A "Letter" Slope puzzle?

Your brother has been forcing you to some read some of his frankly ridiculous poetry, so, knowing that he was never a numbers man, you decide to vex him with a Number Slope puzzle. In fact, you are so confident that he won't be able to solve it that you bet all your PSE rep on it.

The next day, as you work through one of your countless puzzle books, you find a sheet of paper inside. It's not another poem this time, thank goodness, but it is slightly concerning...

Thought you could defeat me with your pitiful numbers? Pah! You'll be soiling yourself!

What on earth is that supposed to mean? Why didn't he just send the completed grid? Or is he teasing you, hiding the solution in a puzzle of his own? Is all your precious, hard-earned rep now gone? What is your brother really saying?

Note:

The message, as well as the rule which provides the enumeration of the letters SOIL, have been correctly found. However, there is one more unidentified, intentional, confirming detail in the puzzle related to the additional letter. I will leave this question open until that aspect of it is found.

Edit: I have edited the image to provide a hint to the aforementioned detail.

• Your brother is saying SOOL.SIIL, of course :) Jan 11, 2017 at 7:54
• Does the secret letter fit the same numbering theme? Jan 12, 2017 at 12:32
• @humn Not quite, unfortunately, but there should be another detail in the puzzle that confirms that the pattern/rule is correct. Jan 12, 2017 at 12:39
• Does a priori enumeration of S O I L mean that it has nothing to do with the given layout? Guessing that the confirming detail of the layout is circled in row 3 of my picture. Jan 16, 2017 at 5:09
• @humn Yes, nothing to do with the given layout. Your layout circled in row 3 is part, but not all, of the confirming detail. Jan 16, 2017 at 5:28

To bare the mechanics of the self-inflicted insult suffered by our brother, as uncovered in TheGreatEscaper’s solution:

Row 1.   Taking the Number Slope™ puzzle hint assumes that S, O, I and L are equivalent to 1, 2, 3 and 4, in a consistent order observed within each set of linear squares within a tetromino.   The circled cells determine that O is 1 or 4 while S and L are in the middle. Thus I is also 1 or 4.

Row 2.   This is the direct result of that partial ordering. The circled cell group determines the order as O S L I or its reverse.

Row 3.   The 4×4 grids proceed to specify all but two inconsequentially ambiguous areas. Letters at dotted corners can now be copied to the 5×5 grid.

Row 3 -T.   Giant upside-down T  tetromino sighting — as the dotted corners are brought together.

Row 4.   A new mystery letter ▯ presumably is equivalent to 5 in the order.   The circled subsequence L S ▯ establishes the complete ordering as...

... I L S O ▯.

The 5×5’s bottom row’s direction is unknown initially but S is in the middle either way.   This forces L into the center cell and the rest follows readily enough.   At the top of the 5×5 grid we at last see our brother’s embarrassing confession.

“ I   L O S T

Subtle reinforcing hints.

Why is the mystery letter T when E could just as well spell an equivalent message?   Continuing on TheGreatEscaper’s reasoning, there are 5 tetromino shapes known to science, T being the one not in play here, while polyominologists simply have not yet discovered an E-shaped tetromino or even pentomino.   As subliminal reinforcement, a gigantic upside-down T  tetromino is formed within the 8×8 combined grid of row 3 -T above.

How were numbers and letters matched up?   The 4 tetrominoes in play enumerate the 4 known grid letters, appropriately enough, by counting slopes.

This order is also found, in reverse, among the circled letters in row 3 above. But the puzzle’s poser has mentioned that yet one more reinforcing detail hides in plain sight among the grids:

The letters in the dotted corners of each 4×4 grid are I, L, S, and O in that order. (This means that the grids are ordered by the number of slopes of their corresponding letters.) But the 5×5 grid seems to be "paired" with the second grid: the space between them is shortened!

Just as the 5×5 grid is an "exception", the letter T is an exception to the rule about number of slopes: it has two slopes, like the L it is paired with, but it is the fifth letter in the sequence. Also, it must be moved to the end of the solving order, just as its letter must be moved to the end of the sequence order.

Assuming that these are number slope puzzles with numbers replaced with letters, it's possible to deduce from puzzles 2 and 4 that 'L' and 'S' are in between 'I' and 'O', in some order.

