Under what scenario would the following chained inequality be true?

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


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

2 Answers 2


These are obviously

Poker hands, sorted by which hands are better!


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)


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

  • 2
    $\begingroup$ Usually inequalities are not very applicable in modular arithmetic since it depends on your residue system. But nice answer anyway! $\endgroup$
    – Riley
    May 11, 2018 at 18:21
  • 4
    $\begingroup$ 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. $\endgroup$ May 11, 2018 at 18:42
  • 2
    $\begingroup$ @Acccumulation Algebraically, it's a coset, but I don't think puzzle answers need always delve into mathematical precision :) $\endgroup$
    – noedne
    May 11, 2018 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 $\mathbb{Z}/n\mathbb{Z}$. $\endgroup$ May 12, 2018 at 16:03

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