MathDB

You are given a balance and one copy of each of ten weights of

1, 2, 4, 8, 16, 32, 64, 128, 256

and

512

grams. An object weighing

M

grams, where

M

is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than

89

ways. (b) Find a value of

M

such that an object weighing

M

grams can be balanced in

89

ways.
(A Shapovalov, A Kulakov)

Problem Statement