cej przedstawiam Tabel\u0119 1, w kt\u00f3rej zestawi\u0142em cz\u0119stotliwo\u015b\u0107 wyst\u0119powania liter (od najpopularniejszych do najrzadszych) na podstawie r\u00f3\u017cnych \u017ar\u00f3de\u0142. Przy por\u00f3wnywaniu wynik\u00f3w trzeba pami\u0119ta\u0107, \u017ce w odr\u00f3\u017cnieniu od pozosta\u0142ych statystyk, s\u0142ownik scrabblisty nie uwzgl\u0119dnia wyraz\u00f3w jednoliterowych (a, i, o, u, w, z) oraz d\u0142u\u017cszych ni\u017c pi\u0119tnastoliterowe.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
li-
\ntera<\/span><\/span><\/td>\n		p\u0142y-
\ntek<\/span><\/span><\/td>\n		udzia\u0142 <\/span><\/span><\/td>\npkt<\/span><\/span><\/td>\nOSPS<\/span><\/span><\/td>\nIPI<\/span><\/span><\/td>\nLitera<\/span><\/span><\/td>\nKrypto<\/span><\/span><\/td>\nP\u0119tla<\/span><\/span><\/td>\nGG<\/span><\/span><\/td>\nZM<\/span><\/span><\/td>\n<\/tr>\n
<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
A<\/span><\/span><\/td>\n\n

