## 22 (Clues) / 7 (Columns) / Boolean Algebra

Post puzzles for others to solve here.

### 22 (Clues) / 7 (Columns) / Boolean Algebra

Code: Select all
+-------+-------+-------+
| . . . | . . . | 3 1 . |
| . 4 . | . 1 . | . 5 . |
| . . 9 | 2 . . | . 6 . |
+-------+-------+-------+
| . . 5 | . . . | . . . |
| . . 3 | 5 . 8 | 9 . . |
| . . . | . . . | 7 . . |
+-------+-------+-------+
| . 9 . | . . 3 | 2 . . |
| . 3 . | . 8 . | . 4 . |
| . 6 2 | . . . | . . . |
+-------+-------+-------+
......31..4..1..5...92...6...5........35.89........7...9...32...3..8..4..62......

And a harder one with 25 clues:

Code: Select all
+-------+-------+-------+
| . . 3 | 1 . . | 4 . . |
| . 1 . | . . 5 | . 9 . |
| . . . | 2 . 6 | 5 . . |
+-------+-------+-------+
| . 3 5 | . . . | . . . |
| . 8 . | . 9 . | . 7 . |
| . . . | . . . | 9 3 . |
+-------+-------+-------+
| . . 2 | 3 . 8 | . . . |
| . 4 . | 6 . . | . 2 . |
| . . 6 | . . 4 | 3 . . |
+-------+-------+-------+
..31..4...1...5.9....2.65...35.......8..9..7.......93...23.8....4.6...2...6..43..
Last edited by mith on Sat Sep 05, 2020 1:36 pm, edited 1 time in total.
mith

Posts: 301
Joined: 14 July 2020

### Re: 22 (Clues) / 7 (Columns)

First puzzle:
Code: Select all
+------------------+------------------+-----------------+
|  2    5     67*  |  8    47   467*  |  3    1    9    |
|  3    4     67*  |  67*  1    9     |  8    5    2    |
|  1    8     9    |  2    3    5     |  4    6    7    |
+------------------+------------------+-----------------+
|  49   127   5    |  3    29   147   |  6    8    14   |
|  6    17    3    |  5    47   8     |  9    2    14   |
|  49   12    8    |  46*  29   46+1* |  7    3    5    |
+------------------+------------------+-----------------+
|  5    9     4    |  1    6    3     |  2    7    8    |
|  7    3     1    |  9    8    2     |  5    4    6    |
|  8    6     2    |  47*  5    47*   |  1    9    3    |
+------------------+------------------+-----------------+

MUG(467)r12c3, b2p34, r69c46 using single internal => +1 r6c6; ste

Second puzzle, two steps:
Code: Select all
+----------------------+----------------------+------------------------+
|  5       26   3      |  1     78      9     |  4     68     2678     |
|  26      1    478    |  478   3478    5     |  278   9      23678    |
|  478     79   4789   |  2     3478    6     |  5     18     1378     |
+----------------------+----------------------+------------------------+
|  9       3    5      |  478  d4678-1  127   |  128   1468   12468    |
| B1246    8   B14     |  45    9       3     | B12    7      456-12   |
| A12467 Ab26  A147    |  458  c468-1 Ca12    |  9     3      4568-12  |
+----------------------+----------------------+------------------------+
|  17      59   2      |  3     17      8     |  6     45     49       |
|  3       4    1789   |  6     5       17    |  178   2      1789     |
|  178     57   6      |  9     2       4     |  3     158    178      |
+----------------------+----------------------+------------------------+

1. (1=2)r6c6 - (2=6)r6c2 - r6c5 = (6)r4c5 => -1 r4c5
2. Sue de Coq: AALS (12467)r6c123, ALS1 (1246)r5c137, ALS2 (12)r6c6, restricted commons 4,6 and 1,2 => -1 r6c5, -12 r56c9; ste

Added: one step solution, but finish with basics.
Hidden Text: Show
Code: Select all
+----------------------+----------------------+------------------------+
|  5       26   3      |  1    a78      9     |  4     6-8    2678     |
|  26      1    478    |  478   3478    5     |  278   9      23678    |
|  478     79   4789   |  2     3478    6     |  5    e18     1378     |
+----------------------+----------------------+------------------------+
|  9       3    5      |  478   14678   127   |  128   1468   12468    |
|  1246    8    14     |  45    9       3     |  12    7      12456    |
|  12467   26   147    |  458   1468    12    |  9     3      124568   |
+----------------------+----------------------+------------------------+
| c17      59   2      |  3    b17      8     |  6     45     49       |
|  3       4    1789   |  6     5       17    |  178   2      1789     |
|  178    d57   6      |  9     2       4     |  3    e158    178      |
+----------------------+----------------------+------------------------+

