Suppose $n\in\mathbb{N}.$ Partition $\{1,2,3,\ldots,2n\}$ into $n$ pairs. Is it true that the maximum pair product is $\geq n^2 ?\ $ Or $\geq n(n+1)?$

score 294 · Answer 7 · 2026-03-26 12:35:06

294

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Proving Pigeonhole Principle is Unsatisfiable

Published on 26 Mar 2026 - 12:35

score 77 · Answer 8 · 2024-01-27 12:45:05

77

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How many different ways are there to choose of $n$ subsets $\{i,j\},$ each of length $2,$ from $[2n]:= \{1,2,3,\ldots, 2n \}$?

Published on 27 Jan 2024 - 12:45

score 73 · Answer 9 · 2026-03-26 21:10:22

73

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Is there a Hamiltonian cycle of $m$ x $n$ rectangular lattice points (these are the vertices) in $\mathbb{R}^2$ such that no two edges are parallel?

Published on 26 Mar 2026 - 21:10

score 92 · Answer 10 · 2026-03-25 13:55:36

92

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Question on counting the number of triangles formed by 1999 points in a square

Published on 25 Mar 2026 - 13:55

score 84 · Answer 11 · 2024-02-28 15:38:49

84

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$2n$ knights around a table with namecards, is it possible that for every rotation there is exactly one person with a correct namecard?

Published on 28 Feb 2024 - 15:38

score 39 · Answer 12 · 2024-03-01 04:51:01

39

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Proving that the Set of Outputs of a Function f(n) over Natural Numbers Has a Power of 2

Published on 01 Mar 2024 - 4:51

score 93 · Answer 13 · 2024-03-08 16:59:34

93

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Problem understanding pigeonhole principle

Published on 08 Mar 2024 - 16:59

score 127 · Answer 14 · 2024-03-19 19:03:21

127

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Is there a permutation $\sigma(n)$ of $\{0,1,2,\ldots,m-1\},$ such that $\{(n+\sigma(n))\pmod m:0\leq n\leq m-1\}=\{0,1,2,\ldots m-1\}?$

Published on 19 Mar 2024 - 19:03

score 48 · Answer 15 · 2026-03-26 06:31:47

48

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$A\subset\mathbb{N},$ natural density $1/2.$ Half the members of $A$ are even, half are odd. Is $A$ an (eventual) additive basis of $\mathbb{N}?$

Published on 26 Mar 2026 - 6:31

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Suppose $n\in\mathbb{N}.$ Partition $\{1,2,3,\ldots,2n\}$ into $n$ pairs. Is it true that the maximum pair product is $\geq n^2 ?\ $ Or $\geq n(n+1)?$

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Is there a Hamiltonian cycle of $m$ x $n$ rectangular lattice points (these are the vertices) in $\mathbb{R}^2$ such that no two edges are parallel?

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Is there a permutation $\sigma(n)$ of $\{0,1,2,\ldots,m-1\},$ such that $\{(n+\sigma(n))\pmod m:0\leq n\leq m-1\}=\{0,1,2,\ldots m-1\}?$

$A\subset\mathbb{N},$ natural density $1/2.$ Half the members of $A$ are even, half are odd. Is $A$ an (eventual) additive basis of $\mathbb{N}?$

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