calculation puzzle - How to fill a honeymoon

       

How can a honeybee visit all cells exactly once in this crescent shaped honeycomb, beginning at the bottom tip and ending at the top?

• 
  

  The starting cell, at lower left, has 1 drop of honey. All other cells begin empty.

  
  


• 
  
  

  Each step consists of moving to an adjacent cell and filling it with 0, 1 or 2 honeydrops, based on how many total drops its (1 to 6) neighboring cells, combined, contain at that moment.

  
  


• 
  

  The number of new honeydrops is the remainder of the surrounding total when divided by 3.

  
  

$$\small \begin{matrix} \textsf{Total adjacent drops} ~&~ 0,3,6,9,12 ~&~ 1,4,7,10 ~&~ 2,5,8,11 \\ \textsf{Number of new drops} ~&~ 0 ~&~ 1 ~&~ 2 \\ \end{matrix}$$

• 
  

  The top cell, marked (1), is empty at first but should receive 1 honeydrop when it is reached.

  
  
  


• 
  

  The six cells with 0 should receive 0 drops when they are reached. (The sums of their neighbors’ drops should be multiples of 3 at those moments.) Other cells may also receive 0 drops.

  
  

No need to spoilerize a text solution. Site implementation makes that unduly onerous.

The following sequence of eight steps demonstrates how the bottommost 0 cell might be reached.

 

In this example, division by 3 comes into play when the last cell to receive 1 drop, on the seventh step, has a total of 1+2+2+2 = 7 drops in its adjacent cells, giving a remainder of 1 when divided by 3. The eight step correctly reaches the 0 cell as 1+2 = 3, which leaves 0 remainder when divided by 3.

This puzzle forthrightly, though incompletely, imitates Two honeycomb hints by Yuriy S.

This is meant to be convenient on paper and in a text editor.   Here is a template for

...

:

                                                         ___
                                                        /(1)\___
                                                        \___/   \
                                                            \___/
                                                            /   \___
                                                            \___/   \
                                                            /   \___/
                                                            \___/   \___

                                                            /   \___/ 0 \
                                                            \___/   \___/
                                                         ___/   \___/   \
                                                        / 0 \___/   \___/
                                                        \___/   \___/   \
                                                     ___/   \___/   \___/
                                                    /   \___/   \___/ 0 \
                                                    \___/   \___/   \___/
                                                 ___/   \___/   \___/
                                                / 0 \___/   \___/   \

                                                \___/   \___/   \___/
                                             ___/   \___/   \___/
                                         ___/   \___/   \___/ 0 \
                             ___     ___/   \___/   \___/   \___/
                            / 1 \___/   \___/   \___/   \___/
                            \___/   \___/   \___/ 0 \___/
                                \___/   \___/   \___/

And this is how the eight-step example could begin to resemble a maze:

                                                \___/
                                             ___/   \___/
                                         ___/ 2 \___/   \___/
                             ___     ___/ 2  ___  2 \___/   \_
                            / 1 \___/ 1 \   / 1  ___/   \___/
                            \___  1  ___  1 \___  0 \___/
                                \___/   \___/   \___/

Answer

I noticed that things got a lot easier the more zeroes you are able to stick in there, and with that in mind, here's my solution:

Sorry it's not in text form, but I solved it on a tablet without a nice text editor. Lines between cells indicate where the bee has to travel.

If you'd like, I can update with a text version when I get to a computer, but I hope this is sufficiently readable on its own.