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Conway and Soifer showed that an equilateral triangle $T$ of side $n +\varepsilon$ with sufficiently small $\varepsilon > 0$ can be covered by $n^2 +2$ unit equilateral triangles. They conjectured that it is impossible to cover$T$ with $n^2 + 1$ unit equilateral triangles no matter how small $\varepsilon$is.  We show that if we require all sides of the unit equilateral triangles to beparallel to the sides of $T$ (e.g. $\bigtriangleup$ and $\bigtriangledown$),then it is impossible to cover $T$ of side $n + \varepsilon$ with $n^2 + 1$unit equilateral triangles for any $\varepsilon > 0$. As the coverings of $T$by Conway and Soifer only involve triangles with sides parallel to $T$, ourresult determines the exact minimum number $n^2+2$ of unit equilateraltriangles with all sides parallel to $T$ that cover $T$. We also determine thelargest value $\varepsilon = 1/(n + 1)$ (resp. $\varepsilon = 1 / n$) of$\varepsilon$ such that the equilateral triangle $T$ of side $n + \varepsilon$can be covered by $n^2+2$ (resp. $n^2 + 3$) unit equilateral triangles withsides parallel to $T$, where the first case is achieved by the construction ofConway and Soifer.

$n^2 + 1$ unit equilateral triangles cannot cover an equilateral triangle of side $> n$ if all triangles have parallel sides