9<\/span><\/span><\/p>\n<\/td>\n

\n

9,18%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

9,50%<\/span><\/span><\/p>\n<\/td>\n

\n

8,91%<\/span><\/span><\/p>\n<\/td>\n

\n

8,49%<\/span><\/span><\/p>\n<\/td>\n

\n

9,02%<\/span><\/span><\/p>\n<\/td>\n

\n

8,70%<\/span><\/span><\/p>\n<\/td>\n

\n

10,46%<\/span><\/span><\/p>\n<\/td>\n

\n

8,87%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

I<\/span><\/span><\/td>\n\n

8<\/span><\/span><\/p>\n<\/td>\n

\n

8,16%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

9,40%<\/span><\/span><\/p>\n<\/td>\n

\n

8,21%<\/span><\/span><\/p>\n<\/td>\n

\n

8,26%<\/span><\/span><\/p>\n<\/td>\n

\n

8,09%<\/span><\/span><\/p>\n<\/td>\n

\n

8,98%<\/span><\/span><\/p>\n<\/td>\n

\n

8,27%<\/span><\/span><\/p>\n<\/td>\n

\n

7,99%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

E<\/span><\/span><\/td>\n\n

7<\/span><\/span><\/p>\n<\/td>\n

\n

7,14%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

7,57%<\/span><\/span><\/p>\n<\/td>\n

\n

7,66%<\/span><\/span><\/p>\n<\/td>\n

\n

7,44%<\/span><\/span><\/p>\n<\/td>\n

\n

7,51%<\/span><\/span><\/p>\n<\/td>\n

\n

7,75%<\/span><\/span><\/p>\n<\/td>\n

\n

9,91%<\/span><\/span><\/p>\n<\/td>\n

\n

7,27%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

O<\/span><\/span><\/td>\n\n

6<\/span><\/span><\/p>\n<\/td>\n

\n

6,12%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

7,47%<\/span><\/span><\/p>\n<\/td>\n

\n

7,75%<\/span><\/span><\/p>\n<\/td>\n

\n

7,70%<\/span><\/span><\/p>\n<\/td>\n

\n

7,90%<\/span><\/span><\/p>\n<\/td>\n

\n

6,82%<\/span><\/span><\/p>\n<\/td>\n

\n

8,09%<\/span><\/span><\/p>\n<\/td>\n

\n

7,58%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

N<\/span><\/span><\/td>\n\n

5<\/span><\/span><\/p>\n<\/td>\n

\n

5,10%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

6,94%<\/span><\/span><\/p>\n<\/td>\n

\n

5,52%<\/span><\/span><\/p>\n<\/td>\n

\n

5,46%<\/span><\/span><\/p>\n<\/td>\n

\n

5,81%<\/span><\/span><\/p>\n<\/td>\n

\n

4,59%<\/span><\/span><\/p>\n<\/td>\n

\n

5,15%<\/span><\/span><\/p>\n<\/td>\n

\n

5,57%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

Z<\/span><\/span><\/td>\n\n

5<\/span><\/span><\/p>\n<\/td>\n

\n

5,10%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

4,34%<\/span><\/span><\/p>\n<\/td>\n

\n

5,64%<\/span><\/span><\/p>\n<\/td>\n

\n

5,80%<\/span><\/span><\/p>\n<\/td>\n

\n

5,17%<\/span><\/span><\/p>\n<\/td>\n

\n

6,34%<\/span><\/span><\/p>\n<\/td>\n

\n

6,11%<\/span><\/span><\/p>\n<\/td>\n

\n

6,06%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

W<\/span><\/span><\/td>\n\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

4,08%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

4,12%<\/span><\/span><\/p>\n<\/td>\n

\n

4,65%<\/span><\/span><\/p>\n<\/td>\n

\n

4,50%<\/span><\/span><\/p>\n<\/td>\n

\n

4,78%<\/span><\/span><\/p>\n<\/td>\n

\n

3,79%<\/span><\/span><\/p>\n<\/td>\n

\n

3,39%<\/span><\/span><\/p>\n<\/td>\n

\n

5,04%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

R<\/span><\/span><\/td>\n\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

4,08%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

3,78%<\/span><\/span><\/p>\n<\/td>\n

\n

4,69%<\/span><\/span><\/p>\n<\/td>\n

\n

4,51%<\/span><\/span><\/p>\n<\/td>\n

\n

5,06%<\/span><\/span><\/p>\n<\/td>\n

\n

3,72%<\/span><\/span><\/p>\n<\/td>\n

\n

3,53%<\/span><\/span><\/p>\n<\/td>\n

\n

4,49%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

S<\/span><\/span><\/td>\n\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

4,08%<\/span><\/span><\/p>\n<\/td>\n

1 <\/span><\/span><\/td>\n\n

3,03%<\/span><\/span><\/p>\n<\/td>\n

\n

4,32%<\/span><\/span><\/p>\n<\/td>\n

\n

4,52%<\/span><\/span><\/p>\n<\/td>\n

\n

4,46%<\/span><\/span><\/p>\n<\/td>\n

\n

4,07%<\/span><\/span><\/p>\n<\/td>\n

\n

4,57%<\/span><\/span><\/p>\n<\/td>\n

\n

4,26%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

Y<\/span><\/span><\/td>\n\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

4,08%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

5,27%<\/span><\/span><\/p>\n<\/td>\n

\n

3,76%<\/span><\/span><\/p>\n<\/td>\n

\n

3,78%<\/span><\/span><\/p>\n<\/td>\n

\n

3,70%<\/span><\/span><\/p>\n<\/td>\n

\n

3,82%<\/span><\/span><\/p>\n<\/td>\n

\n

3,25%<\/span><\/span><\/p>\n<\/td>\n

\n

4,35%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

C<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

4,13%<\/span><\/span><\/p>\n<\/td>\n

\n

3,96%<\/span><\/span><\/p>\n<\/td>\n

\n

3,81%<\/span><\/span><\/p>\n<\/td>\n

\n

3,96%<\/span><\/span><\/p>\n<\/td>\n

\n

3,79%<\/span><\/span><\/p>\n<\/td>\n

\n

3,93%<\/span><\/span><\/p>\n<\/td>\n

\n

3,94%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

M<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

4,11%<\/span><\/span><\/p>\n<\/td>\n

\n

2,80%<\/span><\/span><\/p>\n<\/td>\n

\n

2,57%<\/span><\/span><\/p>\n<\/td>\n

\n

2,73%<\/span><\/span><\/p>\n<\/td>\n

\n

3,41%<\/span><\/span><\/p>\n<\/td>\n

\n

3,73%<\/span><\/span><\/p>\n<\/td>\n

\n

2,59%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

K<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

2,98%<\/span><\/span><\/p>\n<\/td>\n

\n

3,51%<\/span><\/span><\/p>\n<\/td>\n

\n

3,02%<\/span><\/span><\/p>\n<\/td>\n

\n

3,54%<\/span><\/span><\/p>\n<\/td>\n

\n

3,50%<\/span><\/span><\/p>\n<\/td>\n

\n

3,42%<\/span><\/span><\/p>\n<\/td>\n

\n

3,74%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

P<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

2,78%<\/span><\/span><\/p>\n<\/td>\n

\n

3,13%<\/span><\/span><\/p>\n<\/td>\n

\n

2,93%<\/span><\/span><\/p>\n<\/td>\n

\n

2,92%<\/span><\/span><\/p>\n<\/td>\n

\n

3,19%<\/span><\/span><\/p>\n<\/td>\n

\n

2,66%<\/span><\/span><\/p>\n<\/td>\n

\n

3,02%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

L<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

2,34%<\/span><\/span><\/p>\n<\/td>\n

\n

2,10%<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

\n

2,36%<\/span><\/span><\/p>\n<\/td>\n

\n

1,85%<\/span><\/span><\/p>\n<\/td>\n

\n

2,53%<\/span><\/span><\/p>\n<\/td>\n

\n

1,91%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

T<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

2,24%<\/span><\/span><\/p>\n<\/td>\n

\n

3,98%<\/span><\/span><\/p>\n<\/td>\n

\n

3,54%<\/span><\/span><\/p>\n<\/td>\n

\n

3,94%<\/span><\/span><\/p>\n<\/td>\n

\n

3,72%<\/span><\/span><\/p>\n<\/td>\n

\n

4,53%<\/span><\/span><\/p>\n<\/td>\n

\n

3,62%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

D<\/span><\/span><\/td>\n\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

3,06%<\/span><\/span><\/p>\n<\/td>\n

2 <\/span><\/span><\/td>\n\n

2,21%<\/span><\/span><\/p>\n<\/td>\n

\n

3,25%<\/span><\/span><\/p>\n<\/td>\n

\n

3,36%<\/span><\/span><\/p>\n<\/td>\n

\n

3,23%<\/span><\/span><\/p>\n<\/td>\n

\n

3,44%<\/span><\/span><\/p>\n<\/td>\n

\n

3,12%<\/span><\/span><\/p>\n<\/td>\n

\n

3,55%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

U<\/span><\/span><\/td>\n\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

3 <\/span><\/span><\/td>\n\n

2,78%<\/span><\/span><\/p>\n<\/td>\n

\n

2,50%<\/span><\/span><\/p>\n<\/td>\n

\n

1,87%<\/span><\/span><\/p>\n<\/td>\n

\n

2,59%<\/span><\/span><\/p>\n<\/td>\n

\n

2,22%<\/span><\/span><\/p>\n<\/td>\n

\n

2,03%<\/span><\/span><\/p>\n<\/td>\n

\n

2,81%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u0141<\/span><\/span><\/td>\n\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

3 <\/span><\/span><\/td>\n\n

2,70%<\/span><\/span><\/p>\n<\/td>\n

\n

1,82%<\/span><\/span><\/p>\n<\/td>\n

\n

1,90%<\/span><\/span><\/p>\n<\/td>\n

\n

1,82%<\/span><\/span><\/p>\n<\/td>\n

\n

3,55%<\/span><\/span><\/p>\n<\/td>\n

\n

1,19%<\/span><\/span><\/p>\n<\/td>\n

\n

2,23%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

B<\/span><\/span><\/td>\n\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

3 <\/span><\/span><\/td>\n\n

2,32%<\/span><\/span><\/p>\n<\/td>\n

\n

1,47%<\/span><\/span><\/p>\n<\/td>\n

\n

1,26%<\/span><\/span><\/p>\n<\/td>\n

\n

1,39%<\/span><\/span><\/p>\n<\/td>\n

\n

1,77%<\/span><\/span><\/p>\n<\/td>\n

\n

1,98%<\/span><\/span><\/p>\n<\/td>\n

\n

1,17%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

J<\/span><\/span><\/td>\n\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

3 <\/span><\/span><\/td>\n\n

1,92%<\/span><\/span><\/p>\n<\/td>\n

\n

2,28%<\/span><\/span><\/p>\n<\/td>\n

\n

2,32%<\/span><\/span><\/p>\n<\/td>\n

\n

2,26%<\/span><\/span><\/p>\n<\/td>\n

\n

2,12%<\/span><\/span><\/p>\n<\/td>\n

\n

2,89%<\/span><\/span><\/p>\n<\/td>\n

\n

1,96%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

G<\/span><\/span><\/td>\n\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

3 <\/span><\/span><\/td>\n\n

1,36%<\/span><\/span><\/p>\n<\/td>\n

\n

1,42%<\/span><\/span><\/p>\n<\/td>\n

\n

1,38%<\/span><\/span><\/p>\n<\/td>\n

\n

1,54%<\/span><\/span><\/p>\n<\/td>\n

\n

1,34%<\/span><\/span><\/p>\n<\/td>\n

\n

1,11%<\/span><\/span><\/p>\n<\/td>\n

\n

1,36%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

H<\/span><\/span><\/td>\n\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

2,04%<\/span><\/span><\/p>\n<\/td>\n

3 <\/span><\/span><\/td>\n\n

1,14%<\/span><\/span><\/p>\n<\/td>\n

\n

1,08%<\/span><\/span><\/p>\n<\/td>\n

\n

1,19%<\/span><\/span><\/p>\n<\/td>\n

\n

1,25%<\/span><\/span><\/p>\n<\/td>\n

\n

0,93%<\/span><\/span><\/p>\n<\/td>\n

\n

1,12%<\/span><\/span><\/p>\n<\/td>\n

\n

1,01%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u0104<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

5 <\/span><\/span><\/td>\n\n

1,43%<\/span><\/span><\/p>\n<\/td>\n

\n

0,99%<\/span><\/span><\/p>\n<\/td>\n

\n

0,84%<\/span><\/span><\/p>\n<\/td>\n

\n

0,95%<\/span><\/span><\/p>\n<\/td>\n

\n

1,24%<\/span><\/span><\/p>\n<\/td>\n

\n

0,33%<\/span><\/span><\/p>\n<\/td>\n

\n

1,13%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u015a<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

5 <\/span><\/span><\/td>\n\n

1,31%<\/span><\/span><\/p>\n<\/td>\n

\n

0,66%<\/span><\/span><\/p>\n<\/td>\n

\n

0,76%<\/span><\/span><\/p>\n<\/td>\n

\n

0,58%<\/span><\/span><\/p>\n<\/td>\n

\n

0,93%<\/span><\/span><\/p>\n<\/td>\n

\n

0,47%<\/span><\/span><\/p>\n<\/td>\n

\n

0,65%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u0118<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

5 <\/span><\/span><\/td>\n\n

0,80%<\/span><\/span><\/p>\n<\/td>\n

\n

1,11%<\/span><\/span><\/p>\n<\/td>\n

\n

1,15%<\/span><\/span><\/p>\n<\/td>\n

\n

0,98%<\/span><\/span><\/p>\n<\/td>\n

\n

1,81%<\/span><\/span><\/p>\n<\/td>\n

\n

0,71%<\/span><\/span><\/p>\n<\/td>\n

\n

1,22%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u017b<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

