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Solve this diabolic Square Number Sudoku.

Normal Sudoku Apply. Additionally, there are some rectangled area, in each area should be read as 1 number. The numbers must be SQUARE NUMBERS. There are: 1 1-digit, 18 2-digits, and 5 3-digits square numbers. Horizontal numbers is read from left to right. Vertical numbers is read from top to bottom.

Edit : I miss a rectangle, at the 2nd bottom row. Which make the solution not unique. Sory fot the inconvenience. I have replace the picture with the right one.

enter image description here

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  • $\begingroup$ Assuming I do not have any fundamental misunderstanding of the rules rot13(vf gur erpgnatyr va ebj gjb erqhaqnag)? $\endgroup$ – happystar Jul 17 at 8:28
  • $\begingroup$ @happystar: rot13(lrf, V guvax lbh haqrefgnaq gur ehyr ) $\endgroup$ – Jamal Senjaya Jul 17 at 8:45
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    $\begingroup$ I made a lot of progress and nearly reached the end but must have made an error somewhere as it ended up unsolvable. This is very hard but doable, I'm going to start over and see if I can get it right this time. $\endgroup$ – Beastly Gerbil Jul 17 at 13:02
  • $\begingroup$ I wrote a computer program to solve this, I got three solutions. I will post them as soon as one manual solution is posted. I tried solving it by hand, got not that far. Now found out that I took 18 to be a square number :P $\endgroup$ – daw Jul 17 at 15:34
  • $\begingroup$ @daw nvm, I've either made a mistake or its insolvable :P $\endgroup$ – Beastly Gerbil Jul 17 at 16:05
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Blimey, this took ages by hand!

I've got three solutions, which stem from this (sorry for the terrible drawing):Sudoku with Missing Bits

If it doesn't load, the structure is (left to right sweep):

It's 817493625539.264.8642..83.9975834162168..2943324961857456387291293615784781249536

Solution 1:

Solution 1

817493625539726418642158379975834162168572943324961857456387291293615784781249536

Solution 2:

Solution 2

817493625539126478642578319975834162168752943324961857456387291293615784781249536

Solution 3:

enter image description here

817493625539126478642758319975834162168572943324961857456387291293615784781249536

Method/Hints

I'm not going to go through every single step I did, because that would take AGES, and would be very boring. So, I'm going to describe the "key points"
(It saved a huge amount to write down a list of all squares with no repeating numbers)

1. As there are no numbers, it's impossible to solve this like a normal sudoku, so I only concentrated on the complex rectangular bits to start with

2. The first one was on row F, because there is so much overlapping. To solve this bit, I concentrated on the middle 2-digit square, which can't be 25, 36 or 81 (squares can't end in 2, 3 or 8), and 64, because even though some squares can start in 4 (400, 441, 484), they all have repeating digits. This leaves 16 and 49, which can only be paired like: 841625, 784961 and 324961 (no repeats)

3. The top cluster of rectangles interlocking in a square formation look interesting, so I investigated that next. From the list of 2-digit squares, I created a list to show which square could go in A(across) and D(down):
Square Lists AD
, which gives the possibilities 16+64, 36+64, 81+16 and 64+49:
Square Possibilities

4. The top row contains 4 2-digit squares, which is also interesting. 7 is not included in them, so it must be in A3 - Wahey, first number on the board!. 2, 3 and 9 can only be used in the squares 25, 36 and 49, which leaves 81. Therefore, the square mention in step 3 cannot be 16+64 or 64+49

5. Cell D7 is the only 1-digit square, which can only be 1 or 4 (no 2-digit squares start in 9)

6a. Try combining the outcomes of the top right square arrangement in step 3 (with 5), with the F row of squares in step 2

6b. Let's try grid arrangement 81+16. This means (D7,D8) are (4,9). Neither row F combinations 784961 or 324961 will work, so 841625 is inserted. What happens to the 3-digit in the E row?

6c. There are no 3-digit squares that fall into the parameters
a. 1st digit != 2,5,6
b. 2nd digit != 1,2,5,6,8
c. 3rd digit != 1,4,6,9

Therefore the top square arrangement can't be 81+16

Therefore it must be 36+64

Therefore D7,D8=16

Therefore F3,F4,F5,F6=4961

Now try working out the 3-digit square on the E row again...

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    $\begingroup$ Could you post your solving method/explanation (even just some high-level thoughts) & hide your solution behind spoiler tags? Also, this is GREAT! $\endgroup$ – bobble Jul 17 at 17:44
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    $\begingroup$ Oh - sorry, I'll hide it now. The method is rather lengthy, so I'll try to think of some key steps $\endgroup$ – Oliver Jul 17 at 18:54
  • $\begingroup$ How do I hide pictures? As you can see, it's a bit of a mess $\endgroup$ – Oliver Jul 17 at 19:01
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    $\begingroup$ I edited the formatting for you to hide your pics. If it's not how you want it to look, you can rollback the edit. Or you can edit again to see what I did. $\endgroup$ – MacGyver88 Jul 17 at 19:41
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    $\begingroup$ Wow amazing. I must have made mistakes when created and analized my puzzle, so I didn"t realize the puzzle have up to 3 solutions. Great work !! $\endgroup$ – Jamal Senjaya Jul 17 at 21:51

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