Equilateral triangles of sidelengths 1, $$2^{-t}$$
2
-
t
, $$3^{-t}$$
3
-
t
, $$4^{-t},\ldots \ $$
4
-
t
,
…
can be packed perfectly into an equilateral triangle, provided that $$\ 1/2<t \le 37/72$$
1
/
2
<
t
≤
37
/
72
. Moreover, for t slightly greater than 1/2, squares of sidelengths 1, $$2^{-t}$$
2
-
t
, $$3^{-t}$$
3
-
t
, $$4^{-t},\ldots \ $$
4
-
t
,
…
can be packed perfectly into a square $$S_t$$
S
t
in such a way that some squares have a side parallel to a diagonal of $$S_t$$
S
t
and the remaining squares have a side parallel to a side of $$S_t$$
S
t
.