5 <\/span><\/span><\/td>\n\n

0,78%<\/span><\/span><\/p>\n<\/td>\n

\n

0,83%<\/span><\/span><\/p>\n<\/td>\n

\n

0,69%<\/span><\/span><\/p>\n<\/td>\n

\n

0,66%<\/span><\/span><\/p>\n<\/td>\n

\n

0,96%<\/span><\/span><\/p>\n<\/td>\n

\n

0,53%<\/span><\/span><\/p>\n<\/td>\n

\n

0,77%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

F<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

5 <\/span><\/span><\/td>\n\n

0,35%<\/span><\/span><\/p>\n<\/td>\n

\n

0,30%<\/span><\/span><\/p>\n<\/td>\n

\n

0,37%<\/span><\/span><\/p>\n<\/td>\n

\n

0,41%<\/span><\/span><\/p>\n<\/td>\n

\n

0,19%<\/span><\/span><\/p>\n<\/td>\n

\n

0,22%<\/span><\/span><\/p>\n<\/td>\n

\n

0,28%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u00d3<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

5 <\/span><\/span><\/td>\n\n

0,35%<\/span><\/span><\/p>\n<\/td>\n

\n

0,85%<\/span><\/span><\/p>\n<\/td>\n

\n

1,22%<\/span><\/span><\/p>\n<\/td>\n

\n

0,89%<\/span><\/span><\/p>\n<\/td>\n

\n

0,72%<\/span><\/span><\/p>\n<\/td>\n

\n

0,34%<\/span><\/span><\/p>\n<\/td>\n

\n

0,93%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u0106<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

6 <\/span><\/span><\/td>\n\n

0,19%<\/span><\/span><\/p>\n<\/td>\n

\n

0,40%<\/span><\/span><\/p>\n<\/td>\n

\n

0,47%<\/span><\/span><\/p>\n<\/td>\n

\n

0,26%<\/span><\/span><\/p>\n<\/td>\n

\n

0,51%<\/span><\/span><\/p>\n<\/td>\n

\n

0,31%<\/span><\/span><\/p>\n<\/td>\n

\n

0,29%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u0143<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

7 <\/span><\/span><\/td>\n\n

0,25%<\/span><\/span><\/p>\n<\/td>\n

\n

0,20%<\/span><\/span><\/p>\n<\/td>\n

\n

0,19%<\/span><\/span><\/p>\n<\/td>\n

\n

0,21%<\/span><\/span><\/p>\n<\/td>\n

\n

0,15%<\/span><\/span><\/p>\n<\/td>\n

\n

0,06%<\/span><\/span><\/p>\n<\/td>\n

\n

0,22%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\u0179<\/span><\/span><\/td>\n\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

9 <\/span><\/span><\/td>\n\n

0,09%<\/span><\/span><\/p>\n<\/td>\n

\n

0,06%<\/span><\/span><\/p>\n<\/td>\n

\n

0,10%<\/span><\/span><\/p>\n<\/td>\n

\n

0,07%<\/span><\/span><\/p>\n<\/td>\n

\n

0,07%<\/span><\/span><\/p>\n<\/td>\n

\n

0,04%<\/span><\/span><\/p>\n<\/td>\n

\n

0,12%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
polskie<\/span><\/span><\/td>\n<\/td>\n\n

10,20%<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

7,90%<\/span><\/span><\/p>\n<\/td>\n

\n

6,92%<\/span><\/span><\/p>\n<\/td>\n

\n

7,32%<\/span><\/span><\/p>\n<\/td>\n

\n

6,42%<\/span><\/span><\/p>\n<\/td>\n

\n

9,94%<\/span><\/span><\/p>\n<\/td>\n

\n

4,00%<\/span><\/span><\/p>\n<\/td>\n

\n

7,56%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

samo-
\ng\u0142oski<\/span><\/span><\/td>\n		\n

39,80%<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

44,59%<\/span><\/span><\/p>\n<\/td>\n

\n

41,74%<\/span><\/span><\/p>\n<\/td>\n

\n

40,75%<\/span><\/span><\/p>\n<\/td>\n

\n

41,63%<\/span><\/span><\/p>\n<\/td>\n

\n

42,05%<\/span><\/span><\/p>\n<\/td>\n

\n

43,40%<\/span><\/span><\/p>\n<\/td>\n

\n

42,16%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

W kolejnych kolumnach tabeli podane s\u0105: litera, liczba p\u0142ytek z t\u0105 liter\u0105 w zestawie Scrabble, udzia\u0142 procentowy w zestawie, punktacja tej litery, a nast\u0119pnie frekwencja liter na podstawie r\u00f3\u017cnych \u017ar\u00f3de\u0142:<\/p>\n

– OSPS \u2013 Oficjalny S\u0142ownik Polskiego Scrabblisty<\/p>\n

– NKJP \u2013 Narodowy Korpus J\u0119zyka Polskiego, opracowany przez IPI PAN<\/p>\n

– Litera \u2013 artyku\u0142 Barbary Kowalskiej w czasopi\u015bmie \u201eLitera\u201d [1]\n

– Krypto \u2013 strona http:\/\/practicalcryptography.com\/cryptanalysis\/letter-frequencies-various-languages\/<\/p>\n

– P\u0119tla \u2013 opowiadanie Marka H\u0142aski \u201eP\u0119tla\u201d [2]\n

– GG \u2013 zbi\u00f3r zapis\u00f3w rozm\u00f3w z komunikatora Gadu-Gadu [2]\n

– ZM \u2013 artyku\u0142 L. Zubrzyckiej z czasopisma Zastosowania Matematyki [3]\n

Dwa dolne wiersze zawieraj\u0105 procentowy udzia\u0142 \u201epolskich liter\u201d (\u0105, \u0107, \u0119, \u0142, \u0144, \u00f3, \u015b, \u017a, \u017c) oraz samog\u0142osek.<\/p>\n

Gar\u015b\u0107 informacji na temat \u017ar\u00f3de\u0142 (poza przedstawionym wcze\u015bniej artyku\u0142em z \u201eLitery\u201d). Wykorzysta\u0142em pierwsze wydanie OSPS, zawieraj\u0105ce ok. p\u00f3\u0142tora miliona wyraz\u00f3w, co daje ok. 15 milion\u00f3w liter.<\/p>\n

Narodowy Korpus J\u0119zyka Polskiego to zr\u00f3wnowa\u017cony j\u0119zykowo i tematycznie zbi\u00f3r polskich tekst\u00f3w. Lista \u017ar\u00f3de\u0142 korpusu zawiera nie tylko klasyk\u0119 literatury polskiej, ale te\u017c pras\u0119 codzienn\u0105 i specjalistyczn\u0105, nagrania rozm\u00f3w, teksty ulotne i internetowe. Warto doda\u0107, \u017ce wyniki z tej kolumny nie sumuj\u0105 si\u0119 do 100%, poniewa\u017c w statystyce uwzgl\u0119dnione s\u0105 tak\u017ce litery, nie wyst\u0119puj\u0105ce w j\u0119zyku polskim, a pojawiaj\u0105ce si\u0119 w nazwach w\u0142asnych, m.in. Q, V oraz X.<\/p>\n

Strona \u201ePracticalCryptography\u201d podaje statystyki wyst\u0119powania poszczeg\u00f3lnych liter, a tak\u017ce cz\u0119stotliwo\u015b\u0107 wyst\u0119powania najpopularniejszych zestaw\u00f3w dwuliterowych, trzyliterowych i czteroliterowych w r\u00f3\u017cnych j\u0119zykach. Statystyka dla j\u0119zyka polskiego zosta\u0142a opracowana na podstawie z\u0142o\u017conego z 90 milion\u00f3w znak\u00f3w korpusu j\u0119zyka polskiego z zasob\u00f3w uniwersytetu w Lipsku.<\/p>\n

