Károly, László and Mihály went hunting for deer, foxes, and hares over three days. Each of them shot at least one of each type of game every day, and in total they shot 86 animals. On the first day, they shot 12 foxes and 14 deer, and on the second day, they shot a total of 44 animals. László shot an even number of each type of game every day, except for the second day when he shot 5 foxes. At the end of the hunt, Károly counted that they had shot 12 hares in total. Question: How many foxes did Mihály shoot on the third day?
I. solution. The number of pheasants shot on the first day and the total catch on the third day, on the one hand, provide a lower limit, as the three hunters collectively shot at least 4 of each type of game every day - Károly and Mihály one each, and László two each; thus, the value of the mentioned total is at least $4+3 \cdot 4=16$. On the other hand, the total is obtained by subtracting the number of deer and foxes shot on the first day, as well as the total number of game shot on the second day, from the entire catch: $86-(12+14)-44=16$. This means that on the third day, their catch of each type of game was the smallest number according to the conditions, so Mihály shot 1 fox on that day.

Imre Bárány (Budapest, Corvin Mátyás Gymn. II. grade)

II. solution. The group collectively shot at least 4 pheasants each day, - Mihály and Károly one each, and László two, so the total number of pheasants shot during the entire hunt was at least 12. On the other hand, this number is exactly 12, so no hunter shot more pheasants on any day than the minimum number indicated by the data. Thus, on the first day, they shot 4 pheasants, and the entire catch was $4+12+14=30$ game. Therefore, the result on the third day is $86-(30+44)=12$ game. The same reasoning applies to the minimum number of deer and foxes shot each day, so on the third day, no hunter shot more of any type of game than the minimum number; thus, Mihály shot 1 fox on that day.

György Darvas (Budapest, Radnóti M. Gymn. II. grade)

Note. This solution also used the total number of pheasants shot, which was not necessary in the first solution. Neither solution used the fact that László shot 5 foxes on the second day. According to this, fewer data would have been sufficient for the solution, but the additional data were consistent with the necessary ones for the answer.