# H-one-one-oneycomb

What path could a honeybee follow to fill all cells with honey, beginning and ending at the center and visiting every cell exactly once?

• At first the only honey is 1 drop in the center cell.

• Each step consists of moving to an adjacent cell and filling it with as many honeydrops as its 6 neighbors, combined, contain at that moment.

• The cell marked (1) is empty at first but should receive 1 honeydrop when it is reached. (Thus, the previous cell in the path would have 1 drop and no other adjacent cells would have been visited yet.)

• The cell marked (6) should receive 6 drops when the path gets there.

• The path ends back at the center cell, whose 6 adjacent cells have a total of 241 drops by then.

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

The following sequence of six steps demonstrates how the rightmost cells might be filled first.

Unknown at an hour after post time: Is there a path that ends with a smaller number than 241?
What is the smallest possible ending number without the constraint of (6)? Already ffao and Jonathan Allan have found paths under 140.

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>:

                 ___                                         ___
___/   \___                                 ___/ ? \___
___/   \___/   \___                         ___/ ? \___/ ? \___
___/(6)\___/   \___/(1)\___                 ___/ 6 \___/ ? \___/ 1 \___
/   \___/   \___/   \___/   \               / ? \___/ ? \___/ ? \___/ ? \
\___/   \___/ 1 \___/   \___/     --->      \___/ ? \___/241\___/ ? \___/
/   \___/   \___/   \___/   \               / ? \___/ ? \___/ ? \___/ ? \
\___/   \___/   \___/   \___/               \___/ ? \___/ ? \___/ ? \___/
\___/   \___/   \___/                       \___/ ? \___/ ? \___/
\___/   \___/                               \___/ ? \___/
\___/                                       \___/


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

                 ___
___/   \___
___/   \___/   \___
___/   \___/   \___/(1)\___
/   \___/   \___/ 1  ___  1 \
\___/   \___/ 1  ___/ 4 \   /
/   \___/   \___/   \     1 \
\___/   \___/   \___/ 5 \___/
\___/   \___/   \___/
\___/   \___/
\___/

• The one on the side of the grid, does that mean that cell needs to be filled with a 1? Commented Oct 6, 2016 at 5:31
• @humn Like, the original publication had the first letter in a larger font size. (comments deleted) Commented Oct 6, 2016 at 5:37
• Puzzle statement now has a clearer description of the (1) cell, @gtwebb, sorry about the ambiguity
– humn
Commented Oct 6, 2016 at 6:01
• "Is there a path that ends with a smaller number than 241?" I believe so: i.sstatic.net/hZFyG.png
– ffao
Commented Oct 6, 2016 at 6:43
• @humn this one is even better at 136 (or worse, because I'm getting farther from the actual 241 goal!): i.sstatic.net/uPsa5.png
– ffao
Commented Oct 6, 2016 at 6:52

Here is a way to 241:

                 ___
___/ c \___
___/ b \___/ d \___
___/ p \___/ v \___/ e \___
/ o \___/ a \___/ u \___/ g \
\___/ q \___/ w \___/ f \___/
/ n \___/ r \___/ t \___/ h \
\___/ m \___/ s \___/ i \___/
\___/ l \___/ j \___/
\___/ k \___/
\___/
                 ___
___/ 1 \___
___/ 1 \___/ 1 \___
___/ 6 \___/61 \___/ 1 \___
/ 4 \___/ 1 \___/56 \___/ 2 \
\___/19 \___/241\___/ 1 \___/
/ 4 \___/29 \___/52 \___/ 3 \
\___/ 4 \___/42 \___/ 4 \___/
\___/ 4 \___/ 4 \___/
\___/ 4 \___/
\___/

And here is the smallest I can make:

136

                 ___
___/ c \___
___/ b \___/ d \___
___/ o \___/ v \___/ e \___
/ n \___/ a \___/ u \___/ f \
\___/ p \___/ w \___/ t \___/
/ m \___/ q \___/ s \___/ g \
\___/ l \___/ r \___/ h \___/
\___/ k \___/ i \___/
\___/ j \___/
\___/
                 ___
___/ 1 \___
___/ 1 \___/ 1 \___
___/ 3 \___/48 \___/ 1 \___
/ 1 \___/ 1 \___/43 \___/ 1 \
\___/ 7 \___/136\___/22 \___/
/ 1 \___/11 \___/18 \___/ 1 \
\___/ 1 \___/15 \___/ 1 \___/
\___/ 1 \___/ 1 \___/
\___/ 1 \___/
\___/

• How do I get that in a spoiler? It all gets messed up! Commented Oct 6, 2016 at 6:49
• Makes no difference it removes loads of the underscores and backslashes. Commented Oct 6, 2016 at 6:55
• @JonathanAllan you have to escape all of the backslashes with \\
– ffao
Commented Oct 6, 2016 at 6:56
• Got 241... now to edit it in. Commented Oct 6, 2016 at 7:44
• So I was right when I got 251! Faith in my arithmetic restored.
– ffao
Commented Oct 7, 2016 at 22:23