Statystyki dla opowiadania \u201eP\u0119tla\u201d i zapisu rozm\u00f3w na Gadu-Gadu (po oko\u0142o 80 tysi\u0119cy liter) pochodz\u0105 z artyku\u0142u Marka Osiewicza i Krzysztofa Skibskiego z Uniwersytetu Adama Mickiewicza w Poznaniu. Autorzy por\u00f3wnali frekwencj\u0119 liter w dw\u00f3ch tekstach z XVI wieku z napisanym wsp\u00f3\u0142czesnym j\u0119zykiem (no mo\u017ce prawie wsp\u00f3\u0142czesnym, bo z roku 1956) opowiadaniem Marka H\u0142aski i zapisem rozm\u00f3w na Gadu Gadu (artyku\u0142 zosta\u0142 napisany w roku 2012, kiedy ten komunikator by\u0142 do\u015b\u0107 powszechnie u\u017cywany). Oczywi\u015bcie tych starych tekst\u00f3w nie bra\u0142em pod uwag\u0119. Nie tylko dlatego, \u017ce gra Scrabble powsta\u0142a znacznie p\u00f3\u017aniej ale przede wszystkim z powodu wyst\u0119powania w nich innego zestawu liter ni\u017c we wsp\u00f3\u0142czesnej polszczy\u017anie.<\/p>\n

Praca Zubrzyckiej mia\u0142a na celu przygotowanie za\u0142o\u017ce\u0144 do reformy systemu stenografii. Chodzi\u0142o o to, \u017ceby najcz\u0119\u015bciej wyst\u0119puj\u0105cym literom odpowiada\u0142y najszybsze do napisania znaki stenograficzne. Autorka analizowa\u0142a 4 teksty: polityczno-gospodarczy, naukowo-historyczny, handlowy i literacki. Ka\u017cdy z nich liczy\u0142 po oko\u0142o 6 tysi\u0119cy liter. Okaza\u0142o si\u0119, \u017ce rozk\u0142ad liter, szczeg\u00f3lnie tych rzadziej wyst\u0119puj\u0105cych, by\u0142 w nich bardzo r\u00f3\u017cny (np. w pierwszym tek\u015bcie litera \u0143 wyst\u0105pi\u0142a tylko raz, a \u0179 14 razy, za\u015b w ostatnim \u0143 by\u0142o 15, a \u0179 3. Dlatego wykorzysta\u0142em \u015bredni\u0105 dla tych czterech tekst\u00f3w, opart\u0105 na zbiorze 24 tysi\u0119cy liter.<\/p>\n

\"\"<\/a><\/p>\n

W polskiej wersji Scrabble 7 p\u0142ytek z woreczka mo\u017cna wyci\u0105gn\u0105\u0107 na 11.268.199 r\u00f3\u017cnych<\/strong> sposob\u00f3w. Specjalnie wyt\u0142u\u015bci\u0142em s\u0142owo r\u00f3\u017cnych, bo oczywi\u015bcie wz\u00f3r na kombinacje, okre\u015blaj\u0105cy liczb\u0119 mo\u017cliwych wybor\u00f3w 7 element\u00f3w ze 100, daje warto\u015b\u0107 du\u017co wy\u017csz\u0105 – ponad 1,6 x 1010<\/sup>. Ale, jak napisa\u0142em wcze\u015bniej, tylko niekt\u00f3re litery w zestawie Scrabble nie powtarzaj\u0105 si\u0119, a np. litera A wyst\u0119puje a\u017c w 9 egzemplarzach. Jak mo\u017cna otrzyma\u0107 liczb\u0119 r\u00f3\u017cnych rozwi\u0105za\u0144? Ot\u00f3\u017c ka\u017cdej literze, kt\u00f3ra wyst\u0119puje w zestawie tylko raz, przypisujemy wielomian 1+x, tym, co wyst\u0119puj\u0105 po dwa razy wielomian 1+x+x2 <\/sup>itd., a nast\u0119pnie wszystkie te wielomiany przez siebie mno\u017cymy (nie polecam robienia tego r\u0119cznie na kartce) czyli wykonujemy dzia\u0142anie:<\/p>\n

(1+x)9 <\/sup>(1+x+x2<\/sup>)7 <\/sup>(1+x+x<\/span>2<\/sup><\/span>+x<\/span>3<\/sup><\/span>)<\/span>7<\/sup><\/span> (1+x+x<\/span>2<\/sup><\/span>+x<\/span>3<\/sup><\/span>+x<\/span>4<\/sup><\/span>)<\/span>4<\/sup><\/span> (1+x+x<\/span>2<\/sup><\/span>+x<\/span>3<\/sup><\/span>+x<\/span>4<\/sup><\/span>+x<\/span>5<\/sup><\/span>)<\/span>2 <\/sup><\/span>(1+x+x<\/span>2<\/sup><\/span>+x<\/span>3<\/sup><\/span>+x<\/span>4<\/sup><\/span>+x<\/span>5<\/sup><\/span>+x<\/span>6<\/sup><\/span>)<\/span>(1+x+x<\/span>2<\/sup><\/span>+x<\/span>3<\/sup><\/span>+x<\/span>4<\/sup><\/span>+x<\/span>5<\/sup><\/span>+x<\/span>6<\/sup><\/span>+x<\/span>7<\/sup><\/span>) (1+x+ x<\/span>2<\/sup><\/span>+x<\/span>3<\/sup><\/span>+x<\/span>4<\/sup><\/span>+x<\/span>5<\/sup><\/span>+x<\/span>6<\/sup><\/span>+x<\/span>7<\/sup><\/span>+x<\/span>8<\/sup><\/span>)<\/span> (1+x+x2<\/sup>+x3<\/sup>+x4<\/sup>+x5<\/sup>+x6<\/sup>+x7<\/sup>+x8<\/sup>+x9<\/sup>)<\/p>\n

W rezultacie dostajemy d\u0142ugi wielomian stopnia 100, w kt\u00f3rym interesuje nas tylko jedna liczba \u2013 wsp\u00f3\u0142czynnik stoj\u0105cy przy wyrazie x7 <\/sup>(7 \u2013 bo tyle p\u0142ytek losujemy) i to jest w\u0142a\u015bnie podana wcze\u015bniej warto\u015b\u0107: 11.268.199. Oczywi\u015bcie dla ka\u017cdej wersji j\u0119zykowej ta warto\u015b\u0107 mo\u017ce by\u0107 inna, bo liczby liter w poszczeg\u00f3lnych alfabetach s\u0105 r\u00f3\u017cne, a w dodatku poszczeg\u00f3lne wydania mog\u0105 si\u0119 r\u00f3\u017cni\u0107 \u0142\u0105czn\u0105 liczb\u0105 p\u0142ytek.<\/p>\n

Intersuj\u0105cy mo\u017ce by\u0107 procentowy udzia\u0142 r\u00f3\u017cnych struktur w wylosowanym uk\u0142adzie p\u0142ytek, np. jak cz\u0119sto si\u0119 zdarza, \u017ce na ka\u017cdej z wylosowanych p\u0142ytek b\u0119dzie inna litera. W Tabeli 2 przedstawiam takie zestawienie dla kilku wersji j\u0119zykowych \u2013 polskiej, angielskiej, czeskiej i \u0142aci\u0144skiej. Dlaczego akurat te wybra\u0142em te j\u0119zyki? Ot\u00f3\u017c zestaw dla ka\u017cdego z nich sk\u0142ada si\u0119 ze 100 p\u0142ytek, wi\u0119c \u0142atwiej mo\u017cna dokona\u0107 por\u00f3wna\u0144, a zestaw czeski i \u0142aci\u0144ski maj\u0105 odpowiednio najwi\u0119cej i najmniej r\u00f3\u017cnych liter.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Rozk\u0142ad liter<\/td>\nPolski<\/span><\/span><\/td>\nAngielski<\/span><\/span><\/td>\nCzeski<\/span><\/span><\/td>\n\u0141aci\u0144ski<\/span><\/span><\/td>\n<\/td>\nPolski<\/span><\/span><\/td>\nAngielski<\/span><\/span><\/td>\nCzeski<\/span><\/span><\/td>\n\u0141aci\u0144ski<\/span><\/span><\/td>\n<\/tr>\n
\n

