# Sets and subsets [closed]

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?

## closed as off-topic by Shahriar Mahmud Sajid, rhsquared, Chowzen, Jaap Scherphuis, JonMark PerryOct 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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