Problem C. 842.

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C. 842. We built a large solid cube out of more than ten wooden cubes of unit edge, each. Then we painted all faces of the large cube. Is it possible that the number of unit cubes that have some faces painted is a multiple of the number of the unpainted ones?

(5 points)

Deadline expired.

Sorry, the solution is published in Hungarian only.

Megoldás: Ha k³ darab (a feltétel miatt k>2) kiskockából raktuk össze a nagyot, akkor a festetlen kockák száma (k-2)³, a festetteké pedig k³-(k-2)³. Ez utóbbi pontosan akkor többszöröse az előbbinek, ha hányadosuk egész, vagyis

${k^3-(k-2)^3\over(k-2)^3}=\left({k\over k-2}\right)^3-1$

egész, ami csak akkor lehet, ha ${k\over k-2}=1+{2\over k-2}$ egész, vagyis ha k=3 vagy 4.

Statistics on problem C. 842.

344 students sent a solution.

5 points: 221 students.

4 points: 58 students.

3 points: 14 students.

2 points: 11 students.

1 point: 12 students.

0 point: 7 students.

Unfair, not evaluated: 21 solutions.

Problems in Mathematics of KöMaL, February 2006

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