# Zed's Garden: Nothing But Flowers

My friend Etta and I constantly pay visits to each other. Etta runs a little flower shop down the street from my house, so naturally I try to stop on by while I gad about town, and sometimes I'll even offer help whenever she needs it — provided it's not too much trouble.

Just last week, she received a massive shipment of flowers all squared away in her backroom for the upcoming season but because her usual stockperson called in sick, yours truly had to lend a hand.

She started by handing me two boxes, each containing one kind of plant, and asked me to pull out each one. To my left was a box with the shipping label IM-74-Q9-668-64C-5KK, and to my right was another box but with shipping label AZ-240-1oF-BY-38o-25A. I figured I'd start with the box on left, and so opened it up.

Much to my dismay, I saw not a flower, but... a number! It was the strangest thing: it was the number 20, sitting stiff in a terracotta pot, looking... well, rather not plant-like.

Naturally, I picked it up without much thought, only to have part of the number brush against my forearm. Suddenly, pins-&-needles crept up my arm and I yowled in pain, startling Etta who immediately rushed to my aid. She looked at the box's label and frowned. "Oh dear, you need to be more careful! Those spines will tear you to shreds!" My eyes were watering from the pain.

She got some gauze and wrapped the tender spot of my arm until I felt fit enough to continue my work. Of course, I nearly forgot I left the other box untouched, so I hastily went back to my station.

Opening the second box, I was surprised yet again: this time, the number 10 sat rather low in the soil. Warily, I quickly grabbed the number — but carefully enough not to make the same mistake twice — and practically threw it down onto the table; Etta turned around just in time to see me do this and scolded me. "What are you doing, you brute?! That there is a delicate one! Just because it sounds like the last one doesn't mean you get to brutalize it." She rushed passed me and cupped the number gingerly in her palm without so much of a scratch on her fingers — all the while, flashing me the most bitter scowl imaginable.

I was exasperated. "Honestly, how was I supposed to know? I'm no botanist, but I know a number when I see one!" Her frown looked more serious.

"Well, maybe if you read the label once in a while, you'd have figured it out!" A lightbulb suddenly went off in Etta's head. "Fine, since you're being clueless at the moment, I'll make a little game out of this." She turned around and lugged two more boxes in front of me, each with new labels.

She opened up the first box on my left to reveal a shrubby number 21. She pointed emphatically at the label, which read 1QF-89-635-4V0-156-3T3-4o0-36C. "This guy you do not want to just grab and thrash about like before. See the little thorns on the bark?" Her finger strayed back to the number. I tilted my head a few times looking at the thing, but I didn't see anything noticeable: just a squat, earthy number. Still, I didn't wanted to disappoint Etta, so I pretended I understood and nodded my head; it seemed that was enough to convince her.

She then turned to the box on my right, whose label read PXC-3CD-2G7A-Y80-1WY3-7K8-7AI-56o. As she opened the box, I peered in and — lo and behold — another number: this time it was a fat, low-lying 420. "This one, however, is safe to handle. See for yourself." I hesitated, of course, but Etta reassured me with a hush of her voice; I reached out, and I was surprised to feel small hairs tickling my fingertips. "Sure, they both sound related, but this guy's closer to dandelions than 'fairy trees'."

She crunched up her face again, though this time more worried than irritated. "Hmm... I'm not totally convinced you understand yet, so I'll try two more". Once again, she pulled out two more boxes, the one on the left reading 2K8-Ao1-4U8-DUR-AJ6-CBC-1CM and the one on the right reading 54G-LC2-9oG-ZEL-ID4-FE6-1Q8. She ripped open both boxes at the same time, the left one with an aromatic and rather beautiful 70 and the right one with a hardy and wooden 140.

"Now, you'd think these guys are related, right?" I could do little but nod unconvincingly, but Etta didn't seem to mind my confusion. "This guy on the left, however, is normally quite thorny, but I got one of the only cultivars with no thorns, so you're safe." I let out a shaky breathe before she turned to the plant on the right. "This guy is also pretty harmless, although you could make some sharp things out of it when it's fully grown." At this point, Etta seemed quite satisfied with her tutorial. "Now then, it's you're turn." Two new boxes sat in front of me. She opened them up for both of us to see their contents.

