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- #1

- Feb 14, 2012

- 3,802

- Thread starter anemone
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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,802

- Admin
- #2

- Mar 5, 2012

- 8,999

\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 $174,\, 175,\, 339,\, 505$.

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}

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 $174,\, 175,\, 339,\, 505$.

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