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1 | unfinished:Group | ||
2 | <p> | ||
3 | Fill in the grid with the letters shown to the top and left of it, so | ||
4 | that the full grid is a valid | ||
5 | <a href="http://en.wikipedia.org/wiki/Cayley_table">Cayley table</a> | ||
6 | for a | ||
7 | <a href="http://en.wikipedia.org/wiki/Group_(mathematics)">group</a>. | ||
8 | <p> | ||
9 | If you don't already know what a group is, I don't really recommend | ||
10 | trying to play this game. But if you want to try anyway, the above is | ||
11 | equivalent to saying that the following conditions must be satisfied: | ||
12 | <ul> | ||
13 | <li> | ||
14 | <strong>Latin square</strong>. Every row and column must contain | ||
15 | exactly one of each letter. | ||
16 | <li> | ||
17 | <strong>Identity</strong>. There must be some letter <i>e</i> such | ||
18 | that, for all <i>a</i>, the letter in row <i>e</i> column <i>a</i> and | ||
19 | the one in row <i>a</i> column <i>e</i> are both <i>a</i>. In the | ||
20 | default mode, this letter is always <i>e</i> and its row and column | ||
21 | are filled in for you; by reconfiguring the game using the Type menu, | ||
22 | you can select a mode in which you have to work out which letter is | ||
23 | the identity. | ||
24 | <li> | ||
25 | <strong>Inverses</strong>. For every letter <i>a</i>, there must be | ||
26 | some letter <i>b</i> (which may sometimes be the same letter | ||
27 | as <i>a</i>) such that the letters in row <i>a</i> column <i>b</i> and | ||
28 | in row <i>b</i> column <i>a</i> are both the identity letter (as | ||
29 | defined above). | ||
30 | <li> | ||
31 | <strong>Associativity</strong>. For every combination of | ||
32 | letters <i>a</i>, <i>b</i>, and <i>c</i>, denote the letter in | ||
33 | row <i>a</i> column <i>b</i> by <i>d</i>, and the one in row <i>b</i> | ||
34 | column <i>c</i> by <i>e</i>. Then the letters in row <i>d</i> | ||
35 | column <i>c</i> and in row <i>a</i> column <i>e</i> must be the same. | ||
36 | </ul> | ||
37 | <p> | ||
38 | To place a letter, click in a square to select it, then type the | ||
39 | letter on the keyboard. To erase a letter, click to select a square | ||
40 | and then press Backspace. | ||
41 | <p> | ||
42 | Right-click in a square and then type a letter to add or remove the | ||
43 | number as a pencil mark, indicating letters that you think | ||
44 | <em>might</em> go in that square. | ||
45 | <p> | ||
46 | You can rearrange the order of elements in the rows and columns by | ||
47 | dragging the column or row headings back and forth. (The rows and | ||
48 | columns will stay in sync with each other.) Also, | ||
49 | left-clicking <em>between</em> two row or column headings will add or | ||
50 | remove a thick line between those two rows and the corresponding pair | ||
51 | of columns (which is useful if you're considering a subgroup and its | ||
52 | cosets). | ||