I can’t believe what I’m about to say, but can we get back to solving puzzles?
Gallus knew better than to voice his thoughts to his professor. In fact, he has become rather burned out at the whole notion, even as adept as he and the others had become. But more to the point: fashion never really inspired him the way it did others, such as his professor. And given a choice, he’d rather toil through another puzzle than—
“Gallus?”
The griffin looks up with a start, now aware of Rarity staring back at him. “Uh … could you repeat the question?”
“How much impact would you say fashion has had on griffin culture?”
“Fashion? Uh …” Gallus’s eyes dart around the room as he searches for an acceptable response. “Well? … er …” He sighs before concluding: “Not much.”
Rarity frowns at the response. “Surely there must be some emphasis made.”
“Griffinstone isn’t much for finery,” he sheepishly replies.
“Well perhaps we can change that. Come on up and let me take your measurements: we’re going to design an outfit specifically for griffins!”
Gallus gulps inaudibly as he gets up and walks towards the front of the class. He catches a glimpse of various photos on the teacher’s desk. “So what are these?” he asks, pointing out the pictures.
Rarity sighs. “A failed attempt at pleasing the unpleasable, mixed with a tinge of revenge, no doubt.” Her response draws the attention of the rest of the class.
Gallus takes a second look at the photos. “Really? All I see are scarves … six of them, each with some strange markings … are those exes and ohs?”
Rarity’s horn illuminates as the photos separate and spread themselves atop her desk. The students get up and gather around the pictures, each of which is of a scarf trimmed with five gems of various colors.
“Who are all these for?” Smolder asks.
“Chroma-chaotica, the avant-garde fashion designer,” Rarity replies. “Every year, the stallion gets in his head *THE* color combination that is ‘in’, and every year designers from all of Equestria follow his lead. This year, I had hoped to get a jump on the crowd.”
Ocellus takes a closer look at one of the photos. The scarf displays a series of five gemstones: diamond, ruby, another diamond, onyx, and a third diamond. In the bottom-right corner, she discerns an ‘x’ and an ‘o’. “Miss Rarity … what do these symbols at the bottom mean?”
“I had my trusted Manehattan assistant Coco take six samples to Chroma’s representatives; these photos bear those samples. For each sample, she received feedback as to what truly worked, what worked in a minor sense, and what didn’t work at all.”
“Is that what ‘x’ and ‘o’ for?” Yona asks.
“The ‘x’ signifies that Chroma fully agrees with the color and placement of exactly one of those gems,” Rarity explains. “So either he thinks a ruby should be in the second spot, an onyx should be in the fourth spot, or one of the diamonds is correctly placed.”
“And the ‘o’?” Sandbar asks.
“The ‘o’ means Chroma likes the idea of one of the gems, but not its placement. So maybe he believes the ruby should come first, instead. Or maybe last. Or maybe one of the diamonds should be fourth. Unfortunately, there’s no way to know for sure.”
“But you have six of these photos,” Silverstream notes. “That’s enough information, isn’t it?”
As the hippogriff speaks, Ocellus picks up a piece of chalk and notes each photo’s gem ensemble and associated marks on the board:
D R D O D – x o
D R S R R – o o o
O S S E O – x x
R O O S T – x o o
S E T R E – o o
T S O T D – x o
Gallus scans the board. “I guess ‘D’ stands for diamond. What about the other letters?”
“ ‘R’ is for ruby, ‘O’ is for onyx, ‘S’ is for sapphire, ‘E’ is for emerald, and ‘T’ is for topaz,” Ocellus replies. “Really pretty colors: white, red, black, blue, green, and yellow.”
“So … this like puzzle,” Yona points out. “That means we can solve.”
“If only it were that simple,” Rarity sullenly responds. “I’m not particularly adept at figuring things like this out, as you have seen.”
“We are,” Smolder asserts.
“Yes, the whole school has witnessed your capabilities, but I’m afraid there’s an added issue, for which you can thank Suri.”
“Suri?” Ocellus asks.
