MathDB

n

is a given positive integer, such that it’s possible to weigh out the mass of any product weighing

1,2,3,\cdots ,ng

with a counter balance without sliding poise and

k

counterweights, which weigh

x_ig(i=1,2,\cdots ,k),

respectively, where

x_i\in \mathbb{N}^*

for any

i \in \{ 1,2,\cdots ,k\}

and

x_1\leq x_2\leq\cdots \leq x_k.

(1)

Let

f(n)

be the least possible number of

k

. Find

f(n)

in terms of

n.

(2)

Find all possible number of

n,

such that sequence

x_1,x_2,\cdots ,x_{f(n)}

is uniquely determined.

Problem Statement