1. (8=7)r1c5 - r7c5 = r7c1 - (7=5)r9c2 - (5=18)r39c8 => -8 r1c8; lclste
Cenoman
Cenoman

Posts: 1520
Joined: 21 November 2016
Location: Paris, France

### Re: 22 (Clues) / 7 (Columns)

Code: Select all
*----------------------------------------------------------------------*
|  5       26   3      |  1     78      9     |  4     68     2678     |
|  26      1    478    |  478   3478    5     |  278   9      23678    |
|  478     79   4789   |  2     3478    6     |  5     18     1378     |
|----------------------+----------------------+------------------------|
|  9       3    5      |  478  g14678 ga12-7  | f128  f1468  f12468    |
| d1246    8    14     |  45    9       3     | e12    7     e12456    |
| c12467  c26   147    |  458   1468   b12    |  9     3      124568   |
|----------------------+----------------------+------------------------|
|  17      59   2      |  3     17      8     |  6     45     49       |
|  3       4    1789   |  6     5       17    |  178   2      1789     |
|  178     57   6      |  9     2       4     |  3     158    178      |
*----------------------------------------------------------------------*

2r4c6 = r6c6 - r6c12 = 2r5c1 - (2=4,5,1,6)r5c3479 - (1|6)r4c789 = 16r4c56 => -7r4c6, stte
eleven

Posts: 2461
Joined: 10 February 2008

### Re: 22 (Clues) / 7 (Columns)

Code: Select all
+---------------+---------------+-----------------+
| 5     26 3    | 1   78    9   | 4   68   2678   |
| 26    1  478  | 478 3478  5   | 278 9    23678  |
| 478   79 4789 | 2   3478  6   | 5   18   1378   |
+---------------+---------------+-----------------+
| 9     3  5    | 478 14678 127 | 128 1468 12468  |
| 1246  8  14   | 45  9     3   | 12  7    12456  |
| 12467 26 147  | 458 1468  12  | 9   3    124568 |
+---------------+---------------+-----------------+
| 17    59 2    | 3   17    8   | 6   45   49     |
| 3     4  1789 | 6   5     17  | 178 2    1789   |
| 178   57 6    | 9   2     4   | 3   158  178    |
+---------------+---------------+-----------------+

7r8c6=r4c6-r4c4=r2c4-(7=8)r1c5-(8=6)r1c8-6r1c2=r2c1-r5c1=(6-4*5) r5c9=(5-48)r6c9=48r4c789-(48=7)r4c4=>-7r4c6=>stte
Ajò Dimonios

Posts: 201
Joined: 07 November 2019

### Re: 22 (Clues) / 7 (Columns)

Hi Eleven

The final part of the chain of your resolution is not clear to me. In my opinion instead of
(1 | 6) r4c789 = 16r4c56 => - 7r4c6, stte you should have written (1 | 6) r4c789 = (16 | 26) r4c56 => - 7r4c6, stte.
If 6r4c789 is false, after several singles we obtain the solution that has r4c5 = 6 and r4c6 = 2 and not r4c5 = 6 and r4c6 = 1.

Paolo
Ajò Dimonios

Posts: 201
Joined: 07 November 2019

### Re: 22 (Clues) / 7 (Columns)

First puzzle only...need the CoALS practice.
Code: Select all
+--------------+----------------+-------------+
| 2   5   67   |  8    4-7  467 | 3   1   9   |
| 3   4   67   | b67   1    9   | 8   5   2   |
| 1   8   9    |  2    3    5   | 4   6   7   |
+--------------+----------------+-------------+
| 49  127 5    |  3    29   147 | 6   8   14  |
| 6   17  3    |  5   a47   8   | 9   2   14  |
| 49  12  8    | b46   29  b146 | 7   3   5   |
+--------------+----------------+-------------+
| 5   9   4    |  1    6    3   | 2   7   8   |
| 7   3   1    |  9    8    2   | 5   4   6   |
| 8   6   2    |  47   5    47  | 1   9   3   |
+--------------+----------------+-------------+

Myth's CoALS rule applied to the two overlapping ALS(b):
(7=4)r5c5 - (46=17)r26c4,r6c6 => -7 r1c5; stte

SteveC
Sudtyro2

Posts: 730
Joined: 15 April 2013

### Re: 22 (Clues) / 7 (Columns)

Ajò Dimonios wrote:Hi Eleven

The final part of the chain of your resolution is not clear to me. In my opinion instead of
(1 | 6) r4c789 = 16r4c56 => - 7r4c6, stte you should have written (1 | 6) r4c789 = (16 | 26) r4c56 => - 7r4c6, stte.
If 6r4c789 is false, after several singles we obtain the solution that has r4c5 = 6 and r4c6 = 2 and not r4c5 = 6 and r4c6 = 1.

