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Assume a set of numbers $S$ which must satisfy four conditions.

(1) Partition this set $S$ into three subsets $U_1$, $U_2$, and $U_3$ such that the sum of the largest subset is as small as possible (no other partition would yield a largest subset smaller than this).

(2) Partition this same set $S$ into three subsets $V_1$, $V_2$, and $V_3$ such that the difference between the sum of the largest and smallest subsets is as small as possible (again, no other partition would yield a difference smaller than this).

(3) The solution of (1) with subsets $U$ is strictly better that the solution of (1) with subsets $V$.

(4) The solution of (2) with subsets $V$ is strictly better that the solution of (2) with subsets $U$.

Can such a set $S$ exist?

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closed as off-topic by Shahriar Mahmud Sajid, rhsquared, Chowzen, Jaap Scherphuis, JonMark Perry Oct 20 '18 at 6:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – Shahriar Mahmud Sajid, rhsquared, Chowzen, Jaap Scherphuis, JonMark Perry
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