Last year I presented here two new exciting open problems. This is a report on substantial advance obtained in each of them over the past year. Problem 1. Find the smallest number A(n) of unit equilateral triangles that can cover an equilateral triangle of side length n + ε. John Conway and I [CS1] showed that n~2 +1 ≤ Δ(n) Δ n~2 + 2 . Recently Dmytro Karabash and I have shown that equilaterality is essential in this problem: Result 1.1 [KS1]. Any non-equilateral triangle T can be covered by n~2 +1 triangles similar to T, whose corresponding sides are n + e times smaller. We then considered a more general object, a trigon. We define n-trigon T_n to be the union of n triangles from the standard triangulation of the plane such that triangular rook can find a path between any two triangles of T_n i.e., the union of n edge-connected triangles. If the triangulation is equilateral, we will say that the n-trigon is equilateral. Let T' and T be congruency classes of the triangles from which the trigon T_n is composed and the triangles with which we will cover T_n, respectively. T'-triangles and T-triangles are similar with the corresponding sides in a ratio (l + ε) : 1.
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