{"status": "success", "data": {"description_md": "A *Fano table* is a table with three columns where\n\n- each entry is an integer taken from the list $1, 2, 3, \\ldots, n$, and\n- each row contains three different integers, and\n- for each possible pair of distinct integers from the list $1, 2, 3, \\ldots, n$, there is exactly one row that contains both of these integers.\n\nThe number of rows in the table will depend on the value of $n$. For example, the table shown is a Fano table with $n = 7$. (Notice that 2 and 6 appear in the same row only once, as does every other possible pair of the numbers $1, 2, 3, 4, 5, 6, 7$.) For how many values of $n$ with $3 \\le n \\le 12$ can a Fano table be created?\n\n| 1 | 2 | 4 |\n| 2 | 3 | 5 |\n| 3 | 4 | 6 |\n| 4 | 5 | 7 |\n| 5 | 6 | 1 |\n| 6 | 7 | 2 |\n| 7 | 1 | 3 |\n\n$\\text{(A)}\\ 2\\qquad\\text{(B)}\\ 3\\qquad\\text{(C)}\\ 5\\qquad\\text{(D)}\\ 6\\qquad\\text{(E)}\\ 7$\n\n___\n\n**[Cayley](https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf), by [CEMC](https://cemc.uwaterloo.ca/), is licensed under the [CC BY-NC 3.0 DEED](https://creativecommons.org/licenses/by-nc/3.0/). They are not affiliated with or endorse TopsOJ in any way.**\n", "description_html": "

A *Fano table* is a table with three columns where

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- each entry is an integer taken from the list 1, 2, 3, \\ldots, n, and\n- each row contains three different integers, and\n- for each possible pair of distinct integers from the list 1, 2, 3, \\ldots, n, there is exactly one row that contains both of these integers.

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The number of rows in the table will depend on the value of n. For example, the table shown is a Fano table with n = 7. (Notice that 2 and 6 appear in the same row only once, as does every other possible pair of the numbers 1, 2, 3, 4, 5, 6, 7.) For how many values of n with 3 \\le n \\le 12 can a Fano table be created?

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| 1 | 2 | 4 |\n| 2 | 3 | 5 |\n| 3 | 4 | 6 |\n| 4 | 5 | 7 |\n| 5 | 6 | 1 |\n| 6 | 7 | 2 |\n| 7 | 1 | 3 |

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\\text{(A)}\\ 2\\qquad\\text{(B)}\\ 3\\qquad\\text{(C)}\\ 5\\qquad\\text{(D)}\\ 6\\qquad\\text{(E)}\\ 7

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\n

Cayley, by CEMC, is licensed under the CC BY-NC 3.0 DEED. They are not affiliated with or endorse TopsOJ in any way.

\n", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2011 Cayley Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/11_cayley_p24", "prev": "/problem/11_cayley_p22"}}