A Special Number: 3367

 Figure 1 shows an interesting result via Cliff Pickover.

To see why 3367 can be written as the various fractions shown, one approach is to consider a number of the form \( 0 \, a \, 0 \, a \, 0 \, a\) where \(1 \geq a \geq 99\). Let's manipulate this number in the following way:$$  \begin{align} \frac{0 \, a \, 0 \, a \, 0 \, a}{a+a+a} &= \frac{a \times 0 \, 1 \, 0 \, 1 \, 0 \, 1}{3 \, a}\\ \\ &= \frac{0 \, 1 \, 0 \, 1 \, 0 \, 1}{3} \\  \\ &=3 \, 3 \, 6 \,7 \end{align}$$Thus when \(a=11\) we have:$$\frac{0 \, a \, 0 \, a \, 0 \, a}{a+a+a} = \frac{1 \, 1 \, 1 \, 1 \, 1 \, 1}{11 + 11 +11}$$If \(a>99\), then the pattern breaks down.