Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 842. (February 2006)

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 pont)

Deadline expired on March 16, 2006.


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

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:

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