# A puzzling puzzle

The following images represent the same board:

This image looks similar to the Empty sudoku puzzle but is not a sudoku. The only similarity are the > and < signs between the cells.

This grid is divided into regions (marked by thick borders). The sum of the cells in each region is given. Also, the numbers in each of the regions are distinct - that means no region contains two identical numbers. The colored tiles identify specific numbers. However, not all positions of a specific number are given (with one exception).

Here there are some equations - marked by thick borders and the signs +, - and =.

The numbers at the end of each column/row represent the sum of the particular column/row.

I have edited the second image - there was a mistake in transposing the colors, particularly the yellow/purple cells in C9,D10/D9, E10 - they have moved down one square each. And a second edit - moved the black cell on I3, I4 to the right (J3,J4).

PS. All numbers are positive, i.e. the smallest number is 1. I will let you figure out what is the largest.

• These cannot represent the same board unless there's additional information you're not telling us. In the top board, 5K contains "23"; in the second, 5K contains "79". – Jeff Zeitlin Oct 25 '17 at 13:40
• 79 in the second board is the sum of the area it is in (marked by thick borders), i.e. 5I + 5J + 5K + 6I + 6J + 6K. 23 in the first board is the value of the cell. Maybe I did not explain it right. – Maria Ivanova Oct 25 '17 at 14:01
• OK, that clarifies things. Thank you. – Jeff Zeitlin Oct 25 '17 at 14:04
• I assume that the numbers at the end of the columns and rows in the first two boards are only listed as placeholders and are not reflective of some equation as is the case in the third board, correct? – John Eisbrener Oct 25 '17 at 15:15
• The numbers represent the same equation on each board - the sum of each row/column. – Maria Ivanova Oct 25 '17 at 15:40

In order to solve it,

I created a spreadsheet showing the different layouts and added some formulas in order to check the totals at all time. For the grid with the > and < signs, I made the background of the image transparent and added it on top of the numbers.

Then

I proceeded to narrow down the possible values for each cell to the smallest interval taking into account as many constraints as possible (right hand side table). As this was not enough to determine many values with certainty, I then proceeded to making hypothesis and testing values back and forth until everything finally fitted in!!

But (wait for the plot twist)

Although I had managed to respect all the constraints of the puzzle, my solution was refused! Having been caught in the numbers I had completely missed the big picture! Deusovi came to the rescue, pointing out that the numbers were probably encoding a message. So I proceeded to convert the numbers into letters, but not much came out immediately. After a closer look though, I could find the word 'PUZZLE' in the grid. I realized that some tag names were hidden in the rows and columns, and decided to look for more. Most of them were missing one or two letters, which helped improve my solution.

After resolving using that last clue, this is what my grid looked like

Encoding horizontally: MYSTERY, PATTERN, PUZZLE, GRID, CIPHER, STORY, CARDS, vertically: PALINDROME, VISUAL / REBUS, STRATEGY, MAZE / CRYPTIC, LIARS, STEGANOGRAPHY, ENIGMATIC / CHESS, RIDDLE, LETTERS, ALPHAMETIC, CROSSWORD, and diagonally ANAGRAM, RHYME, leaving the following message for all to meditate: CALL NO MAN HAPPY UNTIL HE IS DEAD (thanks again Deusovi for the last 3 tags)

• That looks amazing that you found a solution for the grids, but I'm not sure it works with what the OP stated in the section after the 2nd grid (colour grid). "Not all positions of a specific number are given (with one exception)" With your answer, there would be TWO exceptions. No 24 appears outside of a dark green square, and no 26 appears outside of a pale orange square. – Timoris Oct 27 '17 at 9:08
• damn it! I found this statement quite ambiguous and dismissed it when working on a solution... – sousben Oct 27 '17 at 9:12
• Well, that is really amazing and there are enough correct numbers on it. Also, there is one more method you haven't used to find the correct numbers. And I believe, that if you use this method, you would easily correct the wrong numbers. Also, for the colored squares - only one of the colors is incorrect (the rest of the colors are good). – Maria Ivanova Oct 28 '17 at 22:18
• @sousben, you are almost there. Just a few more tags are hidden and then, yes, the leftover letters do form a message, :) I will grant you now the bounty, so I don't miss the deadline, but the tick would be given once the puzzle is completely solved. :) – Maria Ivanova Nov 8 '17 at 18:57
• "CALL NO MAN HAPPY UNTIL HE IS DEAD". Missing ANAGRAM reading down and to the right (hitting the right wall at the M), LIARS (straight down to the bottom wall), and RHYME (roughly in the center going northwest). – Deusovi Nov 8 '17 at 19:31