As soon as you make a sketch to the problem, the solution will come into your mind instantly.


The band has the form of a polygon with 12 sides.


Six of these 12 sides have the length equal to the length of the side of a section of the pencil, which is half of its diameter,

i.e. 3.5 mm.


Six other sides have the length   mm.


Thus the length of the elastic band is equal to the perimeter of this 12-gon


    P =  =  =  = 57.37 mm (approximatyely, correct with two decimal places).

I think somebody is forgetting that the elastic band 
has straight parts and round parts.

OK, maybe they took the pencils to be hexagonal in cross-section
and not circular.  I have both kind in my pencil collection.

Anyway I took the pencils to be circular in cross-section.

The elastic band is in green:



There are 6 straight parts and 6 round parts to the elastic band.

Each straight part is the length of two radii, as we see by this
red rectangle. which is 1 diameter or 7mm.  So the 6 straight
parts amount to 6×7mm or 42mm.



And as you can see by the 60° red sector,



Each round part of the elastic band is 1/6 of a circumference
of one of the circles.  And there are 6 of them, so all 6
total to one circumference of each circle, which is 

So the answer is 6 straight parts which is 42mm plus 6 round
parts which is  mm.

Answer  mm.

Edwin