# Non-Congruent Fillomino

Here's a Fillomino puzzle (rules can be found here).

The gimmick is that no two areas can be congruent. Enjoy!

Got the same answer as @Kruga. Here is the explanation.

The first thing to notice is that the 3's cannot connect since doing so would create an area of bigger than 3. Thus, the top 3 cannot move down and can only move up. This lets us place a 3 and a 5 in the top middle.

--6--     --6--
--5-- --> -355-
637-4     637-4


Since the areas cannot be congruent, we know that one 3 section must be an "L" shape, and the other a straight line. This also means that there are no other sections of 3 in the graph. Also, all sections of 2 are congruent, and since we know one exists on the right middle, no other 2s can exist.

4s can form 5 non-congruent shapes. The only 4 that can make a square is the top one and the 4 below it cannot extent up more than one, meaning it must go down at least one.

----6----     ----6----
---355--- --> ---35544-
--637-4--     --637-44-
------6--     ------6--
456---524     456---524
--4------     --4-----4


Now we have two choices for which 3 is straight. Lets say it is the top one. This forces the top 6 completely, and the middle 5 and 7. Then we would have the following:

---366666
---355446
--637544-
----756--
453--5524
--4-----4
--8-475--
----7----
----4----
----7----


The two 5s end up touching and this is a region larger than 5.

Thus, the top 3 is a corner and the other three is straight.

----6----
--335544-
--637-44-
------6--
45333-524
--4-----4
--8-475--
----7----
----4----
----7----


Now, lets say the 6 on the right middle takes one square to its left. This forces the 7 to fill out to the left and the 5 out the top. This makes the top 6 go left too. At this point, there is not enough space to finish both 6s on the top left.

----6----    -666655--
--335544-    --335544-
--637-44-    --637-44-
-----66--    -777766--
45333-524 -> 45333-524
--4-----4    --4-----4
--8-475--    --8-475--
----7----    ----7----
----4----    ----4----
----7----    ----7----


Thus, the middle 6 must go right and up to completion. This means the top 6 needs to go right at least 4. And the 2 and 4 on the right are now mostly completed.

-6666---6
--3355446
--637-446
------666
45333-524
--4----24
--8-475-4
----7----
----4----
----7----


The top 5 can not complete by going downwards without either connecting to the other 5 and making too large an area, or by blocking the 7. Thus, it must go up at least 1. The other 6 on the top needs to extend towards the 7 at least 2 since there is insufficient room for it to complete on the left. This forces the 7 down, and also forces the top 5 up even more.

-666655-6
--3355446
--637-446
--66-7666
453337524
--4--7-24
--8-475-4
----7----
----4----
----7----


We know we will need something to keep the 6s separated on the top left. We cannot use a 2, 3, or 4 since those shapes are all used up. Thus, we can only use a 1. Therefore, the 5 on top must fill the remaining space so as not to use up the 1 needed elsewhere, and the 7 need to fill the lone gap there as well. The middle right 5 can then extend down to connect with the one below it.

-66665556
--3355446
--6377446
--66-7666
453337524
--4--7524
--8-475-4
----7----
----4----
----7----


Now that the middle 7 has six pieces, we cannot connect with the other 7. Thus, it must go left. The two middle 4s are then also forced to connect, but there are a couple ways they can do that.

-66665556
--3355446
--6377446
--66-7666
453337524
--44-7524
--8-475-4
---77----
----4----
----7----


Quick math tells us the left 5 can only extend up 2 before it will impact the ability of the 6s to finish. Just it must also go down. Similar to the 4 beside it. This forces the 8 and then the 7 down as well. The bottom 4 must then to right.

-66665556
--3355446
--6377446
--66-7666
453337524
4544-7524
-58-475-4
--877----
---744---
----7----


Since we can't use more than one 1, the top left has only one solution. This forces the middle 7s and the adjacent 4.

666665556
613355446
466377446
466667666
453337524
454477524
-584475-4
--877----
---744---
----7----


Counting the gaps in the bottom left, there are 10. Two are needed to complete the 5, six are needed for the 8. This leaves 2 that must go to the 7. So, the two 7s must be connected and it goes one to the right on the bottom.

In order to maintain non-congruence, the 4 on the right must be an 'L'. This leaves the remaining 4 to be a 'T'. The 5 in the middle is then a long 'L'. The remaining gaps are 6 and we can fit that shape in.

666665556
613355446
466377446
466667666
453337524
454477524
-58447544
--8774556
---744466
---777666


For the 5 on the left, we've already used the long 'L' an the 'S' shape. This leaves a 'T' or straight. If straight, the 8 must go the long way around leaving no room for the 7 to finish. Thus, it must be a 'T'.

666665556
613355446
466377446
466667666
453337524
454477524
558447544
-58774556
---744466
---777666


The 8 is then forced, forcing the 7, and then the remaining spaces are 8s.

666665556
613355446
466377446
466667666
453337524
454477524
558447544
858774556
888744466
887777666

• Your second step has 456 instead of 453 at the bottom. – w l Mar 3 '17 at 7:56

Only possible solution

I'll add an explanation later. When I get time. Maybe.