This one took a while.

Always remembering to approach these problems from at least two angles: top-down AND bottom-up. We know from page 2:

“…Theorem[‘]s…solutions will always be a word or phrase. …”

So after piecing together some puzzle answer fragments from top-down puzzle solving, you can try to bottom-up reverse-engineer a coherent answer word/phrase from the available answer fragments.

My first round guess at groups answers (before hints):

1) lost dog / lost weight / lost [runner’s] leg / lost ‘n found:

clearly I lost the difference

2) in shape / in love / strike out / out house:

50/50 chance of in or out

3) a music note / Q-tip / T-shirt / X-ray:

only one item without a letter in it’s proper colloquialism.

4) wind-borne steam / wind-born water / wind-born mill / water-adjacent chain:

clearly Wind for the win! (Wrong)

5) LUXury suit/ good LUCK charm / kindle / LUSstrous sheen:

Luck or Lux clearly meant to be a fragment

6). Cup of coffee (expresso-latte?) / neon / plane / office (empty)?

The hints verified 1’s lost weight so clearly the difference dichotomy would have to be lose vs something else so I had to reconsider all pics for something different.

In 2) in shape became work out so in became the odd man out.

In 3) turns out I guessed correctly for the wrong reason

In 4) after reconsideration the commonality was clearly a rhyme scheme: train / rain / windmill / chain. One of these was not like the others

In 5) hints-by-hand revealed 3 piece suit so my out-of-luck -schema was rendered from luck-lover to numbers: 3-piece suit / 4-leaf-clover / kindle / 6-pack abs, and one of those didn’t count the same.

In 6) by-hand clues I reveled in revealed take off which didn’t put me off, and got me off to the races identifying

cOFFee / neON / take OFF / OFFice.

On of these was not like the others.

Lieu and so I inverted from top down to bottom up, and started to reverse engineer for a coherent winning word or phrase answer from the candidate fragments:

1) Won

2) in/out

3) a note

4) windmill

5) e-reader

6) on

A little selective pruning this crop yielded:

*Won-in-a-mill-e-on”

Or

One in a million.

Enjoy

I agree that it was difficult, but certainly not impossible. I was able to solve this one without hints. The most important thing to do is

Figure out the six different images. Then read out the phrase you have. Even if you have some images wrong, the phrase will sound familiar. I too had "windmill" instead of "mill", but when I read out "won in a windmill", it was clear what the phrase should be. This also helped distinguish between "kindle" and "e-reader"

I found this impossible without counterhash’s post. Even with the hints, I wasn’t even close.

On 1, I thought three started with Ws - wanted/waist/wash so the answer would be race. The hint gave me won but it wasn’t entirely clear that was the right word.

2 had three pix of people so I was going with outhouse. The hint helped but again, answer could have been sunset! Argh!

3 I had music or note. The only way I could’ve had A sharp was thanks for the hint. And that just made it harder until I got the hint about consonants versus vowels.

4 and I figured out the rhyming but that left me with windmill. The hint tried to help, but just left me somewhat confused.

5 I had 3 piece suit, four leaf clover, and six pack abs.. But I was thinking the misfit waskindle. The hint was helpful here.

Six. This was the hardest one. I kept thinking latte. The hint was helpful here in connecting all of the “off“ sounds in coffee, takeoff and office.

Still, there was no way I could’ve connected them into a coherent phrase without the help in the forum.

Agreed

Thank you counterhash for your post. I could not figure out how to type the solution in once I had it, but now I do.

Haha! I was wait off on #2, I was trying to get them to rhyme, like #4. I thought it was fit, hit, and sh@$.

This puzzle took years off of our lives, but I still loved it.

I got stuck because I misunderstood what to do with the different pictures. I did not realize they were opposites. I actually thought that they had to be shuffled. For example, the e-reader is not a number like the 3 piece suit, 4 leaf clover and 6 pack. However, there was a section of things starting with letters. So clearly the ereader was supposed to fit with the Tshirt, Qtip, and Xray. I thought the race picture was a 5k and was supposed to match with the numbers. So then it looked like a shift 4! Should I start by reading what the different thing matches with? 5K with numbers so it's 5. Sunset or date something something. Sheet music and windmill something. Then e-reader matches the category of words that start with letters. The plane fits with the rhyming caregory (train rain chain plane). So I must be looking for a word with FIVE letters, starts with E, rhymes with PLANE.

I have all of the differences, but I have no idea how to combine them into a phrase. If I could get any help, that would be great.

my question is how you can make this all into a phrase when you are done?

I think this theorem should have been called opposites. I couldn't get it until reading all the hints and reading the forum. Thanks to the previous posts who didn't completely give it a way.

I really did like this one. I was able to figure out most of the different pictures besides group 6. That wasn't clicking for me.

I do think there were too many combinations of words to check. Some pictures might have more than 1 word.

I'm pissed I needed the hints and more help. It took me over a day to give in and look at the hints.

Too many interpretations of what is different. M's perception differs from mine. Some were a stretch. But it is M's challenge.

The challenge is that there wasn't a consistent theme to determine what is different in the groupings. There are a number of differences that could be found. But not having the right interpretation means, not solving the puzzle.

Isn't that part of this "game"? The code originator sets the rules for their coded message. It's up to us to figure out the deciphering methodology. Even with obscure clues.

Continuing to build skills to deduct different type of keys or starting points.

Someone else had the same issue below, too. Guess hint 2 is confusing. I don't think it's saying those are the "different" ones. It's just helping you identify some of the hard-to-identify images.

The hints do not match up with the solution in the back of the book. The hint said in the first group, the top right pic (lose weight) was the oddball, but it’s not. It’s the winning picture. So that threw me off the correct solution.

I get the clues, but I don't understand how to turn this into a phrase

Really liked the concept of this puzzle--however, there were too many ways to interpret each image. For example, in box #2 we interpreted workout, strike out, night out for the first three, and for sure thought that outhouse was the odd one out (because of the placement of the word "out").

Overall, fun idea but too tricky.

My issue was that I would get the correct different image in the wrong way. For Group 5, for instance I interpreted it as numbers - 3 piece suit, 4 leaf clover, 6 pack - to get the correct odd one out, but my reasoning for why it was wrong didn't fit any kind of solution I could see.

Yes, I would agree with this.

I found the toughest part to be how they relate to each other, but after writing out the specific solutions that I could make sense of the overall solution became clear. I disagree that this is about personal interpretations. It’s just that the different images are not all different in the same way. More specifically, the articulation Of the differences is not all the same.

My first interpretations:

1. "Missing" is the odd one out because it looks like the winners ribbon in the footrace is slack, indicating having been broken already by someone just ahead of the visible runner. I guess he's got his arms up in a victory gesture because he is stoked to be taking second.

2. "Strike" is out because the others are compound words: push-up, sunset, outhouse.

3. "A sharp" isn't always written with a hyphen while the others are, so it is out.

4. I was with him on this one, "Mill" doesn't rhyme with the others.

5. I thought "clover" was out because watch, kindle, and muscle are all words that are both nouns and verbs.

6. I agreed here also that "neon" is "on" and not "off" .

Of course, these conclusions resulted in gibberish as far as a final solution phrase, so I knew I had to re-think some of them.

This was a frustrating and disappointing one. Too much room for interpreting the images different ways and then the translation of each into the phrase was inconsistent from image set to image set. (i.e. the first one you take “won” - descriptive word of what is different - whereas for the 3rd set, you have to look at what is different (vowels) but then take the specific vowel for the image that is different (“a”)).