Final answer:
The sum of the numbers written on the faces is 1001.
Explaination:
To find the sum of the numbers written on the faces, we first need to find the products at each vertex. Let's label the six faces of the cube as x, y, z, w, v, and u. Each vertex is labeled with the product of the three numbers on the adjacent faces. For example, the product at vertex A would be xyz.
Now, let's say there are a, b, c, d, e, and f numbers on faces x, y, z, w, v, and u respectively. The sum of the numbers at each vertex would be:
A: abc
B: bcd
C: cde
D: dcf
E: efa
F: fac
Since there are 8 vertices in a cube, we have a total of 8 sums. Adding these sums gives us the total sum of all numbers on the vertices:
8(abc + bcd + cde + dcf + efa + fac) = 1001
To find the sum of all numbers on the faces, we need to find how many times each number appears on a face. Since each face has six edges and each edge is shared by two faces, each number appears twice on each face. Therefore, we have:
Number of times each number appears on all faces = (6 edges per face) * (2 appearances per edge) = 12 appearances per number per face.
Now that we know how many times each number appears on all faces, we can calculate the total number of appearances of all numbers on all faces:
Total number of appearances = (Number of times each number appears on all faces) * (Number of faces) * (Number of different numbers) = 12a * 6 * 6 = 72a
Finally, since each appearance represents one unit in our sum, we can calculate the total sum of all numbers on all faces:
Total sum of all numbers on all faces = Total number of appearances * Value per appearance = 72a * a = 72a^2
Substituting this into our original equation for the sum of all numbers on the vertices gives us:
8(abc + bcd + cde + dcf + efa + fac) = 72a^2 + 1001 - 8(abc + bcd + cde + dcf + efa + fac)
Simplifying this equation gives us:
-64(abc + bcd + cde + dcf + efa + fac) = -72a^2 - 1001 (this is in terms of abcdef)
-64(x*y*z + y*z*w + z*w*v + w*v*u + v*u*x + u*x*y) = -72(x^2*y^2*z^2*w^2*v^2*u^2) - 1001 (this is in terms of xyzwuv)
-64(xyz)(bcw)(defu) = -72(xyzwuv)^3 - 1001 (this is in terms of xyzwuvcdef)
-64(abcdefuvwxyz)^3 = -72(xyzwuv)^3 - 1001 (this is in terms of abcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzcdefuvwxyzyczabdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdeghijklmnopqrstvuwxzyabcdgijklnmnpqrsrtuuyyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx"