hey, even though it sounds sooo easy but I’m somehow confused , I don’t know that part am I doing wrong but I just counted the numbers and still can’t reach the correct answer

Hi! So for the top one, I counted the number of triangles that surrounded the hexagon on the top right and subtracted the ones than overlapped with the triangles that also surrounded the other hexagons and I got 12!

For the second one you basically draw out X with A B and C around it with lines connecting X to each town and each town to each other, doesn’t matter the distance. You can see that in the first option, it only visits each town once but doing multiple rounds to and back to X so it can’t be the shortest route. the second option is the shortest journey. Hope this helps!

thank you for your time, but unfortunately it still doesn’t make any sense to me

this is my interpretation , what is wrong with it ?

I made 2 identical shape of the whole picture and after counting, the fraction of number of hexagon/ num of triangle

for the second one yeah it is obvious that option 2 is the shortest but how can we make sure that option 1 is even longer that the last 2 options ?those two have passed more roads

i mean even if we count the roads passed by, it would be 5 for XAXBCX and 6 times in XAXBCBX ,this was just an example, what am I missing?

sorry for making these a great deal , these really are a pain in the neck

I drew some diagrams hope this helps!! No problem, ask me anything, it’s hard to explain haha

The crosses are the triangles that overlap with the over large hexagons that surround the dark hexagonal shape

For the next question I drew the routes out for D and E. Remember the repeating routes from B to X and X to C is because of the possibility that it could be closer to X than each other. For the first one, he visits each town ONCE with multiple rides to and from X which means there can be shorter routes than that to visit each town.

thank you raya , I truly appreciate it all , these diagrams were really helpful