# Just how many logs can two woodchucks chuck in a minute?

Disaster has struck on your tree farm! You drove up yesterday from the city to find that just over half of your trees were down! Your trees had been set up in a 21x21 grid, and it looked like a random assortment of them had beed felled and demolished over the past few weeks since you'd been here last. As an excellent tracker, you scoured the area for tracks, and discovered that woodchucks had been responsible. In a panic, you decided to cut your losses, and ordered all the rest of your trees be harvested while you decided what to do about the woodchucks.

You spent that night in your office at the tree farm. The next morning you awoke to find a note on the doorstep.

Just How Many Logs Can Two Woodchucks Chuck in a Minute?

You fool! We spent weeks setting this up to answer the age old question about how many logs we can chuck! We had the answer, but when you cut down the rest of the trees, our work was destroyed, and now you'll never know... unless you can recreate our work!

All we can give you are these notes our woodchuck cartographers made while they were perparing to create a map of which trees we felled. Their vocabulary isn't very extensive, so we think you have your work cut out for you. If you can make any sense of it, you should be able to figure out how many logs us two chuckers can chuck in a minute.

Signed, The Woodchucks

Underneath that note, you found another piece of paper with two columns:

EDIT: I'M SO SORRY! There was an error in this table. The second line under West side was missing one letter A. It is correct now: A A A Oak A A

West Side: North Side:
Sapling Oak Sapling Sapling A Ax Sapling
A A A Oak A A A A Ax A A
A Oak A A A A Oak A A Oak A A A Oak A
A Oak A Tree A Oak A A Oak A A A A Oak A
A Oak A A Ax A Oak A A Oak A A A Oak A
A A A A A A A A A A A A
Sapling A A A Sapling Sapling A A A Sapling
Ax Ax Tree
A A Ax Oak Oak A Ax Stick Oak
A Tree Ax A Ax Oak A A Ax Oak
A Forest Ax Oak Ax Oak Ax A A Ax
Ax A A A A Tree Oak A A Stick
Oak A A A Oak A Ax Forest Ax
A A A A Oak Ax A Tree
Sapling A A A Ax A A Oak A A Sapling
A A A Ax Oak A Ax A A A A A
A Oak A Oak Ax Ax Ax A Ax A Oak A
A Oak A Ax A Ax Oak A A Ax A A Oak A
A Oak A A Rainforest Ax Oak Ax A A Oak A
A A A A A Ax A A A Ax A A A A
Sapling Ax A A A A Ax A Sapling

Can you answer the woodchucks' question?

• I thought I had a nice plausible theory for what was going on, and it still looks broadly plausible, but it seems to me like it lands me with a contradiction rather quickly. @Stevish would you be so kind as to double-check the first three entries in the left column, and the last seven in the right column? (Note: this is not the exact set of things needed for the contradiction; I am adding a bit of fuzz in the hope of making this comment less informative to others who would prefer not to be spoiled :-).) Jan 10 at 23:59
• For the avoidance of doubt: I bet the error is in fact mine (most likely my nice plausible theory is wrong). Jan 11 at 0:00
• my revised version is looking quite reasonable, but after a while it still runs into contradictions. I've double-checked my transcriptions and sought computerized help. @Stevish despite the above the error is still probably mine, but I'd be glad if you'd double-check the puzzle... Jan 11 at 0:51
• Yes, @GarethMcCaughan, you did find an error. As if puzzles weren't hard enough when they were written correctly! I was only able to find one error, though, and that's on the second line of the west side. Jan 11 at 17:01

It seems worthy of note that

of our repertoire of short tree-related words, no two have the same length. You'd think there might be "wood" as well as "tree" or "branch" as well as "forest" or something, but no.

This suggests

interpreting those words as numbers according to their lengths

which in turn suggests

that what we have here is a nonogram.

The most obvious way to try to make that work is

by taking the "west side" entries in the table to describe run-lengths from west to east, and the "north side" entries to describe run-lengths from north to south -- i.e., all in the same order as nonogram clues are usually written.

Unfortunately this runs into two kinds of trouble.

The first is an outright contradiction at the top right, which one arrives at very quickly. The second is that all those 7s and 1-3-1s rather quickly start to look like we're drawing a QR code (I think 21x21 cells may in fact be the minimal size for a valid QR code), but the synchronization-pattern squares are in the wrong places: they should be at top left, top right, and bottom left.

