Triangle Friendly Sequences
Source: Canadian Repêchage 2013: Problem 7
May 20, 2014
Asymptotecombinatorics proposedcombinatorics
Problem Statement
Consider the following layouts of nine triangles with the letters in its interior.[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(200);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = 1.740000000000003, xmax = 8.400000000000013, ymin = 3.500000000000005, ymax = 9.360000000000012; /* image dimensions */
draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286)--cycle);
/* draw figures */
draw((5.020000000000005,8.820000000000011)--(2.560000000000003,4.580000000000005));
draw((2.560000000000003,4.580000000000005)--(7.461947712046029,4.569577506690286));
draw((7.461947712046029,4.569577506690286)--(5.020000000000005,8.820000000000011));
draw((3.382989341689345,5.990838871467448)--(4.193333333333338,4.580000000000005));
draw((4.202511849578174,7.405966442513598)--(5.828619600041468,4.573707435672692));
draw((5.841878190157451,7.408513542990484)--(4.193333333333338,4.580000000000005));
draw((6.656214943659867,5.990342259816768)--(5.828619600041468,4.573707435672692));
draw((4.202511849578174,7.405966442513598)--(5.841878190157451,7.408513542990484));
draw((3.382989341689345,5.990838871467448)--(6.656214943659867,5.990342259816768));
label("A",(4.840000000000007,8.020000000000010),SE*labelscalefactor,fontsize(22));
label("B",(3.980000000000006,6.640000000000009),SE*labelscalefactor,fontsize(22));
label("C",(4.820000000000007,7.000000000000010),SE*labelscalefactor,fontsize(22));
label("D",(5.660000000000008,6.580000000000008),SE*labelscalefactor,fontsize(22));
label("E",(3.160000000000005,5.180000000000006),SE*labelscalefactor,fontsize(22));
label("F",(4.020000000000006,5.600000000000008),SE*labelscalefactor,fontsize(22));
label("G",(4.800000000000007,5.200000000000007),SE*labelscalefactor,fontsize(22));
label("H",(5.680000000000009,5.620000000000007),SE*labelscalefactor,fontsize(22));
label("I",(6.460000000000010,5.140000000000006),SE*labelscalefactor,fontsize(22));
/* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]A sequence of letters, each letter chosen from is said to be triangle-friendly if the first and last letter of the sequence is , and for every letter except the first letter, the triangle containing this letter shares an edge with the triangle containing the previous letter in the sequence. For example, the letter after must be either , or . For example, is triangle-friendly, but and are not.[*] (a) Determine the number of triangle-friendly sequences with letters.[*] (b) Determine the number of triangle-friendly sequences with exactly letters.