7
\n<\/span><\/span><\/p>\n<\/td>\n

\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

0<\/span><\/span><\/p>\n<\/td>\n

\n

7<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0000%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0001%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0000%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0010%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

61<\/span><\/span><\/p>\n<\/td>\n

\n

128<\/span><\/span><\/p>\n<\/td>\n

\n

182<\/span><\/span><\/p>\n<\/td>\n

\n

39<\/span><\/span><\/p>\n<\/td>\n

\n

140<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0011%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0057%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0001%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0193%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

52<\/span><\/span><\/p>\n<\/td>\n

\n

138<\/span><\/span><\/p>\n<\/td>\n

\n

147<\/span><\/span><\/p>\n<\/td>\n

\n

112<\/span><\/span><\/p>\n<\/td>\n

\n

144<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0012%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0046%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0003%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0198%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

511<\/span><\/span><\/p>\n<\/td>\n

\n

2976<\/span><\/span><\/p>\n<\/td>\n

\n

2275<\/span><\/span><\/p>\n<\/td>\n

\n

2964<\/span><\/span><\/p>\n<\/td>\n

\n

1520<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0264%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0711%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0073%<\/span><\/span><\/p>\n<\/td>\n

\n

0,2091%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

43<\/span><\/span><\/p>\n<\/td>\n

\n

160<\/span><\/span><\/p>\n<\/td>\n

\n

121<\/span><\/span><\/p>\n<\/td>\n

\n

136<\/span><\/span><\/p>\n<\/td>\n

\n

143<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0014%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0038%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0003%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0197%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

421<\/span><\/span><\/p>\n<\/td>\n

\n

7130<\/span><\/span><\/p>\n<\/td>\n

\n

5775<\/span><\/span><\/p>\n<\/td>\n

\n

8512<\/span><\/span><\/p>\n<\/td>\n

\n

3762<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0633%<\/span><\/span><\/p>\n<\/td>\n

\n

0,1805%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0208%<\/span><\/span><\/p>\n<\/td>\n

\n

0,5174%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

4111<\/span><\/span><\/p>\n<\/td>\n

\n

49600<\/span><\/span><\/p>\n<\/td>\n

\n

28600<\/span><\/span><\/p>\n<\/td>\n

\n

73112<\/span><\/span><\/p>\n<\/td>\n

\n

12540<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,4402%<\/span><\/span><\/p>\n<\/td>\n

\n

0,8938%<\/span><\/span><\/p>\n<\/td>\n

\n

0,1791%<\/span><\/span><\/p>\n<\/td>\n

\n

1,7248%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

331<\/span><\/span><\/p>\n<\/td>\n

\n

4216<\/span><\/span><\/p>\n<\/td>\n

\n

1650<\/span><\/span><\/p>\n<\/td>\n

\n

5814<\/span><\/span><\/p>\n<\/td>\n

\n

1729<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0374%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0516%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0142%<\/span><\/span><\/p>\n<\/td>\n

\n

0,2378%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

322<\/span><\/span><\/p>\n<\/td>\n

\n

4301<\/span><\/span><\/p>\n<\/td>\n

\n

2520<\/span><\/span><\/p>\n<\/td>\n

\n

6804<\/span><\/span><\/p>\n<\/td>\n

\n

2142<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,0382%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0788%<\/span><\/span><\/p>\n<\/td>\n

\n

0,0167%<\/span><\/span><\/p>\n<\/td>\n

\n

0,2946%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

3211<\/span><\/span><\/p>\n<\/td>\n

\n

181815<\/span><\/span><\/p>\n<\/td>\n

\n

75600<\/span><\/span><\/p>\n<\/td>\n

\n

354312<\/span><\/span><\/p>\n<\/td>\n

\n

43092<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

1,6135%<\/span><\/span><\/p>\n<\/td>\n

\n

2,3627%<\/span><\/span><\/p>\n<\/td>\n

\n

0,8677%<\/span><\/span><\/p>\n<\/td>\n

\n

5,9270%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

31111<\/span><\/span><\/p>\n<\/td>\n

\n

611320<\/span><\/span><\/p>\n<\/td>\n

\n

179400<\/span><\/span><\/p>\n<\/td>\n

\n

1480518<\/span><\/span><\/p>\n<\/td>\n

\n

67830<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

5,4252%<\/span><\/span><\/p>\n<\/td>\n

\n

5,6067%<\/span><\/span><\/p>\n<\/td>\n

\n

3,6258%<\/span><\/span><\/p>\n<\/td>\n

\n

9,3295%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

2221<\/span><\/span><\/p>\n<\/td>\n

\n

60720<\/span><\/span><\/p>\n<\/td>\n

\n

36960<\/span><\/span><\/p>\n<\/td>\n

\n

135198<\/span><\/span><\/p>\n<\/td>\n

\n

17442<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,5389%<\/span><\/span><\/p>\n<\/td>\n