“Suri Polomare … and old rival of mine. Shortly after receiving these photos, I received a letter from Coco that Suri caught wind of my plans. She was able to gain access to the photos and changed the comments on one of them, and I don’t know which one.”
“Why just one?” Gallus asks. “Why not just alter all of them?”
“It’s her way of toying with me, like baiting a cat with a piece of string. In any event, I daresay the situation is now beyond even your abilities. Only five of these six photos can be trusted as having useful information, and there’s no way to figure out which photo is the wrong one.”
“Some might consider that a challenge,” Gallus retorts.
“No, no, no … I appreciate all you’ve shown myself and the others, but even if it were possible to solve this conundrum, I’m afraid it would take too much time out of our already abbreviated day. But, if you wish to stay late and inspect the photos on your own time, they’re of little use to me now. Now then, let’s discuss proper griffin regalia …”
As early evening comes over Ponyville, the students reflect upon another day’s work …
“Miss Rarity can really work fast under a deadline.” Silverstream notes.
Ocellus turns to Gallus. “So what do you think Griffinstone will make of your new modeling duties?” she asks with a chuckle.
He replies with a laugh. “I don’t want to be the one to disillusion our resident fashion guru, but her first trip to Griffinstone would probably be her last. It’s not exactly Manehattan where I’m from.”
“Maybe she’ll get the idea to create outfits for Yakyakistan next,” Sandbar says as he turns to Yona.
“Yona not think other yaks interested in attire … but then, Yona not think other yaks interested in others at all, so first for everything?”
“Ooh, maybe we’ll design outfits for dragons next,” Silverstream notes. “By the way … shouldn’t Smolder be back by now?”
Ocellus muses for a bit. “She said she wanted a few minutes to take a longer look at those photos. Honestly, I’m surprised she wanted to; she never showed a big interest in puzzle-solving before.”
“Even if she did,” Gallus adds, “there’s no way she’d still be at it. I just assume she caught up with Spike afterwards. Not like any of us have fellow students of our own species around here.”
Sandbar opens his mouth to speak, when the door to their foyer bursts open. The group turn to face Smolder, her eyes wide open and twitching with retinas bulging.
“Smolder,” Yona exclaims. “We just talk about you!”
“Where’ve you been,” Ocellus adds. “It’s been hours since we last saw you.”
“You were with Spike, right?” Gallus asks. “I mean, it’s not like you spent all that time on Miss Rarity’s photos … did you? …”
Smolder replies stiltedly: “Four hours … thirty-seven minutes … twenty-five seconds … but I did it.”
“You did what?” Sandbar asks.
“I figured … it … out …”
Yeah, I'd just be guess-and-checking on this one. I think I'll have to defer unless more hints are provided.
This was fun. For the record, it took me about an hour and a half to plot all this out. Full solution below, but brace yourself; this one's a ride.
Suppose that equation (3) is legitimate. The possibilities for the correct gems from (3) are:
1. O, O
2. E, O
3. S, S
4. S, E
5. S, O
We will take these cases one at a time.
Case 1:
If (O,O), then (2) implies two R's (due to not having an S). However, then (1) implies no D, and thus (2) implies three R's... which contradicts itself, as if the correct solution contains three R's, then (2) must contain an x. Therefore, (1) or (2) is the false statement. However, (4) implies that we have only one of R and T in the correct solution. But this contradicts (5), as now it is impossible to get two valid gems. ((3) implies no S and no E, but that means the only possible answer would be having T and R both being valid.) Therefore, (4) or (5) must also contain a false statement. Therefore, (O,O) cannot happen.
Case 2:
If (E, O), then (2) implies two R's (due to not having an S). However, then (1) implies no D, and thus (2) implies three R's... which contradicts itself, as if the correct solution contains three R's, then (2) must contain an x. Therefore, (1) or (2) is the false statement. However, the three valid options from (4) must be R, O, and T, due to no S and no second O. However, this contradicts (5), since we would then have an R, an E, and a T in the solution, compared to two valid gems. Therefore, (4) or (5) must also contain a false statement. Therefore, (E,O) cannot happen.
