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Difficult 4x4 Analysis, Monday June 20 2016 https://www.calcudoku.org/forum/viewtopic.php?f=16&t=801 
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Author:  marblevolcano [ Tue Jun 21, 2016 1:30 am ] 
Post subject:  Difficult 4x4 Analysis, Monday June 20 2016 
This is my first analysis, by the way... To begin, first we look at column a. The only possibility for the 6+ cage is {2,4} so a1 and a2 must be 1 and 3. Next, we find the values in the 1 cages. Taking into account that the sum of columns b, c, and d is 30 and that the 14+ cage already has 1 and 3 (1+3=4), you can subtract the remaining amount of the 14+ cage and the three rightmost columns (c1+d1+b4+c4=301091=10). This means the 1 cages could {1,2}{3,4} or {2,3}{2,3}. If it was {2,3}{2,3}, then the a column would be as follows: a1=1, a2=3, a3=2, and a4=4. Then looking in the 9+ cage, we can quickly find that {2,3,4} is not possible. This would mean that the 9+ cage is {4,1,4}, which is impossible due to the 1 in d4. Thus, the lower 1 cage is {3,4} and the upper one is {1,2}. Due to this, a4=2, a3=4, a2=1, and a1=3. This shows b1 to be 4, and the rest of the puzzle can be solved quickly and easily. Edit by pnm, added puzzle picture: 
Author:  marblevolcano [ Sat Jun 25, 2016 2:17 pm ] 
Post subject:  Difficult 4x4 Analysis, Friday June 24 2016 
Analysis Number 2! (Because I can) Difficult 4x4, Friday June 24 2016 To start, look at the bottom two rows. The sum of these rows is 20, so to get the lone straggler of the 5+ cage subtract b3=2099=2. This means b2 is 3. The next thing to be found is the 2 cage, c1 and c2. Because of the 5+ and 9+ cages in the two rightmost columns, the sum of the numbers in the 2 cage must be 6 (c1+c2=2095=6). The only possible combination for this is {2,4}, so c3c4 must be {1,3}. Also, in the left 9+ cage, b4 must be 4 because if it was 1, the only other available option, then a3 and a4 would both be 4, which is not possible. So b2 is 1, meaning that the a column must be {3,2,4,1} (because if the 0 cage was anything other than {1,2,3}, the 9+ cage would not equal 9). Then the rest of the puzzle is clear. 
Author:  beaker [ Sat Jun 25, 2016 8:20 pm ] 
Post subject:  Re: Difficult 4x4 Analysis, Monday June 20 2016 
Why not just add up all the givens which will total 30 and subtract that from the total possible of 40 which gives you 10. Now you are left with two. 1 cages which will have to be 2,3 and 2,3 or 3,4 and 1,2 and then look at puzzle and decide which will fit best??? 
Author:  marblevolcano [ Sat Jun 25, 2016 10:57 pm ] 
Post subject:  Re: Difficult 4x4 Analysis, Monday June 20 2016 
Yes, of course. I just decided to clarify/separate the a column which would come later. It didn't change the process much anyway 
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