Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem B. 4302. (November 2010)

B. 4302. Soldiers are lined up along a line in the east-west direction, each of them facing north. The officer commands right turn''. Then they should all turn towards the east but, since they are at the very beginning of their military career, some of them get the order wrong and turn towards the west. Every soldier who now faces his neighbour concludes that they have made a mistake (disregarding the possibility of the neighbour making the mistake), and they make a 180-degree turn within 1 second. Then the process continues in the same way: every soldier who faces a neighbour makes a 180-degree turn within 1 second. Prove that the process will end in a finite number of steps.

(Talent Search, U.S.A.)

(3 pont)

Deadline expired on December 10, 2010.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Tegyük fel, hogy az állítással ellentétben van olyan katona, aki végtelen sokszor megfordul. A sor szélén álló katonák azonban csak legfeljebb egyszer fordulnak meg. Ezért kell legyen két szomszédos katona, amelyek egyike végtelen sokszor, másika pedig csak véges sokszor fordul meg, ami ellentmondás, hiszen a végtelen sokszor megforduló katona minden második megfordulásánál a szóban forgó szomszédjának is meg kell fordulnia.

### Statistics:

 187 students sent a solution. 3 points: 101 students. 2 points: 37 students. 1 point: 39 students. 0 point: 8 students. Unfair, not evaluated: 2 solutionss.

Problems in Mathematics of KöMaL, November 2010