Looking at puzzle 2, we know the 'O' in the first column must be at the very bottom. Therefore, R1C1 is 'I'. We can sudoku place the L at R2C1, giving us the order 'ILSO' (or 'OSLI', if you want to reverse it).

Now, solving as much of the puzzles as we can, we can deduce what letters must appear in the corner cells with little quarter-of-circle markings. Map the letters in these corner markings to the centre 5x5 grid around that dot. If you introduce a new letter, let's use ' ' for now, and solve the 5x5 as a letter slope sudoku, you get the unique solution:

ILOS
S IOL
OIL S
LS IO
OSLI


At this point, I seem to have solved as much of these puzzles as I can. So, what now?

EDIT: (Thanks to a nudge from @humn, some explanation from @Deusovi and some more nudging from the OP) I L O S are all the tetronimos. What's missing? T, of course! So, filling in the blank spaces in our 5x5 with T, we can see the final message 'I LOST' written out in the top column!

So why this ordering? T = 5 since sudokus range from 1-n where n is the size of the grid, so T, being the one left out of ILSO in the 4x4s, must be 5.
I L S O are in that order due to the number of cells in the tetronimo with 2 walls at a right angle. I has 0, L has 1, S has 2, and O has 4, giving us the order we have now!

• Now you turn in all of your hard earned reputation :) Jan 11, 2017 at 10:24
• Could the message be just the top line as two words? Jan 11, 2017 at 11:55
• @humn good point! 'I LOSE' or 'I LOST' also fits very well with the flavortext of the quesiton! Jan 11, 2017 at 12:00
• You have found the message, well done! However, the explanation of the ordering is not quite correct (there is a much simpler and more thematic explanation), so I will put off accepting until someone finds the correct one. Jan 11, 2017 at 21:40
• @Volatility is it anything to do with I=1, L=7, O=0, S=5 and E=3? ie. you gave him a puzzle of numbers, he gave you a puzzle that not only insults you, but also uses letters which are similar to numbers?
– Joe
Jan 12, 2017 at 13:30

I tried to solve each slope puzzle as far as possible under the assumption that each letter could only appear once:

+-+-+-+-+   +-+-+-+-*   +-+-+-+-+-+   *-+-+-+-+   +-+-+-+-+
|O|  S| |   | |O   L|   |     |   |   |S  |  I|   |    L|I|
+ +   + +   + +-+-+ +   +   +-+ +-+   +   +   +   +-+-+ + +
| |   | |   |L|  O| |   |   |   | |   |I  |  S|   |   |I|L|
+ +-+-+ +   + + +-+-+   +-+-+ +-+ +   +-+-+-+-+   + +-+-+ +
| |   | |   | |L|  O|   |   | |   |   |  S I| |   |L|I  | |
+ +   + +   + + +   +   + * +-+   +   + +-+-+ +   + +   + +
| |S  |O|   |O| |L  |   |L S  |   |   | |I S  |   |I|L  | |
*-+-+-+-+   +-+-+-+-+   +-+-+-+-+-+   +-+-+-+-+   +-+-+-+-*
|         |
+-+-+-+-+-+
Solving the slope puzzles without knowing which letters were used was really fun!

Why it failed:

After moving the letters around the dots into the middle grid it could not be solved.

The other letters around the dot had to be after L and S respectively and the last remaining letter had to be after S. This means L is the first letter which leads to a contradiction when solving the grid.

I also tried to find common words only fitting the 4 smaller grids that could form a useful sentence, but no luck there either.

Bonus: Middle grid solved with the word slope (that is enough to solve it):

+-+-+-+-+-+   +-+-+-+-+-+
|S P L|O E|   |E L P|O S|
+   +-+ +-+   +   +-+ +-+
|O E|S L|P|   |O S|E P|L|
+-+-+ +-+ +   +-+-+ +-+ +
|L S|P|E O|   |P E|L|S O|
+ * +-+   +   + * +-+   +
|P O E|S L|   |L O S|E P|
+-+-+-+-+-+   +-+-+-+-+-+
|E L O P S|   |S P O L E|
+-+-+-+-+-+   +-+-+-+-+-+