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Full list of counterexamples for the k=3 version of the malicious apprentice problem.
1

%I #12 Jul 14 2017 11:46:01

%S 3,6,27,486

%N Full list of counterexamples for the k=3 version of the malicious apprentice problem.

%C This is the problem of the farmer's helper who, when asked to weigh n bags of grain, does so k at a time and reports the resulting binomial(n,k) combined weights with no indication of the k-tuples that produced them. The problem: is can the weights of the bags be recovered?

%C For k=3 the answer is Yes unless n is one of the four terms of this sequence. For k=2 see A057716.

%C The old entry with this sequence number was a duplicate of A030109.

%C The following references also apply to the general case of the problem.

%D W. W. Rouse Ball, A Short Account of the History of Mathematics.

%D E. Bolker, The finite Radon transform, Contemp. Math., 63 (1987) 27-50.

%D R. K. Guy, Unsolved Problems in Number Theory, C5.

%D Ross A. Honsberger, A gem from combinatorics, Bull. ICA, 1 (1991) 56-58.

%D B. Liu and X. Zhang, On harmonious labelings of graphs, Ars Combin., 36 (1993) 315-326.

%D J. Ossowski, On a problem of Galvin, Congressus Numerantium, 96 (1993) 65-74.

%D P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.

%H I. N. Baker, <a href="http://dx.doi.org/10.4153/CMB-1960-012-8">Solutions of the functional equation (f(x))^2-f(x^2)=h(x)</a>, Canad. Math. Bull., 3 (1960) 113-120.

%H J. Boman, E. Bolker and P. O'Neil, <a href="https://doi.org/10.1016/0196-8858(91)90027-G">The combinatorial Radon transform modulo the symmetric group</a>, Adv. Appl. Math., 12 (1991) 400-411.

%H Jan Boman and Svante Linusson, <a href="http://dx.doi.org/10.7146/math.scand.a-12583">Examples of non-uniqueness for the combinatorial Radon transform modulo the symmetric group</a>, Math. Scand. 78 (1996), 207-212.

%H John A. Ewell, <a href="http://dx.doi.org/10.4153/CJM-1968-059-6">On the determination of sets by sets of sums of fixed order</a>, Canad. J. Math., 20 (1968) 596-611.

%H B. Gordon, A. S. Fraenkel and E. G. Straus, <a href="http://projecteuclid.org/euclid.pjm/1103036716">On the determination of sets by the sets of sums of a certain order</a>, Pacific J. Math., 12 (1962) 187-196.

%H J. Lambek and L. Moser, <a href="http://dx.doi.org/10.4153/CMB-1959-013-x">On some two way classifications of the integers</a>, Canad. Math. Bull., 2 (1959) 85-89.

%H L. Moser and C. F. Pinzka, <a href="http://www.jstor.org/stable/2308470">Problem E1248</a>, Amer. Math. Monthly, 64 (1957) 507.

%H D. G. Rogers, <a href="http://www.jstor.org/stable/2030911">A functional equation: solution to Problem 89-19</a>, SIAM Review, 32 (1990) 684-686.

%H J. L. Selfridge and E. G. Straus, <a href="http://projecteuclid.org/euclid.pjm/1103039706">On the determination of numbers by their sums of a fixed order</a>, Pacific J. Math., 8 (1958) 847-856.

%e For n=27 Boman and Linusson give five examples of which the simplest is {-4,-1^{10},2^{16}} and its negative, where exponents denote repetitions. For n=486 Boman and Linusson give {-7,-4^{56},-1^{231},2^{176},5^{22}} and its negative.

%Y See A057716 for the case k=2.

%K nonn,fini,full

%O 1,1

%A _N. J. A. Sloane_, based on email from _R. K. Guy_, Oct 30 2008