The second bit of trouble suggests a possible fix:

maybe the "north side" numbers need to be understood to run from south to north. (But the "west side" ones still need to run west to east; otherwise we have an equivalent grid rotated 180 degrees and the contradiction is still there, and also that would put the squares in the wrong places again.)

In this case we can

solve the nonogram (there are online solvers, but I did it by hand; I don't think the solution process is so fascinating as to be worth showing all the steps; probably there are ways to use Clever Reasoning to avoid ever having to "case-split" on what's in a particular cell, but I ended up doing that twice) to get this:

                         1                           1   1
1 3 1                         3 1 1
3 1 3 1         2       7 1 1 1 3 1
1 1 1 1 1 7     3 1   2   1 1 3 1 1 1
7 1 1 1 1 1 1   3 2 1 5 6   1 1 1 2 1 2 7
2 2 1 1 1 1 1   5 1 2 1 2 4 3 1 2 1 2 1 1
1 1 3 3 3 1 1   2 1 3 1 1 1 1 1 1 1 3 1 2
7 1 1 1 1 1 7 4 1 3 2 3 3 2 1 2 2 3 2 1 1
+-------------------------------------------+
7 3 7            | # # # # # # #       # # #   # # # # # # # |
1 1 1 3 1 1      | #           #   #   # # #   #           # |
1 3 1 1 1 1 3 1  | #   # # #   #   #     #     #   # # #   # |
1 3 1 4 1 3 1    | #   # # #   #   # # # #     #   # # #   # |
1 3 1 1 2 1 3 1  | #   # # #   #     #   # #   #   # # #   # |
1 1 1 1 1 1      | #           #     #     #   #           # |
7 1 1 1 7        | # # # # # # #   #   #   #   # # # # # # # |
2 2              |                 # #   # #                 |
1 1 2 3 3        | #           #   # #     # # #     # # #   |
1 4 2 1 2        | #   # # # #   # #   #   # #               |
1 6 2 3          |   #         # # # # # #   # #     # # #   |
2 1 1 1 1 4      | # #   #   #   #           #   # # # #     |
3 1 1 1          |             # # #   #   #       #       # |
1 1 1 1 3        |                 #   #   #     #   # # #   |
7 1 1 1 2        | # # # # # # #       #       #       # #   |
1 1 1 2 3 1      | #           #     #       # #   # # #   # |
1 3 1 3 2 2      | #   # # #   #       # # #   # #       # # |
1 3 1 2 1 2      | #   # # #   #       # #   #     # #       |
1 3 1 1 X        | #   # # #   #     #   # # # # # # # # # # |
1 1 1 1 1 2      | #           #     #     #     #   # #     |
7 2 1 1 1        | # # # # # # #   # #     #   #         #   |
+-------------------------------------------+

and then

on turning this QR code into an image and feeding it to an online decoder (one could in principle do this bit by hand as well, but I don't wanna) we get a decode of Title Word 6.

So it seems that the answer to the title question is

two.

• Say, you might know the answer to this. rot13(Vf vg gehr gung perngvat n frg bs ababtenz pyhrf sebz na neovgenel vzntr znl erfhyg va n ababtenz chmmyr jvgu ab havdhr fbyhgvba?) Jan 11 at 2:33
• Yes. Consider rot13(n 2k2 tevq, jvgu gjb funqrq pryyf va bccbfvgr pbearef). This gives you rot13(gur fnzr pyhrf jurgure lbh pubbfr gur AR naq FJ pbearef be gur AJ naq FR pbearef). Jan 11 at 2:36
• It appears to be rot13(zvffvat gur fznyy fdhner ba gur obggbz-evtug), making it invalid. Interestingly, though, the "rot13(gvzvat cnggreaf)" seem to be present. Jan 11 at 13:38
• @codewarrior0, you're right in your comment. There are a lot of ways for these to be non-unique and unsolveable. I had to work through several different versions of this map before finding one that was both solveable and one that could be done with no guessing. It took way more work than I had predicted. That's part of why it was so frustrating that I published it with a mistake in the clues. Jan 11 at 17:08
• @Stevish Not to worry; mistakes happen. Anyway, I've made the appropriate change and indeed everything seems to work now. Thanks! Jan 11 at 23:36