\n

1,1551%<\/span><\/span><\/p>\n<\/td>\n

\n

0,3311%<\/span><\/span><\/p>\n<\/td>\n

\n

2,3990%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

22111<\/span><\/span><\/p>\n<\/td>\n

\n

1240620<\/span><\/span><\/p>\n<\/td>\n

\n

531300<\/span><\/span><\/p>\n<\/td>\n

\n

3425016<\/span><\/span><\/p>\n<\/td>\n

\n

165699<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

11,0099%<\/span><\/span><\/p>\n<\/td>\n

\n

16,6046%<\/span><\/span><\/p>\n<\/td>\n

\n

8,3879%<\/span><\/span><\/p>\n<\/td>\n

\n

22,7907%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

211111<\/span><\/span><\/p>\n<\/td>\n

\n

4833024<\/span><\/span><\/p>\n<\/td>\n

\n

1447160<\/span><\/span><\/p>\n<\/td>\n

\n

16696953<\/span><\/span><\/p>\n<\/td>\n

\n

294576<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

42,8908%<\/span><\/span><\/p>\n<\/td>\n

\n

45,2277%<\/span><\/span><\/p>\n<\/td>\n

\n

40,8908%<\/span><\/span><\/p>\n<\/td>\n

\n

40,5168%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

1111111<\/span><\/span><\/p>\n<\/td>\n

\n

4272048<\/span><\/span><\/p>\n<\/td>\n

\n

888030<\/span><\/span><\/p>\n<\/td>\n

\n

18643560<\/span><\/span><\/p>\n<\/td>\n

\n

116280<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

37,9124%<\/span><\/span><\/p>\n<\/td>\n

\n

27,7533%<\/span><\/span><\/p>\n<\/td>\n

\n

45,6580%<\/span><\/span><\/p>\n<\/td>\n

\n

15,9935%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
<\/td>\n\n

11268199<\/span><\/span><\/p>\n<\/td>\n

\n

3199724<\/span><\/span><\/p>\n<\/td>\n

\n

40833050<\/span><\/span><\/p>\n<\/td>\n

\n

727046<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

100,0000%<\/span><\/span><\/p>\n<\/td>\n

\n

100,0000%<\/span><\/span><\/p>\n<\/td>\n

\n

100,0000%<\/span><\/span><\/p>\n<\/td>\n

\n

100,0000%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Tabela 2<\/p>\n

W lewej kolumnie tabeli podany jest uk\u0142ad liter. Samo 7 oznacza wylosowanie siedmiu takich samych liter, 61 \u2013 sze\u015bciu takich samych i jednej innej itd. a\u017c do 1111111 czyli siedmiu r\u00f3\u017cnych liter. W nast\u0119pnych czterech kolumnach mamy liczb\u0119 mo\u017cliwo\u015bci dla ka\u017cdego j\u0119zyka. Np. w j\u0119zyku polskim mo\u017cna wylosowa\u0107 po 7 liter A, E oraz I, w angielskim po 7: A, E, I oraz O natomiast w czeskim zestawie \u017cadna litera nie wyst\u0119puje w 7 egzemplarzach. Ostatnie 4 kolumny to procentowy udzia\u0142 poszczeg\u00f3lnych uk\u0142ad\u00f3w liter. Jak wida\u0107 z tego zestawienia, tylko w j\u0119zyku czeskim najbardziej prawdopodobne jest wylosowanie siedmiu r\u00f3\u017cnych liter (a to dlatego, \u017ce w czeskim wydaniu Scrabble jest a\u017c 39 r\u00f3\u017cnych liter plus oczywi\u015bcie blanki). W trzech pozosta\u0142ych wersjach zazwyczaj gracz losuje zestaw, w kt\u00f3rym jedna litera si\u0119 powtarza. Oczywi\u015bcie w polskim wydaniu najcz\u0119\u015bciej powtarza si\u0119 litera A (raz na 10 losowa\u0144). Ale raz na 236 partii mo\u017cna na pocz\u0105tku gry wylosowa\u0107 dwa blanki.<\/p>\n

Istotnym zagadnieniem dla gracza jest proporcja samog\u0142osek i sp\u00f3\u0142g\u0142osek. Wed\u0142ug autor\u00f3w ksi\u0105\u017cki [4], \u201eidealny stojak\u201d powinien zawiera\u0107 od dw\u00f3ch do czterech samog\u0142osek. W Tabeli 3 mo\u017cna zobaczy\u0107, ile jest mo\u017cliwo\u015bci i jaki stanowi\u0105 procent uk\u0142ady z dan\u0105 liczb\u0105 samog\u0142osek (pionowo) i blank\u00f3w (poziomo).<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
<\/td>\n\n

0<\/span><\/span><\/p>\n<\/td>\n

\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

2<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0<\/span><\/span><\/p>\n<\/td>\n

\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

2<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

7<\/span><\/span><\/p>\n<\/td>\n

\n

15 380 937<\/span><\/span><\/p>\n<\/td>\n

\n

0<\/span><\/span><\/p>\n<\/td>\n

\n

0<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,096%<\/span><\/span><\/p>\n<\/td>\n

\n

0,000%<\/span><\/span><\/p>\n<\/td>\n

\n

0,000%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

6<\/span><\/span><\/p>\n<\/td>\n

\n

192 494 757<\/span><\/span><\/p>\n<\/td>\n

\n

6 525 246<\/span><\/span><\/p>\n<\/td>\n

\n

0<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

1,203%<\/span><\/span><\/p>\n<\/td>\n

\n

0,041%<\/span><\/span><\/p>\n<\/td>\n

\n

0,000%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

5<\/span><\/span><\/p>\n<\/td>\n

\n

985 120 227<\/span><\/span><\/p>\n<\/td>\n

\n

67 939 326<\/span><\/span><\/p>\n<\/td>\n

\n

575 757<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

6,154%<\/span><\/span><\/p>\n<\/td>\n

\n

0,424%<\/span><\/span><\/p>\n<\/td>\n

\n

0,004%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

2 673 897 759<\/span><\/span><\/p>\n<\/td>\n

\n

281 462 922<\/span><\/span><\/p>\n<\/td>\n

\n

4 852 809<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

16,704%<\/span><\/span><\/p>\n<\/td>\n

\n

1,758%<\/span><\/span><\/p>\n<\/td>\n

\n

0,030%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

4 159 396 514<\/span><\/span><\/p>\n<\/td>\n

\n

594 199 502<\/span><\/span><\/p>\n<\/td>\n

\n

15 636 829<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

25,984%<\/span><\/span><\/p>\n<\/td>\n

\n

3,712%<\/span><\/span><\/p>\n<\/td>\n

\n

0,098%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

3 709 732 026<\/span><\/span><\/p>\n<\/td>\n

\n

674 496 732<\/span><\/span><\/p>\n<\/td>\n

\n

24 089 169<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

23,175%<\/span><\/span><\/p>\n<\/td>\n

\n

4,214%<\/span><\/span><\/p>\n<\/td>\n

\n

0,150%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

1 757 241 486<\/span><\/span><\/p>\n<\/td>\n

\n

390 498 108<\/span><\/span><\/p>\n<\/td>\n

\n

17 749 914<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

10,978%<\/span><\/span><\/p>\n<\/td>\n

\n

2,439%<\/span><\/span><\/p>\n<\/td>\n

\n

0,111%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

0<\/span><\/span><\/p>\n<\/td>\n

\n

341 149 446<\/span><\/span><\/p>\n<\/td>\n

\n

90 114 948<\/span><\/span><\/p>\n<\/td>\n

\n

5 006 386<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

2,131%<\/span><\/span><\/p>\n<\/td>\n

\n

0,563%<\/span><\/span><\/p>\n<\/td>\n

\n

0,031%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n\n

86,424%<\/span><\/span><\/p>\n<\/td>\n

\n

13,152%<\/span><\/span><\/p>\n<\/td>\n

\n

0,424%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Tabela 3<\/p>\n

Jak wida\u0107, najcz\u0119\u015bciej zdarza si\u0119 wylosowanie trzech samog\u0142osek (gdy w uk\u0142adzie nie ma blanka) albo dw\u00f3ch (gdy jest jeden lub dwa blanki). A jak to wygl\u0105da \u201eod drugiej strony\u201d czyli jaki procent stanowi\u0105 samog\u0142oski w wyrazach r\u00f3\u017cnej d\u0142ugo\u015bci? W Tabeli 4, opracowanej na podstawie S\u0142ownika Scrabblisty, kolejne kolumny odpowiadaj\u0105 wyrazom r\u00f3\u017cnej d\u0142ugo\u015bci, a kolejne wiersze liczbom samog\u0142osek.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
<\/td>\n\n

