MathDB

Assume we are given a set of weights,

x_1

of which have mass

d_1

x_2

have mass

d_2

, etc,

x_k

have mass

d_k

, where

x_i,d_i

are positive integers and

1\le d_1<d_2<\ldots<d_k

. Let us denote their total sum by

n=x_1d_1+\ldots+x_kd_k

. We call such a set of weights perfect if each mass

0,1,\ldots,n

can be uniquely obtained using these weights.
(a) Write down all sets of weights of total mass

5

. Which of them are perfect? (b) Show that a perfect set of weights satisfies

(1+x_1)(1+x_2)\cdots(1+x_k)=n+1.

(1+x_1)(1+x_2)\cdots(1+x_k)=n+1

, prove that one can uniquely choose the corresponding masses

d_1,d_2,\ldots,d_k

with

1\le d_1<\ldots<d_k

in order for the obtained set of weights is perfect. (d) Determine all perfect sets of weights of total mass

1993

Problem 1