A stout, little 12 poked out on the left with the shipping label 2XC-4X-2I8-3VC-55C on the box, while the box on the right held a simple 2 in its belly with the label CH-7X-88-69-E8 on the side.

"Tell me," Etta poised, "which of these are safe to touch without gloves?" I immediately broke out in a sweat; both looked rather innocuous. "Now this is a toughy, so don't make any wild assumptions. It's a 50-50 shot." She rested her chin into her hands.

"So, what's it gonna be? Left... or right?"

I really don't want to get hurt again, but I have no clue how Etta expects me to figure this out... perhaps you could help me out?

Just like Etta said, you need to tell me exactly which plant is safe, so don't just guess. Hopefully you can figure out the name of the two plants (along with all the others) to make this easier.

HINTS

I may not yet know what's-what, but I did notice something... peculiar about Etta's behavior: she could neither tell me the identity of the plants without first looking at the shipping label (i.e., my first "prickly" situation) nor without opening the box and looking at the plant-number; she needed both. Why would a botanist not know what plant is which by looks alone? There's got to be a connection... but what is it? What remains to be discovered?

• So, rot13(sbe rnpu cynag V unir fbzr dhnqehcyrf bs fznyyvfu aba-artngvir vagrtref (V gnxr vg gur yrsgbiref pna or qvfpneqrq) bar bs juvpu vf bs pbhefr fcrpvny (naq arire unf nal yrsgbiref gb qvfpneq). Cerfhznoyl gurfr ner zrnag gb lvryq yrggref be fbzrguvat ohg vg frrzf yvxr gurer ner znal znal jnlf va juvpu gung pbhyq or qbar.) The rot13(ynfg fragrapr bs gur uvag fhttrfgf gung znlor V fubhyq or nggraqvat gb erznvaqref naq abg bayl qvivfvovyvgl), but maybe that's overinterpreting. Not necessarily fishing for hints here, but Zed may be interested to know where someone's stuck... – Gareth McCaughan Aug 29 at 15:42
• ROT13("Yrsgbiref")? ROT13("fcrpvny")?! Who said anything about ROT13("yrsgbiref") or being ROT13("fcrpvny")? Why would you believe you'd need to do that? You are closer than you think, though... I would certainly say you are not overinterpreting. – Omicron Zed Aug 29 at 16:15
• I'm not sure how many levels of irony are at work in your last comment, but: The business about rot13(yrsgbiref) was intended to clarify why I have rot13(dhnqehcyrf) rather than, say, arbitrary-length sequences. The rot13(fcrpvny) one is derived from rot13(gur cynag ahzore vgfrys). The most obvious things to do with rot13(erznvaqref) feel unlikely to work, but I'll give them a go. (Probably not right now, though.) – Gareth McCaughan Aug 29 at 19:38

Since this has been here since April, I don't feel too bad about posting a partial. This is basically explaining the cryptic remarks in my comments above.

First of all,

note that each plant's label is made of groups of alphanumeric characters; the letters are all uppercase except that "o" is lowercase and "O" doesn't appear, probably to distinguish it from "0". This suggests, after a little thinking, interpreting these groups in base 36. Here are the results, each preceded by the plant's own number:
20 670-256-945-8000-7932-7220
10 395-2736-2175-430-4200-2782
21 2247-297-7889-6300-1482-4935-6048-4116
420 33600-4333-114310-44352-89355-9800-9450-6720
70 3320-13825-6272-17955-13650-15960-1750
140 6640-27650-12544-45885-23800-19950-2240

What's now apparent is that

many of these are surprisingly round numbers. How do they factorize?
20 2.5.67 - 2^8 - 3^3.5.7 - 2^6.5^3 - 2^2.3.661 - 2^2.5.19^2
10 5.79 - 2^4.3^2.19 - 3.5^2.29 - 2.5.43 - 2^3.3.5^2.7 - 2.13.107
21 3.7.107 - 3^3.11 - 7^3.23 - 2^2.3^2.5^2.7 - 2.3.13.19 - 3.5. 7.47 - 2^5.3^3.7 - 2^2.3.7^3
420 2^6.3.5^2.7 - 7.619 - 2.5.7.23.71 - 2^6.3^2.7.11 - 3.5.7.23.37 - 2^3.5^2.7^2 - 2.3^3.5^2.7 - 2^6.3.5.7
70 2^3.5.83 - 5^2.7.79 - 2^7.7^2 - 3^3.5.7.19 - 2.3.5^2.7.13 - 2^3.3.5.7.19 - 2.5^3.7
140 2^4.5.83 - 2.5^2.7.79 - 2^8.7^2 - 3.5.7.19.23 - 2^3.5^2.7.17 - 2.3.5^2.7.19 - 2^6.5.7
Lots of factors of 2,3,5,7. Way fewer higher factors. And I note that the plant-numbers themselves only have 2,3,5,7 as factors.

