Unit squares with unit line segments (Q6162793)

In this note, the interesting problem from recreational mathematics of finding the minimum number of unit line segments \(T_n\) to construct \(n\) unit squares is considered. To make the problem even more recreational, think of the unit segments as matches. To construct $1,2,3,4,\dots$ squares you need $4,7,10,12,\dots$ matches respectively. The authors give an explicit formula for \(T_n\) and pose the equivalent optimization problem of constructing the maximum number of unit squares with given 1000 unit line segments. They use the explicit formula and show that \(T_{478}=1000\), that is, with 1000 matches one can construct at most 478 squares.