Case 3:
If (S,S) then (2) is immediately contradicted due to the middle slot containing an S in both. However, this also implies that the three valid gems from (4) are R, S, and T (due to the correct solution having no O's), which in turn immediately contradicts (5), which also contains a R, a S, and a T, but does not contain three valid gems. This would imply that (2) is false, but also that one of (4), (5) is false, which is impossible. Therefore, (S,S) cannot happen.
Case 4:
If (S,E), then (5) implies that we have neither T nor R in the correct solution. However, (2) implies the existence of at least one R, therefore either (2) or (5) is false. In addition, (4) implies that the three valid gems are R, S, and T, which also contradicts (5). Therefore, if (S,E), then (5) is the false statement. However, then (6) implies that we have neither D nor O in the correct solution, contradicting (1). This would require multiple false statements. Therefore, (S, E) cannot happen.
Case 5:
If (S, O), then (6) implies no D and no T. As (3) already implies no E, exactly one S, and exactly one O, this would require three R's, which contradicts (2). However, by (5), an R cannot fit in the fourth slot (no x), therefore either R's fill both the second and third slots (contradicting the S in (3)), or the first and fifth slots (contradicting the O). Therefore, (6) must be our false statement. However, by (2), we must have at least one R. Therefore, by (1), we cannot have a D, and by (4) we cannot have a T. Which bring us right back to needing three R's to complete the layout, which we've already seen contradicts (2) anyway. Therefore, (S,O) cannot happen.
Therefore:
Note that no matter what we assume is the correct pair of gems from (3), we reach a set of contradictions that would require multiple bogus statements. Therefore, (3) is the faulty statement in the bunch, and the other five can be assumed legitimate.
Now, we note by (5), that R, S, and T cannot all be in the solution. (Else we'd have at least 3 X's and O's.)
Similarly, by (6), O, S, and T cannot all be in the solution.
However, by (4), if S and T are both in the solution, then the third "present" gem must be an O, or a R. Therefore, S and T cannot both be in the solution. In particular, O must be present in the solution.
By (2), R must be present in the solution, simply because there are three correct gems there and one of them must be an R.
By (1), since we have an R and an O, we cannot have a D, as we've already accounted for the two present gems.
By (2) again, since we do not have a D, and because three R's are impossible (else we'd have at least one x), we must have two R's and an S.
By (5), since we have an R and an S, we cannot have an E or a T. Note that at this point, we've proven that the only possible valid gem types are R, S, and O.
By (4), we cannot have a second O. Therefore, since we can't have three R's and we know we have two R's, the gems must be two R's, an O, and two S's.
By (2), the R's must be in the first and third locations. (Otherwise we would have an x in 2).
This means the x in (4) is already accounted for (by the R in the first position), and therefore the O in the second position and the S in the fourth position are incorrectly placed.
Therefore, the solution is RSROS.
Or, if you'd like it in a weird math/programming hybrid notation...
4,5,6 -> O && (!S || !T)
2 -> R && (S || D)
1,2 -> (!O || !D)
1,2,4,5,6 -> O && !D && S && R && !T && !E (That is, only O, S, and R.)
4 -> !(2O)
2 -> 2R
Only possible with 2 R, 1 O, 2 S.
2 -> R?R??
4 -> RSROS
Anyway, for the record, if you're wondering how I knew to go after (3) as the incorrect statement... I got lucky. I actually started with the notation above, realized that RSROS satisfied every statement except (3), and that meant all I had to do was prove (3) was the bad apple in the bunch.
9510034
I don't know what's more astounding … that you were able to actually solve this behemoth, or that you described the experience as "fun".
Well done!
9512961
To be fair, mathematics is kind of my thing, and this was somewhere on the border between logic puzzle and legitimate math problem.
That, and I enjoy a challenge, so when you said it might not get solved without hints, that was pretty much throwing down the gauntlet. :P