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An Integer Construction Problem

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an-integer-construction-problemNumber Theorymath.COmath.NTposed by Leroy F. Meyersrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

It is conjectured that if division and reciprocation, as well as subtraction and negation, are also made allowed operations, then they are never necessary to achieve a maximizing combination.

Context

Candidate 1 of the open problems stated in "An Integer Construction Problem", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    Problem: Reconstructed conjecture: given a finite labeled multiset of positive integers, a “combination” is an arithmetic expression using each input exactly once. Meyers originally considers +,×+,\times, then −- and unary negation, and conjectures in §5 that adding division and reciprocation x↦1/xx\mapsto 1/x never increases the maximum attainable value; equivalently, some maximizing expression can avoid division/reciprocation. This also covers the motivating digit set {1,2,3,4}\{1,2,3,4\}.

    Result: The conjecture is false, already for the inputs 1,2,3,41,2,3,4.

    First, without division/reciprocation, the maximum is 3636. Let M(S)M(S) be the maximum absolute value obtainable from the elements of a nonempty subset S⊆{1,2,3,4}S\subseteq\{1,2,3,4\} using only +,−,×+,-,\times and unary negation. For any top-level split S=T⊔US=T\sqcup U,

    M(S)≤max⁡{M(T)+M(U), M(T)M(U)}.M(S)\le \max\{M(T)+M(U),\,M(T)M(U)\}.

    A direct recursion gives

    S12341213142324341231241342341234M(S)12343456812912162436.\begin{array}{c|ccccccccccccccc} S&1&2&3&4&12&13&14&23&24&34&123&124&134&234&1234\\ \hline M(S)&1&2&3&4&3&4&5&6&8&12&9&12&16&24&36 . \end{array}

    The last value is attained by

    (1+2)⋅3⋅4=36.(1+2)\cdot 3\cdot 4=36.

    But with reciprocation allowed,

    11 3+11+12 −14\frac{1}{\frac{1}{\,3+\frac{1}{1+\frac12}\,}-\frac14}

    is a valid expression using 1,2,3,41,2,3,4 exactly once. Its value is

    1+12=32,3+13/2=3+23=113,1+\frac12=\frac32,\quad 3+\frac{1}{3/2}=3+\frac23=\frac{11}{3},

    so

    111/3−14=311−14=144,\frac{1}{11/3}-\frac14=\frac{3}{11}-\frac14=\frac{1}{44},

    and the whole expression equals 4444. Thus the enlarged operation set attains a value 44>3644>36, so division/reciprocation can be necessary for a maximizing combination.

    Citation: Original conjecture: Leroy F. Meyers, “An Integer Construction Problem,” American Mathematical Monthly 66 (1959), 556–561, §5. The counterexample above is self-contained.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The solution attacks the correct conjecture: Meyers’ “maximizing combination” is the largest constructible value from a fixed multiset using each input once. For {1,2,3,4}\{1,2,3,4\}, the no-division/no-reciprocation maximum is correctly bounded by 3636 via the finite recursion/table and is attained by (1+2)3⋅4(1+2)3\cdot4. The proposed expression uses each of 1,2,3,41,2,3,4 exactly once and only the newly allowed reciprocation/subtraction operations, and evaluates to 44>3644>36. Thus any maximizer in the enlarged operation set cannot be achieved without division/reciprocation.

      I found the original paper and did not find a prior published resolution/counterexample in the searches performed.

      Novelty assessment

      TYPE1

      Classification rationale: This is a very small explicit counterexample to a low-profile recreational/elementary conjecture. It is mathematically useful as a correction to Meyers’ claim, but it is not substantial enough for a standalone combinatorics paper.

      Literature check: I found no published explicit counterexample matching the 1,2,3,4↦441,2,3,4\mapsto 44 construction or any stronger resolved statement. The main prior lead is Singmaster’s note that Meyers later recalled a student once showing him a counterexample, but with no remembered construction or proof; I would not count that anecdotal report as a published resolution.

      Citation: Leroy F. Meyers, “An Integer Construction Problem,” Amer. Math. Monthly 66 (1959), 556–561, DOI: 10.1080/00029890.1959.11989349.
      David Singmaster, Sources in Recreational Mathematics, 8th preliminary ed. (2004), Puzzle Museum, §7 note on Meyers.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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