4<\/span><\/span><\/p>\n<\/td>\n

\n

5<\/span><\/span><\/p>\n<\/td>\n

\n

6<\/span><\/span><\/p>\n<\/td>\n

\n

7<\/span><\/span><\/p>\n<\/td>\n

\n

8<\/span><\/span><\/p>\n<\/td>\n

\n

9<\/span><\/span><\/p>\n<\/td>\n

\n

10<\/span><\/span><\/p>\n<\/td>\n

\n

11<\/span><\/span><\/p>\n<\/td>\n

\n

12<\/span><\/span><\/p>\n<\/td>\n

\n

13<\/span><\/span><\/p>\n<\/td>\n

\n

14<\/span><\/span><\/p>\n<\/td>\n

\n

15<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

1<\/span><\/span><\/p>\n<\/td>\n

\n

31,36%<\/span><\/span><\/p>\n<\/td>\n

\n

6,09%<\/span><\/span><\/p>\n<\/td>\n

\n

1,04%<\/span><\/span><\/p>\n<\/td>\n

\n

0,15%<\/span><\/span><\/p>\n<\/td>\n

\n

0,02%<\/span><\/span><\/p>\n<\/td>\n

\n

0,00%<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
\n

2<\/span><\/span><\/p>\n<\/td>\n

\n

66,64%<\/span><\/span><\/p>\n<\/td>\n

\n

78,48%<\/span><\/span><\/p>\n<\/td>\n

\n

47,40%<\/span><\/span><\/p>\n<\/td>\n

\n

18,51%<\/span><\/span><\/p>\n<\/td>\n

\n

5,58%<\/span><\/span><\/p>\n<\/td>\n

\n

1,42%<\/span><\/span><\/p>\n<\/td>\n

\n

0,25%<\/span><\/span><\/p>\n<\/td>\n

\n

0,03%<\/span><\/span><\/p>\n<\/td>\n

\n

0,00%<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
\n

3<\/span><\/span><\/p>\n<\/td>\n

\n

2,00%<\/span><\/span><\/p>\n<\/td>\n

\n

15,39%<\/span><\/span><\/p>\n<\/td>\n

\n

49,14%<\/span><\/span><\/p>\n<\/td>\n

\n

66,37%<\/span><\/span><\/p>\n<\/td>\n

\n

51,93%<\/span><\/span><\/p>\n<\/td>\n

\n

28,22%<\/span><\/span><\/p>\n<\/td>\n

\n

10,94%<\/span><\/span><\/p>\n<\/td>\n

\n

3,58%<\/span><\/span><\/p>\n<\/td>\n

\n

0,69%<\/span><\/span><\/p>\n<\/td>\n

\n

0,09%<\/span><\/span><\/p>\n<\/td>\n

\n

0,02%<\/span><\/span><\/p>\n<\/td>\n

\n

0,00%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

4<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n\n

0,03%<\/span><\/span><\/p>\n<\/td>\n

\n

2,41%<\/span><\/span><\/p>\n<\/td>\n

\n

14,82%<\/span><\/span><\/p>\n<\/td>\n

\n

39,66%<\/span><\/span><\/p>\n<\/td>\n

\n

55,39%<\/span><\/span><\/p>\n<\/td>\n

\n

47,05%<\/span><\/span><\/p>\n<\/td>\n

\n

27,73%<\/span><\/span><\/p>\n<\/td>\n

\n

9,43%<\/span><\/span><\/p>\n<\/td>\n

\n

2,20%<\/span><\/span><\/p>\n<\/td>\n

\n

0,59%<\/span><\/span><\/p>\n<\/td>\n

\n

0,24%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

5<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n\n

0,15%<\/span><\/span><\/p>\n<\/td>\n

\n

2,81%<\/span><\/span><\/p>\n<\/td>\n

\n

14,56%<\/span><\/span><\/p>\n<\/td>\n

\n

37,42%<\/span><\/span><\/p>\n<\/td>\n

\n

48,75%<\/span><\/span><\/p>\n<\/td>\n

\n

40,06%<\/span><\/span><\/p>\n<\/td>\n

\n

20,40%<\/span><\/span><\/p>\n<\/td>\n

\n

6,22%<\/span><\/span><\/p>\n<\/td>\n

\n

2,60%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

6<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n\n

0,00%<\/span><\/span><\/p>\n<\/td>\n

\n

0,41%<\/span><\/span><\/p>\n<\/td>\n

\n

4,31%<\/span><\/span><\/p>\n<\/td>\n

\n

18,90%<\/span><\/span><\/p>\n<\/td>\n

\n

41,24%<\/span><\/span><\/p>\n<\/td>\n

\n

44,45%<\/span><\/span><\/p>\n<\/td>\n

\n

29,67%<\/span><\/span><\/p>\n<\/td>\n

\n

15,29%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

7<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n\n

0,03%<\/span><\/span><\/p>\n<\/td>\n

\n

1,02%<\/span><\/span><\/p>\n<\/td>\n

\n

8,36%<\/span><\/span><\/p>\n<\/td>\n

\n

29,70%<\/span><\/span><\/p>\n<\/td>\n

\n

47,48%<\/span><\/span><\/p>\n<\/td>\n

\n

45,51%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

8<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n\n

0,23%<\/span><\/span><\/p>\n<\/td>\n

\n

3,13%<\/span><\/span><\/p>\n<\/td>\n

\n

15,37%<\/span><\/span><\/p>\n<\/td>\n

\n

32,81%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

9<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n\n

0,03%<\/span><\/span><\/p>\n<\/td>\n

\n

0,65%<\/span><\/span><\/p>\n<\/td>\n

\n

3,51%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n

\n

10<\/span><\/span><\/p>\n<\/td>\n

<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n\n

0,03%<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Tabela 4<\/p>\n

UWAGA: 0,00% oznacza, \u017ce takich wyraz\u00f3w jest bardzo ma\u0142o (np. w S\u0142owniku Scrabblisty jest tylko 6 wyraz\u00f3w dziewi\u0119cioliterowych z jedn\u0105 samog\u0142osk\u0105) natomiast puste pole oznacza, \u017ce nie ma np. \u017cadnego wyrazu dziesi\u0119cioliterowego z jedn\u0105 samog\u0142osk\u0105.<\/p>\n

Jak wida\u0107, najcz\u0119\u015bciej samog\u0142oski stanowi\u0105 nieco mniej ni\u017c po\u0142ow\u0119 liter w wyrazie. Oczywi\u015bcie s\u0105 wyrazy o znacznie zaburzonych proporcjach. Z nadmiarem samog\u0142osek: AUDIO, AUCIE, OAZIE, AUREOLE, AUDIONIE, AUREUSIE, OCIENIENIE, AEROBIOLOGIA, NIEUOSOBIENIE, AUTOUTLENIANIE, NIEUNIESIENIAMI albo z nadmiarem sp\u00f3\u0142g\u0142osek: BRZUCH, CHRZEST, WRZESZCZ, CHRZ\u0104SZCZ, FECHTMISTRZ, CHRZ\u0104SZCZACH, BEZWZGL\u0118DNO\u015a\u0106, BEZKSZTA\u0141TNO\u015a\u0106, WSZECHSTRONNO\u015a\u0106.<\/p>\n

Jak wspomnia\u0142em wcze\u015bniej, na stronie na temat kryptografii podane s\u0105 najpopularniejsze uk\u0142ady dw\u00f3ch, trzech i czterech kolejnych liter. Dla kogo\u015b, kto odszyfrowuje teksty, takie informacje s\u0105 bardzo istotne. Ale i dla Scrabblisty cenna mo\u017ce by\u0107 wiedza o tym, kt\u00f3re litery do siebie pasuj\u0105.<\/p>\n

Jak nietrudno zgadn\u0105\u0107, najpopularniejsze pary kolejnych liter (w artykule [2] nazywane diadami, a na stronie kryptolog\u00f3w \u2013 bigramami) to w j\u0119zyku polskim IE oraz NI. Na nast\u0119pnych miejscach kolejno\u015b\u0107 jest r\u00f3\u017cna. Na podstawie S\u0142ownika Scrabblisty reszta pierwszej dziesi\u0105tki to: OW, AN, WA, A\u0141, ZA, BY, CH, CI; wed\u0142ug strony kryptograficznej: OW, ST, NA, AN, PO, CH, CZ, RZ; w opowiadaniu \u201eP\u0119tla\u201d: RZ, PO, A\u0141, SZ, CZ, WI, ZY, ZE; a na GADU-GADU: ZE, NA, CZ, TA, TO, AK, SZ, PO.<\/p>\n

