# 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 <pre>...</pre>:

                                                         ___
/(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 \___/
\___/   \___/   \___/

• If I had the time to devote to this I would figure it out. Oct 7 '16 at 13:25
• Thank you for having devoted time to getting the spoiler to work in Sconibulus's solution, @dcfyj! This is meant to be enjoyable to solve whenever the time is right, not just for posting, because of how the path can be re-kinked fairly locally to work through problems along the way.
– humn
Oct 7 '16 at 21:53

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.

• No need for text, which is just meant as a convenient option when a graphics program isn't handy. Looking forward to checking this out.
– humn
Oct 8 '16 at 1:57
• Nice use of elevator-like segments. I had hoped for something like that but found a different solution first.
– humn
Oct 9 '16 at 8:35

Working out the upper end in reverse was very helpful for finding a solution, before I'd done that, I had tried about half a dozen things, after solving all the forced moves possible there, my second try made it work.

                                                        /(1)\___                                                        \_44/ 1 \                                                            \_43/                                                            / 1 \___                                                            \_42/ 0 \                                                            / 1 \_41/                                                            \_40/ 2 \___                                                            / 2 \_39/ 0 \                                                            \_38/ 0 \_36/                                                         ___/ 2 \_37/ 0 \                                                        / 0 \_29/ 1 \_35/                                                        \_28/ 2 \_30/ 2 \                                                     ___/ 1 \_27/ 1 \_34/                                                    / 2 \_26/ 1 \_31/ 0 \                                                    \_19/ 1 \_25/ 2 \_33/                                                 ___/ 2 \_20/ 0 \_32/                                                / 0 \_18/ 2 \_24/ 0 \                                                \_17/ 2 \_21/ 0 \_23/                                             ___/ 1 \_15/ 1 \_22/                                         ___/ 2 \_16/ 1 \_14/ 0 \                             ___     ___/ 2 \_6_/ 2 \_11/ 0 \_13/                            / 1 \___/ 1 \_5_/ 1 \_7_/ 2 \_12/                            \_1_/ 1 \_3_/ 1 \_8_/ 0 \_10/                                \_2_/   \_4_/   \_9_/

• That was much more of a pain to hide than I thought it was going to be... Oct 7 '16 at 14:41
• Many thanks for making the spoiler work, @dcfyj! You solved one of the many difficult puzzles posed by the website itself. (From a feeling of guilt, though, the puzzle statement has been revised to badmouth the site and to reassure that spoilerizing text is not required.)
– humn
Oct 7 '16 at 21:41
• Very cool solution, especially the reverse approach! Not easy, but more revealing than working forward, and my solution matches exactly in the homestretch (home straight?). There is probably more than one solution but would you mind rechecking the 2 at cell 21? I keep getting 1+2+2+1 = 6 $\to$ 0 and hope to be overlooking something.
– humn
Oct 7 '16 at 21:44