Packing hexagons in an isosceles right triangle
Source: XII Rioplatense Mathematical Olympaid (2003), Level 3
August 9, 2011
geometrycombinatorics unsolvedcombinatorics
Problem Statement
Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size.[asy]
unitsize(.85cm);
draw((0,0)--(1,0)--(1,1)--(2,2)--(-1,2)--(0,1)--(0,0),linewidth(1));
draw((0,2)--(0,1)--(1,1)--(1,2),dashed);
label("\footnotesize ",(0.5,0),S);
label("\footnotesize ",(0,0.5),W);
label("\footnotesize ",(1,0.5),E);
label("\footnotesize ",(0,1.5),E);
label("\footnotesize ",(1,1.5),W);
label("\footnotesize ",(-0.5,2),N);
label("\footnotesize ",(0.5,2),N);
label("\footnotesize ",(1.5,2),N);
[/asy] The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of length of each tile lies between this longest side of the tile and the hypotenuse of the triangle. Furthermore, if the longest side of a tile is farther from the hypotenuse than the longest side of another tile, then the size of the first tile is larger or equal to the size of the second tile. Find the smallest value of such that every such configuration of tiles has a total area less than .