## A rather unique kakuro puzzle

For fans of Kakuro

### A rather unique kakuro puzzle

It is a 13X13 puzzle. Below the first two numbers are vertical cage sum
and last two numbers are for horizontal cage sum. for example

1)403 means vertical sum is 4 and horizontal sum is 3.
2) 700 means vertical sum is 7 and horizontal sum is 0
3) 3100 means vertical sum is 31 and horizontal sum is 0
4) 12 means vertical sum is 0 and horizontal sum is 12

------------The puzzle is below---------

0:0:3100:800:900:2300:0:0:0:0:0:1100:1500
0:11:-:-:-:-:700:0:0:1200:403:-:-
0:625:-:-:-:-:-:1000:15:-:-:-:-
12:-:-:0:2809:-:-:-:1120:-:-:-:-
9:-:-:17:-:-:11:-:-:800:4:-:-
4:-:-:1212:-:-:7:-:-:-:800:0:0
0:14:-:-:-:2200:2600:9:-:-:-:3400:0
0:33:-:-:-:-:-:0:11:-:-:-:3500
0:700:1700:15:-:-:-:0:1900:617:-:-:-
13:-:-:700:2315:-:-:2210:-:-:17:-:-
13:-:-:-:-:16:-:-:-:-:1509:-:-
19:-:-:-:-:0:32:-:-:-:-:-:-
0:0:9:-:-:0:6:-:-:19:-:-:-
Attachments
kakuro puzzle snapshot
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surendra.jain

Posts: 27
Joined: 15 August 2015
Location: India

### Re: A rather unique kakuro puzzle

Instead of "unique", I'd call this puzzle "ill-designed":

* it consists of 3 independent sub-puzzles;

* the lower left sub-puzzle is trivial (solved by singles):
-----
-49--
-2146
-1729
---18

* the upper right sub-puzzle has 2 solutions (so, the global puzzle is not "unique"):
-----
---12
-8124
-4358
---31
and:
-----
---12
-4128
-8354
---31

* the central puzzle requires T&E
here's one solution ( I didn't check if it has more)
-------------
--5213-------
--26845------
-39--126-----
-27-98-38----
-13-57-124---
--176---135--
--45798--128-
----159---179
-----87-73-98
------2851-27
-------962735
-------51-856
denis_berthier
2010 Supporter

Posts: 1276
Joined: 19 June 2007
Location: Paris

### Re: A rather unique kakuro puzzle

denis_berthier wrote:* the upper right sub-puzzle has 2 solutions (so, the global puzzle is not "unique"):
-----
---12
-8124
-4358
---31
and:
-----
---12
-4128
-8354
---31

Make that 3 solutions. The 3rd is:
-----
---21
-4137
-8354
---13

Multiple solutions are a no-no in Kakuroland (and Sudokuland).

Bill Smythe
Smythe Dakota

Posts: 555
Joined: 11 February 2006

### Re: A rather unique kakuro puzzle

I wrote a kakuro solver software that can solve kakuro puzzles with logic and it will not give any solution if there are multiple solutions. My software found the same solution as Smythe. However, it did not find the solutions mentioned by Denis. This is because the two solutions given by Denis gives a unique rectangle situation and my software rejected the solutions.

look at [3,10], [3,13], [4,10], and [4,13] cells
these have 4 and 8 values and you can interchange 4 and 8 and get a different solution (a case of unique rectangle situation found in sudoku puzzles)

If it were a sudoku puzzle, then My software will find only one solution as it would have rejected Denis's solutions owing to unique rectangle situation. However, this was not the case in this kakuro puzzle. I found it amusing because it is one of the rare cases where there exist another solution other than the 2 unique rectangle situations. If the only solutions were Denis's (unique rectangle), then my software would not give any solution at all showing it has multiple solutions. Thus after eliminating the unique rectangle solutions, my software found the same solution as Smythe.