So

it's natural to consider throwing away the primes above 7 and look only at how many copies of the "small" primes divide each number.
2010 1010-8000-0311-6030-2100-2010
1010 0010-4200-0120-1010-3121-1000
0101 0101-0300-0003-2221-1100-0111-5301-2103
2111 6121-0001-1011-6201-0111-3022-1321-6111
1011 3010-0021-7002-0311-1121-3111-1031
2011 4010-1021-8002-0111-3021-1121-6011
Perhaps there's some obvious way to turn these into letters (so, e.g., maybe the first one becomes SPINES or something, though that particular answer is unlikely given that 1010 and 2010 are different), but if so it's far from obvious to me. It's easy enough to think up various schemes by which one could encode letters in quadruples like these, and use the plant-numbers to "twist" them, but it seems like there are dozens of such schemes, all about equally plausible, and I don't think I have the patience :-).

Alternatively,

we could take note of the curious phrase "What remains to be discovered?" in the hint, which suggests that perhaps we are concerned not with exponents but with remainders. The most obvious ways to try to use these are (1) take remainders of group-values modulo 210 (= 2.3.5.7), (2) take remainders of group-values modulo plant-numbers, (3) take remainders of group-values mod 2,3,5,7 separately. Here are the results for 1:
20 40-46-105-20-162-80
10 185-6-75-10-0-52
21 147-87-119-0-12-105-168-126
420 0-133-70-42-105-140-0-0
70 170-175-182-105-0-0-70
140 130-140-154-105-70-0-140
and for 2:
20 10-16-5-0-12-0
10 5-6-5-0-0-2
21 0-3-14-0-12-0-0-0
420 0-133-70-252-315-140-210-0
70 30-35-42-35-0-0-0
140 60-70-84-105-0-70-0
and for 3:
0206 0105-0114-1000-0206-0021-0203
0103 1203-0016-1005-0103-0000-0123
1010 1020-1023-1240-0000-0025-1000-0030-0010
0000 0000-1130-0100-0020-1000-0200-0000-0000
0100 0202-1100-0220-1000-0000-0000-0100
0200 0104-0200-0140-1000-0100-0000-0200
and frankly none of this looks at all promising; too many zeros, aside from anything else.

I'm very confident that

we want to begin by interpreting groups in base 36

and fairly confident that

something to do with the primes 2,3,5,7 is then relevant

but beyond that I'm somewhat stuck; the things that seem most obvious to me don't work, and there are too many not-so-obvious things. Maybe I'm missing some clue lurking in the text. One thing that jumps out at me is

the phrase "all squared away", suggesting that we look at the parity of the exponents (equivalently, divide each number by the largest square that divides it exactly). But this, like the other kinda-obvious idea of just looking at which exponents are nonzero, has the drawback that it gives us only 4 bits per group (along with 4 bits from the plant-number itself, if we treat that the same way).

It's also interesting that

larger plant-numbers seem to go with larger group-values, to some extent. That might be coincidence or not, and if not coincidence might be significant or not...

• Yikes! I didn't expect it would THAT hard to find a plausible answer, but I guess that's what I get for being cheap with hints. I do admit, I underplayed the amount of hints available because I wanted it to be a challenge, but your work shows that I might have left it too open for interpretation. I will say, you are on the right track in finding the ROT13("onfr") of the cipher, but I implore you: do NOT throw away values, as each number is necessary to solve the cipher. There's no special trick, either: no ROT13("ovgf be cevzrf") to consider; it all should rely on generally basic algebra. – Omicron Zed Sep 5 at 18:47
• Hmm, interesting. I'll consider further. – Gareth McCaughan Sep 5 at 19:01