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Can our coinage system be improved? (English) Zbl 1474.11077

Summary: Some forty years ago my wife Christine and I considered the problems in this article, which involves a fair amount of computation. Computing facilities were not good then, so we considered instead the problems in [C. M. Shiu and P. Shiu, “Stamps and coins: two partition problems”, Math. Spect. 13, 49–55 (1980)] in which we showed, without using computers, that there were 64703 ways to make up £1 using coins; this was before the introduction of the 20p and £1 coins, and the \(\frac{1}{2}\)p coin was in circulation. If Christine were still with us, this would have been another piece of joint work. I therefore dedicate this article to her memory.
The design of a coinage system depends on considerations we give to various criteria: for example, the number of denominations for the coins, the maximum number of coins required to deliver any given amount in a range, or the required number of coins averaged over the range; see also §3.

MSC:

11D07 The Frobenius problem
90C90 Applications of mathematical programming
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References:

[1] 1.ShiuC. M. and ShiuP., Stamps and coins: two partition problems, Mathematical Spectrum13 (1980) pp. 49-55.
[2] 2.ShallitJeffrey, What this country needs is an 18 cents piece’. The Mathematical Intelligencer, (2) 25 (2003) pp. 20-23.10.1007/BF02984830
[3] 3.ShiuP., Moment sums associated with binary quadratic forms, Amer. Math. Monthly113 (2006) pp. 545-550.10.1080/00029890.2006.11920334 · Zbl 1113.11018
[4] 4.TripathiA., Formulae for the Frobenius number in three variables, J. Number Theory170 (2017) pp. 368-389.10.1016/j.jnt.2016.05.027 · Zbl 1402.11046
[5] 5.KannanR., Lattice translates of a polytope and the Frobenius problem, Combinatorica12 (2) (1992) pp. 161-177.10.1007/BF01204720 · Zbl 0753.11013
[6] 6.Ramírez-AlfonsínJ. L., The Diophantine Frobenius problem, Oxford University Press (2006). · Zbl 1134.11012
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