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Puzzle 180 | Science and Mathematics

Problem proposed by Bruno Holland and Samuel Vitosa

In the land of truth, where no one lies, friends Marcondes, Francisco and Fernando meet together. Between the three the following conversation took place:

– Marcondes: I will choose two consecutive positive integers and give one to Francisco and the other to Fernando, without you knowing who got the largest number;

After each received their number, Francisco and Fernando continued the conversation.

Francisco: I don’t know the number Fernando received;

Fernando: I don’t know what number Francisco received;

Francisco: I don’t know the number Fernando received;

Fernando: I don’t know what number Francisco received;

Francisco: I don’t know the number Fernando received;

Fernando: I don’t know what number Francisco received;

Francisco: Now I know the number Fernando received;

Fernando: Now I also know the number that Francisco received;

What numbers do each of them get?

(Excerpted from Ciara Mathematical Olympiad)

Send your solution to [email protected] Don’t forget to write your name, city and school, if possible. We will publish the solution next Sunday.

179.

At first, put 64 coins on one board and another 64 on the other. If they are balanced, discard one of the groups. Divide the remaining group into two 32 coins and make a new weight. If they always remain in equilibrium, and repeat this procedure, then after 6 weights we will have only 2 coins, since one must necessarily be heavier than the other, and this ends the problem. Otherwise, assuming the scale hasn’t reached equilibrium in any of these steps, we’ll change the weighing strategy next. Now suppose, for example, that in the first weighing process we did not get the scale. Now choose any 32 coins on the heavier side for the second weight to compare with any 32 coins on the lighter side. If it is balanced, discard it and reduce the problem to account for only the remaining 64, which will necessarily contain coins of both types because the first weight was unbalanced. If they don’t balance, get rid of 64 coins not included in the second weight. Repeat the same procedure as for selecting coins for the second weight in the following steps. With each new weight, we will always make sure to include coins of both types. After the last weight, we will have two coins of different weights.

Bruno Holland Professor of Federal University of Goiás and Samuel Vitosa Professor at the Federal University of Bahia.

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