# Sum of Numbers on Cube Faces to Equal 2004

• MHB
• anemone
In summary, positive integers are written on the faces of a cube and at each corner, the product of the numbers on the faces that meet at the corner is written. The sum of the numbers at all the corners is 2004. The task is to find all the possible values of the sum of numbers on all the faces, denoted by T.
anemone
Gold Member
MHB
POTW Director
Positive integers are written on all the faces of a cube, one on each. At each corner (vertex) of the cube, the product of the numbers on the faces that meet at the corner is written. The sum of the numbers written at all the corners is 2004. If $T$ denotes the sum of the numbers on all the faces, find all the possible values of $T$.

\begin{tikzpicture}[
face/.pic = {
\node {#1};
\draw (-1,-1) rectangle (1, 1);
},
cube/.pic = {
\draw (0,0) pic {face=a};
\draw (1,-1) -- (2,-0.5) -- (2,1.5) -- (1,1) node at (1.5,0.25) {b};
\draw (2,1.5) -- (0,1.5) -- (-1,1) node at (0.5,1.25) {e};
}]
\draw (0,0) pic {face=a} (2,0) pic {face=b} (4,0) pic {face=c} (6,0) pic {face=d} (0,2) pic {face=e} (0,-2) pic {face=f};
\draw (9,0) pic {cube};
\end{tikzpicture}
Let $a,b,c,d,e,f$ be the 6 faces where $(a,c)$, $(b,d)$, and $(e,f)$ are the pairs of opposing faces as shown in the drawing.
The sum of the corners is $(ab+bc+cd+da)e+(ab+bc+cd+da)f=(a(b+d) + c(b+d))(e+f) =(a+c)(b+d)(e+f)=2004$.

The list of suitable factorizations of $2004=2^2\cdot 3\cdot 167$ is $(3\cdot 4\cdot 167,\, 2\cdot 6\cdot 167,\,2\cdot 3\cdot 334,\,2\cdot 2\cdot 501)$.
We are looking for the possibilities for $T=a+b+c+d+e+f$, which is the sum of the 3 factors.
Those possibilities for $T$ are $174,\, 175,\, 339,\, 505$.

Last edited:

## 1. What is the "Sum of Numbers on Cube Faces to Equal 2004" problem?

The "Sum of Numbers on Cube Faces to Equal 2004" problem is a mathematical puzzle that involves finding a combination of numbers on the faces of a cube that add up to a target number of 2004.

## 2. How many solutions are there to the "Sum of Numbers on Cube Faces to Equal 2004" problem?

There are multiple solutions to this problem, but the exact number is not known. It is estimated that there are over 100,000 unique solutions.

## 3. Is there a specific method or strategy to solve this problem?

Yes, there are various strategies and algorithms that can be used to solve the "Sum of Numbers on Cube Faces to Equal 2004" problem. Some common methods include brute force, dynamic programming, and backtracking.

## 4. Can this problem be extended to other target numbers?

Yes, this problem can be extended to any target number, as long as it is a positive integer. The same strategies and methods can be applied to find solutions for different target numbers.

## 5. What real-world applications does the "Sum of Numbers on Cube Faces to Equal 2004" problem have?

This problem has applications in fields such as computer science, cryptography, and game theory. It can also be used as a brain teaser or puzzle for recreational purposes.

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