# Good luck finding the odd-one-out

An entry in Fortnightly Topic Challenge #37: Rare and Endangered 1

## 5

• Does this really qualify for the FTC since one of your two primary tags is enigmatic-puzzle, which is in the top 50? – Riley Jul 9 '18 at 19:11
• @Riley Well each of the 5 puzzles is also for a tag that fits with the FTC, but you can only have 5 on a question. I debated on putting those tags on the question, but then the one that isn't listed could be considered the odd-one-out. – David Starkey Jul 9 '18 at 19:37

The odd one out is

Puzzle 3, because it is the only solvable puzzle. Thanks to @Klyzx for pointing out there were no thick lines, I wasn’t looking closely.

First I’ll need to mention that the puzzles, in order, are

found as the alt text under the images. The 15-puzzle (), the Lights Out game (), “Boxless” Sudoku (), a Rubik’s Cube ()...and the last one is either Minesweeper or Slitherlink ()

The solution:

The Sudoku doesn’t have boxes, so the only rule is that every column and row must contain 1-9.

Rough explanations of the others:

1)

The 15-puzzle has an even number of inversions (56) and the blank is in the third row, which is sufficient to indicate it is unsolvable. That’s the best I can say it, it’s easier to just try it and see that you can’t swap the final 14 and 15.

2)

The Lights Out puzzle has an odd number of lights on in the first and second columns. It must have an even number in all rows and columns to be solvable.

4)

No rigorous proof here. I plugged the Rubik’s cube into a solver, not having one on hand. Its configuration can’t be solved.

5)

The 0 in the Minesweeper conflicts with the 3 right above it. Similarly that causes problems in a Slitherlink, forcing the upper right 2 to be impossible to satisfy.

• Yes, your solver is correct on #4. – Ian MacDonald Jul 9 '18 at 19:59
• The sudoku may be solvable if it's a Squiggly Sudoku, which may be possible due to no thick edges marking boxes. – Klyzx Jul 10 '18 at 5:50
• Hint: The question contains the tags for each puzzle. – David Starkey Jul 10 '18 at 13:14
• Excellent answer. I've added an image and updated with a bit that didn't get found (without which you got the answer anyway). – David Starkey Jul 10 '18 at 20:46
• Cool puzzle. And good to know “Latin Square” is a real name for it. – DoctorXOR Jul 10 '18 at 20:50

#4.

This is because

It is the only shape with a concave angle.

Alternatively, it's something to do with

The puzzle types.
#1 is a sliding puzzle, #3 is a sudoku, #4 is a Rubik's cube... I don't recognize the other two, but perhaps one of the 5 is in an unsolvable state that looks like the kind of puzzle it's supposed to be.

• It has nothing to do with angles. – David Starkey Jul 9 '18 at 19:40

Is it

the second puzzle?

Explanation:

I assumed that the grey squares in the first two puzzles are not filled in, and likewise, the white squares in puzzles $3$ and $5$. Let $x_i$ be the proportion of squares that are filled in (either by text or by colour) in puzzle $i$. Then $$x_1=\frac{15}{16},\quad x_2=\frac{12}{25},\quad x_3=\frac{41}{81},\quad x_4=\frac{54}{54}=1,\quad x_5=\frac{20}{40}=\frac12.$$ Only puzzle $2$ has less than half of the squares filled in.

• This is incorrect. – David Starkey Jul 9 '18 at 15:56

I'll guess

#4

Because

It's the only one that has squares that are self descriptive, e.g. each Blue square describes itself as "B," etc.

• The letters are more for clarity for potential color-blind issues. I also thought about putting it on #2, but I think light/dark is safe for pretty much everyone. – David Starkey Jul 10 '18 at 13:12