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Under what scenario would the following chained inequality be true?

33353 > 48484 > 37465 > 96166 > 71373 > 87285 > 26938

Notes:

  • > means "greater than"
  • The transitive property is in effect
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These are obviously

Poker hands, sorted by which hands are better!

Explanation:

33353 is 4 of a kind
48484 is a full house
37465 is a straight (we'll assume it's not a straight flush)
96166 is 3 of a kind
71373 is 2 pair
87285 is a pair
26938 is 9-high (we'll also assume it's not a flush)

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Modulo $726$: $683>568>439>334>225>165>76$
or
Modulo $727$: $638>502>388>202>127>45>39$

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    $\begingroup$ Usually inequalities are not very applicable in modular arithmetic since it depends on your residue system. But nice answer anyway! $\endgroup$ – Riley May 11 '18 at 18:21
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    $\begingroup$ To expand on @Riley 's comment, while computer programs generally return x mod y as an integer, mathematically it's generally treated as being a member of a distinct mathematical structure. x mod y is the equivalence class of all numbers that, when divided by y, have the same remainder as x does. $\endgroup$ – Acccumulation May 11 '18 at 18:42
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    $\begingroup$ @Acccumulation Algebraically, it's a coset, but I don't think puzzle answers need always delve into mathematical precision :) $\endgroup$ – noedne May 11 '18 at 19:15
  • $\begingroup$ @noedne : It's both. It's an equivalence under the equivalence relation "$ a \sim b \iff a \cong b \pmod{n}$". It's a coset in the quotient $\mathbb{Z}/n\mathbb{Z}$. $\endgroup$ – Eric Towers May 12 '18 at 16:03

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