OlympiadHardLogic5–6

On a balance scale, weights may be placed in both pans. What is the smallest number of weights needed to weigh every whole number of grams from $1$ to $40$ ?

Numa balança de pratos, os pesos podem ser colocados nos dois pratos. Qual é o menor número de pesos necessário para pesar todos os valores inteiros de gramas de $1$ a $40$ ?

Answer options

Solution

Weights of $1$ , $3$ , $9$ and $27$ grams suffice: each may be left off, added to one pan or added to the other, so every value from $1$ to $1 + 3 + 9 + 27 = 40$ is reachable. Three weights give at most $3^{3} = 27$ signed combinations, which cannot cover $40$ values, so $4$ is smallest.

On a balance scale, weights may be placed in both pans. What is the smallest number of weights needed to weigh every whole number of grams from 111 to 404040?

Answer options

On a balance scale, weights may be placed in both pans. What is the smallest number of weights needed to weigh every whole number of grams from $1$ to $40$ ?