Since I was sent a PM asking to explain this problem, I give an example with five segments. Note that $ABCDEF$ is a broken line consisting of five segments $AB, BC, CD, DE, EF$ respectively. The red line doesn't self intersect and it's start and end points $A, F$ are distinct. Extend the five segments to form lines. The number of lines formed in this case is five. Like that, we have to find the minimum number of straight lines for thirty two segments. To get minimum number of straight lines, we have to get maximum number of coincident lines.

Note

[asy][asy] import graph; size(21cm, 24cm); real lsf = 0.5; pen dp = linewidth(0.6) + fontsize(9); defaultpen(dp); pen ds = black; pen ffqqtt = rgb(1,0,0.2); pen ffqqww = rgb(1,0,0.4); draw((2.94,(+10.9+1.34*2.94)/1.74)--(24.47,(+10.9+1.34*24.47)/1.74)); draw((2.94,(+41.26-1.76*2.94)/2.16)--(24.47,(+41.26-1.76*24.47)/2.16)); draw((2.94,(+41.47+0.92*2.94)/4.74)--(24.47,(+41.47+0.92*24.47)/4.74)); draw((2.94,(-27.2+2.14*2.94)/0.42)--(24.47,(-27.2+2.14*24.47)/0.42)); draw((2.94,(+81.26-2.8*2.94)/2.76)--(24.47,(+81.26-2.8*24.47)/2.76)); draw((6.36,11.16)--(8.1,12.5),ffqqtt+linewidth(1.2pt)); draw((8.1,12.5)--(10.26,10.74),ffqqww+linewidth(1.2pt)); draw((10.26,10.74)--(15,11.66),ffqqww+linewidth(1.2pt)); draw((15,11.66)--(15.42,13.8),ffqqww+linewidth(1.2pt)); draw((15.42,13.8)--(18.18,11),ffqqww+linewidth(1.2pt)); dot((6.36,11.16),ds); label("$A$", (6.05,11.48),NE*lsf); dot((8.1,12.5),ds); label("$B$", (8.05,12.86),NE*lsf); dot((10.26,10.74),ds); label("$C$", (10.34,11.04),NE*lsf); dot((15,11.66),ds); label("$D$", (14.49,11.12),NE*lsf); dot((15.42,13.8),ds); label("$E$", (15.9,13.86),NE*lsf); dot((18.18,11),ds); label("$F$", (18.38,11.19),NE*lsf); clip((2.94,4.99)--(2.94,18.34)--(24.47,18.34)--(24.47,4.99)--cycle); [/asy][/asy]