A074894 Full list of counterexamples for the k=3 version of the malicious apprentice problem.

3, 6, 27, 486

Offset: 1

Author

N. J. A. Sloane, based on email from R. K. Guy, Oct 30 2008

Keywords

• nonn
• fini
• full

Comments

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?

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

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

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

Examples

			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.
		

References

• W. W. Rouse Ball, A Short Account of the History of Mathematics.
• E. Bolker, The finite Radon transform, Contemp. Math., 63 (1987) 27-50.
• R. K. Guy, Unsolved Problems in Number Theory, C5.
• Ross A. Honsberger, A gem from combinatorics, Bull. ICA, 1 (1991) 56-58.
• B. Liu and X. Zhang, On harmonious labelings of graphs, Ars Combin., 36 (1993) 315-326.
• J. Ossowski, On a problem of Galvin, Congressus Numerantium, 96 (1993) 65-74.
• P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.

Links

• I. N. Baker, Solutions of the functional equation (f(x))^2-f(x^2)=h(x), Canad. Math. Bull., 3 (1960) 113-120.
• J. Boman, E. Bolker and P. O'Neil, The combinatorial Radon transform modulo the symmetric group, Adv. Appl. Math., 12 (1991) 400-411.
• Jan Boman and Svante Linusson, Examples of non-uniqueness for the combinatorial Radon transform modulo the symmetric group, Math. Scand. 78 (1996), 207-212.
• John A. Ewell, On the determination of sets by sets of sums of fixed order, Canad. J. Math., 20 (1968) 596-611.
• B. Gordon, A. S. Fraenkel and E. G. Straus, On the determination of sets by the sets of sums of a certain order, Pacific J. Math., 12 (1962) 187-196.
• J. Lambek and L. Moser, On some two way classifications of the integers, Canad. Math. Bull., 2 (1959) 85-89.
• L. Moser and C. F. Pinzka, Problem E1248, Amer. Math. Monthly, 64 (1957) 507.
• D. G. Rogers, A functional equation: solution to Problem 89-19, SIAM Review, 32 (1990) 684-686.
• J. L. Selfridge and E. G. Straus, On the determination of numbers by their sums of a fixed order, Pacific J. Math., 8 (1958) 847-856.

Crossrefs

See A057716 for the case k=2.