I have not yet come across any kakuro puzzle with unique rectangle solutions as well as another solution. This is because of the numbers are involved in a sum and in a rare case (as this one) we will have a valid solution in addition to 2 unique rectangle solutions. If it were a sudoku puzzle then we will never find the 2 unique rectangle solutions of Denis and the other solution of Smythe.
surendra.jain

Posts: 27
Joined: 15 August 2015
Location: India

### Re: A rather unique kakuro puzzle

surendra.jain wrote:My software found the same solution as Smythe. However, it did not find the solutions mentioned by Denis. This is because the two solutions given by Denis gives a unique rectangle situation and my software rejected the solutions.

"My" two solution do exist. If your software rejects them, it is buggy.
That's the problem with techniques based on uniqueness. They can be used only if the puzzle is known to have a unique solution. Instead of this, you erroneously use the technique on a non-unique puzzle to conclude it's unique.
BTW, I don't know if there are more solutions than the above 3. As soon as I found 2, I stopped searching.
denis_berthier
2010 Supporter

Posts: 1276
Joined: 19 June 2007
Location: Paris

### Re: A rather unique kakuro puzzle

Puzzle-solving software exists for two reasons. One is to help the solver find a solution. The other is to determine for the creator that the puzzle is valid. When software designed only for the former purpose is used for the latter purpose, there is big trouble in River City.

denis_berthier wrote: .... That's the problem with techniques based on uniqueness. They can be used only if the puzzle is known to have a unique solution. Instead of this, you erroneously use the technique on a non-unique puzzle to conclude it's unique. ....

Amen. By using uniqueness to prove uniqueness, your software is engaged in circular reasoning.

A better name for your "unique rectangle situation" would be "non-unique rectangle problem". The problem needs fixing.

.... BTW, I don't know if there are more solutions than the above 3. As soon as I found 2, I stopped searching.

I did an exhaustive analysis on that upper right corner. Unless I missed something, there are only 3.

It wouldn't surprise me if the larger portion of the puzzle, the middle section, turned out to have hundreds of solutions.

Bill Smythe
Smythe Dakota

Posts: 555
Joined: 11 February 2006

### Re: A rather unique kakuro puzzle

.
Sorry to be so late to the party, I had stopped monitoring the Kakuro forum after several months of inactivity, and by being otherwise occupied with "Min-clue SudokuP" project.

The main section of Surendra's posted puzzle has a unique solution, as does the lower-left section.

The upper-right section has 3 solutions. So there are three solutions for the complete puzzle, listed below.

It's an ugly puzzle - not just because of the non-contiguity of the grid (admittedly this is an aesthetic issue, but most puzzlers would agree), but because of the unforgivable non-uniqueness of solutions. If this was indeed posted online then the web-site authors should be roundly condemned for such sloppiness!

Surendra, you should definitely aim to have your software detect and report multiple solutions, and optionally, if requested, to enumerate all solutions. Only then can you compare your solver with others.

Optionally, you might also consider doing a grid contiguity check, and then solving any sections found separately.

And hello to Denis, long time no see!

Hidden Text: Show
Code: Select all
`.5213.....12.26845..412839..126.835427.98.38..3113.57.124....176...135...45798..128....159...17949..87.73.982146.2851.271729..962735..18..51.856.5213.....12.26845..812439..126.435827.98.38..3113.57.124....176...135...45798..128....159...17949..87.73.982146.2851.271729..962735..18..51.856.5213.....21.26845..413739..126.835427.98.38..1313.57.124....176...135...45798..128....159...17949..87.73.982146.2851.271729..962735..18..51.856`

Mathimagics
2017 Supporter

Posts: 1329
Joined: 27 May 2015
Location: Canberra

### Re: A rather unique kakuro puzzle

Mathimagics wrote: And hello to Denis, long time no see!

Hello Mathimagics,
I'm no longer active on this forum. Sometimes, someone contacts me about something published here and I have a look.
If you're still interested in Kakuro, you should have a look at my solution in the "handmade puzzle" thread. The puzzle there allows a great illustration of g-whips and surface sums.
denis_berthier
2010 Supporter

Posts: 1276
Joined: 19 June 2007
Location: Paris