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The Geometry of Coin-Weighing Problems (Extended Abstract)

Summary: The Geometry of Coin-Weighing Problems
(Extended Abstract)
Noga Alon
Dmitry N. Kozlov
Van H. Vu
Given a set of m coins out of a collection of coins of k unknown distinct weights, we wish
to decide if all the m given coins have the same weight or not using the minimum possible
number of weighings in a regular balance beam. Let m(n, k) denote the maximum possible
number of coins for which the above problem can be solved in n weighings. We show that
m(n, 2) = n( 1
2 +o(1))n
, whereas for all 3 k n + 1, m(n, k) is much smaller than m(n, 2)
and satisfies m(n, k) = (n log n/ log k). The proofs have an interesting geometric flavour, and
combine Linear Algebra techniques with geometric, probabilistic and combinatorial arguments.

Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel
Aviv, Israel. Email address: noga@math.tau.ac.il. Research supported in part by a USA Israeli BSF grant and by the
Fund for Basic Research administered by the Israel Academy of Sciences.


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics