Mathematical and Physical Journal
for High Schools
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Problem K. 340. (September 2012)

K. 340. A large cube is built out of small white cubes, and then the faces of the large cube are painted blue. The large cube is then taken apart again. What is the size of the large cube if the number of small cubes with an even number of blue faces is the same as those with an odd number of blue faces?

(6 pont)

Deadline expired on October 10, 2012.


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

Megoldás. Legyen a kis kockák száma: (n+2)3. A három kék lappal rendelkező kis kockák száma 8. A két kék lappal rendelkező kis kockák száma 12n. Az egy kék lappal rendelkező kis kockák száma 6n2. A nulla kék lappal rendelkező kis kockák száma n3. A feladat szövege szerint: 6n2+8=n3+12n, átrendezve 0=n3-6n2+12n-8, azaz 0=(n-2)3, ahonnan n=2. Tehát a nagy kocka 4×4×4 kiskocka méretű volt.


Statistics:

200 students sent a solution.
6 points:51 students.
5 points:22 students.
4 points:14 students.
3 points:47 students.
2 points:36 students.
1 point:12 students.
0 point:15 students.
Unfair, not evaluated:3 solutions.

Problems in Mathematics of KöMaL, September 2012