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sum of integer weights that come with a two pan balance scale

Source: Indian Postal Coaching 2009 set 3 p3

May 26, 2020
weightscombinatorics

Problem Statement

Let SS be the sum of integer weights that come with a two pan balance Scale, say ω1≤ω2≤ω3≤...≤ωn\omega_1 \le \omega_2 \le \omega_3 \le ... \le\omega_n. Show that all integer-weighted objects in the range 11 to SS can be weighed exactly if and only if ω1=1\omega_1=1 and ωj+1≤2(∑l=1jωl)+1\omega_{j+1} \le 2 \left( \sum_{l=1}^{j} \omega_l\right) +1
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