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 question about the 6x6 difficult of Saturday the 16th 
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Posted on: Mon Jan 18, 2021 11:34 am




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Post question about the 6x6 difficult of Saturday the 16th
I got this question from a puzzler, and thought I'd ask the community first [smile]

pluto wrote:
The "6X6 Difficult" of January 16 has addition cages occupying the entire circumference of the puzzle. I'm trying to make use of this
arrangement but, despite copious linear algebra, I cannot make progress beyond knowing that the sum of the corner squares must be 17.
If you can find time to give me a clue to what else can be revealed from this arrangement I'd be most grateful.


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Last edited by pnm on Mon Jan 18, 2021 8:08 pm, edited 3 times in total.



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Posted on: Mon Jan 18, 2021 12:15 pm




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Joined: Sun Jan 31, 2016 7:52 am
Post Re: question about the 6x6 difficult of Saturday the 16th
The total of the corner cells can be found by adding cages.

First add the circumference cages, this will give the total for all cages in the centre 4x4 group (126-circumference)
Using that total we can find the total for cells b1,c1,d1:e1,b6,c6:d6:e6.
Now double that total and subtract it from the total of the circumference cages, and that is the total of the corner cells.


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Posted on: Mon Jan 18, 2021 12:28 pm




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Post Re: question about the 6x6 difficult of Saturday the 16th
michaele wrote:
Now double that total and subtract it from the total of the circumference cages, and that is the total of the corner cells.


Or 4 x 21 - (total of border cages) ?

But I think the question was if there are any _other_ methods.


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Posted on: Mon Jan 18, 2021 5:05 pm




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Joined: Sun Mar 11, 2012 8:48 pm
Post Re: question about the 6x6 difficult of Saturday the 16th
My strategy is to first solve an "outside" row or column in the center 4x4 section. this time it was b5;e5 which turned out to be 1,2,5,6. Then 17+16+sum of b5;e5 - 42 gives me the sum of a3 and a4 which was 5, making my possibilities for those cells 4+1 and 3+2


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Posted on: Mon Jan 18, 2021 7:04 pm




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Joined: Sun Nov 25, 2012 6:41 am
Post Re: question about the 6x6 difficult of Saturday the 16th
Thanks for your replies so far, but I think I didn't make my difficulty clear.
In the central 4x4 it's not difficult to get: d3,e3=6,4; d4,d5=1,5; e4,e5=5,6, and these narrow down the options for b4,c4, a4,f4, b5,c5, a5,f5.
My difficulty is how to nail down specific cells in the periphery. I can deduce that the sum of the corner cells must be 17, but I haven't been able to progress from that to unequivocally assign values to specific cells around the periphery.
Thanks for considering.


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Posted on: Mon Jan 18, 2021 11:12 pm




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Joined: Sun Jan 31, 2016 7:52 am
Post Re: question about the 6x6 difficult of Saturday the 16th
pnm wrote:
Or 4 x 21 - (total of border cages) ?


I think you missed my point, just finding the total of the corner cells is not much help.
If you add the cages as I described it gives the total for the centre 4x4 group. When you have that total it narrows down your options for the central cages and also helps in adding cages that includes unsolved cells in the centre. For example you can add rows a,b&c to reduce the possible values in c6, even if cages b2 & c2 are not solved we can know the possible totals for those cages.

Also using the cages around the edge we can do things like f5+f6=e1+6, this will be very helpful.

pluto wrote:
My difficulty is how to nail down specific cells in the periphery. I can deduce that the sum of the corner cells must be 17, but I haven't been able to progress from that to unequivocally assign values to specific cells around the periphery.


If you can send me a screenshot of whre you are at with the puzzle I would be happy to help.


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Posted on: Tue Jan 19, 2021 12:15 am




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Joined: Fri May 13, 2011 6:51 pm
Post Re: question about the 6x6 difficult of Saturday the 16th
pluto wrote:
Thanks for your replies so far, but I think I didn't make my difficulty clear.
In the central 4x4 it's not difficult to get: d3,e3=6,4; d4,d5=1,5; e4,e5=5,6, and these narrow down the options for b4,c4, a4,f4, b5,c5, a5,f5.
My difficulty is how to nail down specific cells in the periphery. I can deduce that the sum of the corner cells must be 17, but I haven't been able to progress from that to unequivocally assign values to specific cells around the periphery.
Thanks for considering.


A few tips.
From the point you are, we can continue in this way.
First, we eliminate the pair [2,6] for cage 4-, the reason is this: The sum of the first two columns is 42, so a1 + b1 + a2 = 13 >>> c1 + d1 = 6 and then c1 = 4, d1 = 2, necessarily, but this is impossible because cage 1- (c2-c3) could not be accomplished.

Now we have that cage 4- is the pair [1,5] and then b5 = 2, c5 = 1.
We know that the 4x4 cells inner part sums 59 (126 – the sum of the border, which is 67), this means that cages 1- (c2-c3) and 1- (d2-e2) have a sum of 14.
The options are:
[2,3] + [4,5] invalid
[3,4] + [3,4] invalid (they cross the digits)
[4,5] + [2,3]
[5,6] + [1,2], respectively.

But before deciding about these two cages, we can fill a few more numbers.
Looking to the three bottoms rows, which sum is 63, we have:
17 – a3 + 45 + f4 = 63 >>> f4 = a3 + 1
Only two possibilities: a3 = 1, f4 = 2 or a3 = 3, f4 = 4

The first one is invalid since then: b2 = 1, b3 = 5 and now your cage 1- in c2-c3 can not be either [4,5] or [5,6] as previously obtained, so a3 = 3, f4 = 4
And then: a4 = 2, a5 = 4, f5 = 3

a6 + b6 = 8 >>> a6 = 5, b6 = 3, b4 = 6, c4 = 3, b1 = 4 and also a1-a2 is the pair [1,6].
Observe that c2-c3 can not be [6,5] because then a2 = 1 and it’s not possible [1,2] for cage d2-e2 as previously stated.
So, cage 1- (c2-c3) is [4,5] in this way: c2 = 4, c3 = 5, b2 = 5, b3 = 1, and cage 1- (d2-e2) is the pair [2,3].

The rest is straight forward. So the key for this type of puzzle is first to solve an additional cage in the inner part (defining its candidates) and later to make some algebraic operations considering three lines together, for instance.


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Posted on: Tue Jan 19, 2021 2:26 am




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Joined: Sun Nov 25, 2012 6:41 am
Post Re: question about the 6x6 difficult of Saturday the 16th
Thank you All, particularly michaele, and, even more particularly clm for illustrating a general principle for solving this type of puzzle.
Your quick and informative responses were much appreciated.


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