In a basket there are 13 apples. There are scales, with which you can find out the total weight of any two apples. Think of a way to find out from 8 weighings the total weight of all the apples.
What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.
There are scales without weights and 3 identical in appearance coins, one of which is fake: it is lighter than the real ones (the real coins are of the same weight). How many weightings are needed to determine the counterfeit coin? Solve the same problem in the cases where there are 4 coins and 9 coins.
We have scales without weights and 3 identical in appearance coins. One of the coins is fake, and it is not known whether it is lighter or heavier than the real coins (note that all real coins are of the same weight). How many weighings are needed to determine the counterfeit coin? Solve the same problem in the cases where there are 4 coins and 9 coins.
Jack the goldminer extracted 9 kg of golden sand. Will he be able to measure 2 kg of sand in three goes with the help of scales: a) with two weights of 200 g and 50 g; b) with one weight of 200 g?
From a set of weights with masses 1, 2, ..., 101 g, a weight of 19 grams was lost. Can the remaining 100 weights be divided into two piles of 50 weights each in such a way that the masses of both piles are the same?
Alice the fox and Basilio the cat are counterfeiters. Basilio makes coins heavier than real ones, and Alice makes lighter ones. Pinocchio has 15 identical in appearance coins, but one coin is fake. How can Pinocchio determine who made the false coin – Basilio the cat or Alice the fox – with only 2 weighings?
It is known that “copper” coins that are worth 1, 2, 3, 5 pence weigh 1, 2, 3, 5 g respectively. Among the four “copper” coins (one for each denomination), there is one defective coin, differing in weight from the normal ones. How can the defective coin be determined using scales without weights?
How can we divide 24 kg of nails into two parts of 9 kg and 15 kg with the help of scales without weights?