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authormzuenni <mzuenni@users.noreply.github.com>2024-07-28 22:54:40 +0200
committerGitHub <noreply@github.com>2024-07-28 22:54:40 +0200
commit8d11c6c8213f46f0fa19826917c255edd5d43cb1 (patch)
tree96d75baff33d5a04b5a60f1a41f514a26c716874 /math/tables/nim.tex
parent8c33b4e0d3030cfed17fc64b4fe41133339f6d87 (diff)
Test (#4)
* update * moved content in subdir * rename file * add test setup * add test setup * add github action * automaticly test all cpp files * timeout after 10s * setulimit and dont zero memory * test build pdf * install latexmk * update * update * ngerman * fonts * removed old code * add first test * added tests * test in sorted order * more tests * simplified test * more tests * fix suffix tree * fixes and improvements * done ust lst directly * fix swap * add links to pdf * fix constants * add primorial * add comment * various improvements * more tests * added missing stuf * more tests * fix tests * more tests * more tests * more tests * fix recursion? * test trie * more tests * only use python temporarily for listings * only use python temporarily for listings * more tests * fix longestCommonSubstring * more tests * more tests * made code more similiar * fix? * more tests * more tests * more tests * add ahoCorasick test + limit 4GB stack size * more tests * fix test * add additional test * more tests * more tests * fix? * better fix * fix virtual tree * more tests * more tests * recursive closest pair * more tests * decrease limit * new tests * more tests * fix name * more tests * add test * new test * more tests * more tests * more tests * more tests * new test and content * new code * new code * larger tests * fix and test * new test * new test * update pdf * remove comments * new test * more tests * more testcases * more tests * increased limit * more tests * more tests * more tests * new tests * more tests * shortened code * new test * add basic tests for bigint * more tests * removed old files * new test * ignore some files * more auto more ccw * fix test * more tests * fix * new tests * more tests * more tests * stronger test * actually verify delaunay... * more tests * fix header * more tests * run tests parallel? * test parralel? * add --missing * separate workflows * test * is the pdf checked? * separate workflows * fix workflow * more workflows --------- Co-authored-by: Yidi <noob999noob999@gmail.com>
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-\begin{tabularx}{\linewidth}{|p{0.37\linewidth}|X|}
- \hline
- \multicolumn{2}{|c|}{Nim-Spiele (\ding{182} letzter gewinnt (normal), \ding{183} letzter verliert)} \\
- \hline
- Beschreibung &
- Strategie \\
- \hline
-
- $M = [\mathit{pile}_i]$\newline
- $[x] := \{1, \ldots, x\}$&
- $\mathit{SG} = \oplus_{i = 1}^n \mathit{pile}_i$\newline
- \ding{182} Nimm von einem Stapel, sodass $\mathit{SG}$ $0$ wird.\newline
- \ding{183} Genauso.
- Außer: Bleiben nur noch Stapel der Größe $1$, erzeuge ungerade Anzahl solcher Stapel.\\
- \hline
-
- $M = \{a^m \mid m \geq 0\}$ &
- $a$ ungerade: $\mathit{SG}_n = n \% 2$\newline
- $a$ gerade:\newline
- $\mathit{SG}_n = 2$, falls $n \equiv a \bmod (a + 1) $\newline
- $\mathit{SG}_n = n \% (a + 1) \% 2$, sonst.\\
- \hline
-
- $M_{\text{\ding{172}}} = \left[\frac{\mathit{pile}_i}{2}\right]$\newline
- $M_{\text{\ding{173}}} =
- \left\{\left\lceil\frac{\mathit{pile}_i}{2}\right\rceil,~
- \mathit{pile}_i\right\}$ &
- \ding{172}
- $\mathit{SG}_{2n} = n$,
- $\mathit{SG}_{2n+1} = \mathit{SG}_n$\newline
- \ding{173}
- $\mathit{SG}_0 = 0$,
- $\mathit{SG}_n = [\log_2 n] + 1$ \\
- \hline
-
- $M_{\text{\ding{172}}} = \text{Teiler von $\mathit{pile}_i$}$\newline
- $M_{\text{\ding{173}}} = \text{echte Teiler von $\mathit{pile}_i$}$ &
- \ding{172}
- $\mathit{SG}_0 = 0$,
- $\mathit{SG}_n = \mathit{SG}_{\text{\ding{173},n}} + 1$\newline
- \ding{173}
- $\mathit{ST}_1 = 0$,
- $\mathit{SG}_n = \text{\#Nullen am Ende von $n_{bin}$}$\\
- \hline
-
- $M_{\text{\ding{172}}} = [k]$\newline
- $M_{\text{\ding{173}}} = S$, ($S$ endlich)\newline
- $M_{\text{\ding{174}}} = S \cup \{\mathit{pile}_i\}$ &
- $\mathit{SG}_{\text{\ding{172}}, n} = n \bmod (k + 1)$\newline
- \ding{182} Niederlage bei $\mathit{SG} = 0$\newline
- \ding{183} Niederlage bei $\mathit{SG} = 1$\newline
- $\mathit{SG}_{\text{\ding{174}}, n} = \mathit{SG}_{\text{\ding{173}}, n} + 1$\\
- \hline
-
- \multicolumn{2}{|l|}{
- Für jedes endliche $M$ ist $\mathit{SG}$ eines Stapels irgendwann periodisch.
- } \\
- \hline
-
- \textsc{Moore}'s Nim:\newline
- Beliebige Zahl von maximal $k$ Stapeln. &
- \ding{182}
- Schreibe $\mathit{pile}_i$ binär.
- Addiere ohne Übertrag zur Basis $k + 1$.
- Niederlage, falls Ergebnis gleich 0.\newline
- \ding{183}
- Wenn alle Stapel $1$ sind:
- Niederlage, wenn $n \equiv 1 \bmod (k + 1)$.
- Sonst wie in \ding{182}.\\
- \hline
-
- Staircase Nim:\newline
- $n$ Stapel in einer Reihe.
- Beliebige Zahl von Stapel $i$ nach Stapel $i-1$. &
- Niederlage, wenn Nim der ungeraden Spiele verloren ist:\newline
- $\oplus_{i = 0}^{(n - 1) / 2} \mathit{pile}_{2i + 1} = 0$\\
- \hline
-
- \textsc{Lasker}'s Nim:\newline
- Zwei mögliche Züge:\newline
- 1) Nehme beliebige Zahl.\newline
- 2) Teile Stapel in zwei Stapel (ohne Entnahme).&
- $\mathit{SG}_n = n$, falls $n \equiv 1,2 \bmod 4$\newline
- $\mathit{SG}_n = n + 1$, falls $n \equiv 3 \bmod 4$\newline
- $\mathit{SG}_n = n - 1$, falls $n \equiv 0 \bmod 4$\\
- \hline
-
- \textsc{Kayles}' Nim:\newline
- Zwei mögliche Züge:\newline
- 1) Nehme beliebige Zahl.\newline
- 2) Teile Stapel in zwei Stapel (mit Entnahme).&
- Berechne $\mathit{SG}_n$ für kleine $n$ rekursiv.\newline
- $n \in [72,83]: \quad 4, 1, 2, 8, 1, 4, 7, 2, 1, 8, 2, 7$\newline
- Periode ab $n = 72$ der Länge $12$.\\
- \hline
-\end{tabularx}