W Tabeli 5 dla ka\u017cdej litery poda\u0142em (na podstawie S\u0142ownika Scrabblisty) po 5 liter wyst\u0119puj\u0105cych po niej i przed ni\u0105.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
<\/td>\nPo literze z lewej kolumny<\/td>\nPrzed liter\u0105 z lewej kolumny<\/td>\n<\/tr>\n
A<\/td>\nN, \u0141, C, M, J<\/td>\nW, Z, I, N, \u0141<\/td>\n<\/tr>\n
\u0104<\/td>\nC, D, G, \u0141, T<\/td>\nJ, N, Z, I, C<\/td>\n<\/tr>\n
B<\/td>\nY, I, A, R, O<\/td>\nO, A, Y, \u0141, I<\/td>\n<\/tr>\n
C<\/td>\nH, I, Z, E, Y<\/td>\n\u0104, A, Y, \u015a, E<\/td>\n<\/tr>\n
\u0106<\/td>\nW, M, K, C, P<\/td>\nA, \u015a, I, \u0118, E<\/td>\n<\/tr>\n
D<\/td>\nO, Z, A, R, E<\/td>\nO, A, E, U, Z<\/td>\n<\/tr>\n
E<\/td>\nM, N, R, J, P<\/td>\nI, Z, N, C, L<\/td>\n<\/tr>\n
\u0118<\/td>\nC, T, \u0141, D, B<\/td>\nI, N, R, Z, J<\/td>\n<\/tr>\n
F<\/td>\nI, A, O, R, E<\/td>\nE, A, O, U, Y<\/td>\n<\/tr>\n
G<\/td>\nO, A, R, N, I<\/td>\nE, O, A, Z, Y<\/td>\n<\/tr>\n
H<\/td>\nA, O, U, N, E<\/td>\nC, E, O, A, Z<\/td>\n<\/tr>\n
I<\/td>\nE, A, O, \u0141, N<\/td>\nN, C, M, L, W<\/td>\n<\/tr>\n
J<\/td>\n\u0104, A, E, C, M<\/td>\nA, E, U, I, O<\/td>\n<\/tr>\n
K<\/td>\nA, O, I, R, U<\/td>\nS, E, A, O, Y<\/td>\n<\/tr>\n
L<\/td>\nI, A, E, O, U<\/td>\nA, E, O, I, U<\/td>\n<\/tr>\n
\u0141<\/td>\nA, O, Y, B, E<\/td>\nA, I, Y, P, S<\/td>\n<\/tr>\n
M<\/td>\nI, Y, U, A, O<\/td>\nY, E, A, O, \u015a<\/td>\n<\/tr>\n
N<\/td>\nI, A, O, E, Y<\/td>\nA, O, E, I, Z<\/td>\n<\/tr>\n
\u0143<\/td>\nS, C, M, K, \u017b<\/td>\nA, E, I, O, Y<\/td>\n<\/tr>\n
O<\/td>\nW, D, N, M, B<\/td>\nP, R, \u0141, K, G<\/td>\n<\/tr>\n
\u00d3<\/td>\nW, \u0141, R, C, \u017b<\/td>\nR, P, K, \u0141, T<\/td>\n<\/tr>\n
P<\/td>\nO, R, I, A, L<\/td>\nE, O, A, S, U<\/td>\n<\/tr>\n
R<\/td>\nZ, O, A, Y, U<\/td>\nA, E, P, O, T<\/td>\n<\/tr>\n
S<\/td>\nZ, K, T, I, A<\/td>\nA, E, O, U, Y<\/td>\n<\/tr>\n
\u015a<\/td>\nC, M, L, N, W<\/td>\nY, O, I, E, A<\/td>\n<\/tr>\n
T<\/td>\nA, O, R, Y, E<\/td>\nS, A, O, E, Y<\/td>\n<\/tr>\n
U<\/td>\nJ, S, R, C, P<\/td>\nM, K, R, I, T<\/td>\n<\/tr>\n
W<\/td>\nA, Y, I, E, O<\/td>\nO, E, A, Y, \u00d3<\/td>\n<\/tr>\n
Y<\/td>\nM, \u015a, C, B, W<\/td>\nB, W, M, N, Z<\/td>\n<\/tr>\n
Z<\/td>\nA, E, Y, O, I<\/td>\nR, C, S, D, O<\/td>\n<\/tr>\n
\u0179<\/td>\nN, W, L, C, M<\/td>\nD, E, A, U, O<\/td>\n<\/tr>\n
\u017b<\/td>\nE, A, Y, O, N<\/td>\nE, Y, O, A, J<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Tabela 5<\/p>\n

Jak \u0142atwo zgadn\u0105\u0107, po samog\u0142oskach najcz\u0119\u015bciej wyst\u0119puj\u0105 sp\u00f3\u0142g\u0142oski i na odwr\u00f3t. Ale s\u0105 pewne wyj\u0105tki od tej regu\u0142y. W\u015br\u00f3d samog\u0142osek litera I, po kt\u00f3rej najcz\u0119\u015bciej wyst\u0119puj\u0105 inne samog\u0142oski, a w\u015br\u00f3d sp\u00f3\u0142g\u0142osek C, R i S oraz wszystkie sp\u00f3\u0142g\u0142oski zmi\u0119kczone (\u0106, \u0143, \u015a, \u0179).<\/p>\n

Je\u015bli chodzi o zagadnienie odwrotne, to powy\u017csza prawid\u0142owo\u015b\u0107 te\u017c zachodzi i te\u017c jest kilka od niej wyj\u0105tk\u00f3w. Przed literami E i \u0118 najcz\u0119\u015bciej wyst\u0119puje litera I, przed K i T \u2013 S, przed Z \u2013 R, a przed \u0179 \u2013 D. Ciekawa jest sytuacja z liter\u0105 H. Mniej wi\u0119cej w 6% przypadk\u00f3w pojawia si\u0119 na pocz\u0105tku wyrazu, w 88% po literze C, a w 6% po wszystkich innych literach \u0142\u0105cznie. Druga w kolejno\u015bci para EH wyst\u0119puje ponad 60 razy rzadziej ni\u017c CH.<\/p>\n

 <\/p>\n

Bibliografia:<\/p>\n[1] Barbara Kowalska \u201eCz\u0119sto\u015b\u0107 wyst\u0119powania liter w j\u0119zyku polskim\u201d, Litera, Rok II, nr 13-5\/1967<\/p>\n[2] Marek Osiewicz, Krzysztof Skibski \u201eStruktura statystyczna polskiego systemu graficznego.<\/p>\n

Analiza historyczno-por\u00f3wnawcza\u201d, Pozna\u0144skie Studia Polonistyczne, Seria J\u0119zykoznawcza t. 19 (39), z. 2, 2012<\/p>\n[3] L. Zubrzycka \u201eO wyznaczaniu system\u00f3w stenograficznych\u201d, Zastosowania Matematyki, XI, 3, 1970<\/p>\n[4] Wojciech Usakiewicz, Micha\u0142 Derlacki \u201eGrajmy w Scrabble\u201d, Wydawnictwo KLEKS, Bielsko-Bia\u0142a 1998<\/p>\n

\"\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Na pierwszy rzut oka Scrabble to gra humanistyczna, w kt\u00f3rej decyduj\u0105c\u0105 rol\u0119 odgrywa zas\u00f3b s\u0142ownictwa, a matematyka sprowadza si\u0119 do prostych dzia\u0142a\u0144 \u2013 sumowania punkt\u00f3w za litery i ewentualnie mno\u017cenia, gdy uk\u0142adany wyraz przechodzi przez pole premii literowej lub s\u0142ownej. W tym tek\u015bcie postaram si\u0119 wykaza\u0107, \u017ce do dok\u0142adniejszej analizy gry matematyka jest przydatna, a …<\/p>\n","protected":false},"author":101,"featured_media":806981,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"cybocfi_hide_featured_image":"","footnotes":""},"categories":[2,184],"tags":[381],"post_folder":[],"class_list":["post-806804","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cykle","category-matematyka-w-grach","tag-scrabble"],"_links":{"self":[{"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/posts\/806804","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/users\/101"}],"replies":[{"embeddable":true,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/comments?post=806804"}],"version-history":[{"count":0,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/posts\/806804\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/media\/806981"}],"wp:attachment":[{"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/media?parent=806804"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/categories?post=806804"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/tags?post=806804"},{"taxonomy":"post_folder","embeddable":true,"href":"https:\/\/wp4wpuw.pedagog.uw.edu.pl\/test\/wp-json\/wp\/v2\/post_folder?post=806804"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}