Circle packing in an isosceles right triangle

Circle packing in a right isosceles triangle is a packing problem where the objective is to pack n unit circles into the smallest possible isosceles right triangle.

Minimum solutions (lengths shown are length of leg) are shown in the table below.^[1] Solutions to the equivalent problem of maximizing the minimum distance between n points in an isosceles right triangle, were known to be optimal for n < 8^[2] and were extended up to n = 10.^[3]

In 2011 a heuristic algorithm found 18 improvements on previously known optima, the smallest of which was for n = 13.^[4]

Number of circles	Length
1	${\displaystyle 2+{\sqrt {2}}}$ = 3.414...
2	${\displaystyle 2{\sqrt {2}}}$ = 4.828...
3	${\displaystyle 4+{\sqrt {2}}}$ = 5.414...
4	${\displaystyle 2+3{\sqrt {2}}}$ = 6.242...
5	${\displaystyle 4+{\sqrt {2}}+{\sqrt {3}}}$ = 7.146...
6	${\displaystyle 6+{\sqrt {2}}}$ = 7.414...
7	${\displaystyle 4+{\sqrt {2}}+{\sqrt {2+4{\sqrt {2}}}}}$ = 8.181...
8	${\displaystyle 2+3{\sqrt {2}}+{\sqrt {6}}}$ = 8.692...
9	${\displaystyle 2+5{\sqrt {2}}}$ = 9.071...
10	${\displaystyle 8+{\sqrt {2}}}$ = 9.414...
11	${\displaystyle 5+3{\sqrt {2}}+{\dfrac {1}{3}}{\sqrt {6}}}$ = 10.059...
12	10.422...
13	10.798...
14	${\displaystyle 2+3{\sqrt {2}}+2{\sqrt {6}}}$ = 11.141...
15	${\displaystyle 10+{\sqrt {2}}}$ = 11.414...