Hi Paolo,

there is a hidden pair 16 in r4c56, if both digits cannot be in r4c789. Then the only remaining cells for both are r4c56.
You don't have to check any singles, which in this case might lead to a contradiction to the hidden pair.
eleven

Posts: 2461
Joined: 10 February 2008

### Re: 22 (Clues) / 7 (Columns)

Hi Eleven

OK, now I understand, then for the sake of clarity "(1 | 6) r4c789 = (16) r4c56 => - 7r4c6, stte." Should be "(16) r4c789 = (16) r4c56 => - 7r4c6, stte." since the two candidates 16 are either absent or are simultaneously present in r4c789.

Paolo
Ajò Dimonios

Posts: 201
Joined: 07 November 2019

### Re: 22 (Clues) / 7 (Columns)

Hi Paolo,

I'm breaking my pledge because this is related to our recent discussion.

Ajò Dimonios wrote:OK, now I understand, then for the sake of clarity "(1 | 6) r4c789 = (16) r4c56 => - 7r4c6, stte." Should be "(16) r4c789 = (16) r4c56 => - 7r4c6, stte." since the two candidates 16 are either absent or are simultaneously present in r4c789.

Nope. There's nothing wrong with eleven's chain. (1|6)r4c789 is the only correct way to write the second last node. (16)r4c789 is wrong. I'll let eleven explain the reasons because I don't want any more quagmires.

(It should be obvious, though, if you look at the solution. With your suggestion the last strong link can't be even theoretically valid because the solution has 1r4c9 and 6r4c5. Your link claims that both digits 16 must be in either r4c789 or r4c56, neither of which is true in the solution. A strong link with both sides false is obviously invalid. With eleven's original node the left side with (1|6)r4c789 is true because 1r4c9 is in the solution.)

This is exactly what I was talking about when I said that chains with ANDed and ORed terms aren't trivial. Almost everyone makes mistakes with them at first because they're not exactly intuitive (until they are). I did too. I suggest you accept that both eleven and I know what we're talking about. It's the fastest route to finding your mistake. I'm not entering any more debates about it anyway.
-SpAce-: Show
Code: Select all
*             |    |               |    |    *
*        |=()=|    /  _  \    |=()=|               *
*    |    |   |-=( )=-|   |    |      *
*                     \  ¯  /                   *

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."

SpAce

Posts: 2573
Joined: 22 May 2017

### Re: 22 (Clues) / 7 (Columns)

Hi Space,Hi Eleven.
Space wrote:
I'm breaking my pledge because this is related to our recent discussion.

Ajò Dimonios wrote:
OK, now I understand, then for the sake of clarity "(1 | 6) r4c789 = (16) r4c56 => - 7r4c6, stte." Should be "(16) r4c789 = (16) r4c56 => - 7r4c6, stte." since the two candidates 16 are either absent or are simultaneously present in r4c789.

Nope. There's nothing wrong with eleven's chain. (1|6)r4c789 is the only correct way to write the second last node. (16)r4c789 is wrong. I'll let eleven explain the reasons because I don't want any more quagmires.

(It should be obvious, though, if you look at the solution. With your suggestion the last strong link can't be even theoretically valid because the solution has 1r4c9 and 6r4c5. Your link claims that both digits 16 must be in either r4c789 or r4c56, neither of which is true in the solution. A strong link with both sides false is obviously invalid. With eleven's original node the left side with (1|6)r4c789 is true because 1r4c9 is in the solution.)

This is exactly what I was talking about when I said that chains with ANDed and ORed terms aren't trivial. Almost everyone makes mistakes with them at first because they're not exactly intuitive (until they are). I did too. I suggest you accept that both eleven and I know what we're talking about. It's the fastest route to finding your mistake. I'm not entering any more debates about it anyway.

Nothing to complain about the beautiful resolution of Eleven. Only a clarification on the term "|" which I interpret as "xor", in Latin "aut" and not as "or". I guess you mean "|" be or (inclusive).
"|" or inclusive is not allowed in the step "(2 = 4,5,1,6) r5c3479- (1 | 6) r4c789" because (2 = 4,5,1,6) r5c3479 imposes three independent possibilities -1r4c789; -6r4c789 and -16r4c789 (all 3 can be true), to say that it is enough that only one is true ,you have to prove it. The Eleven chain, if it is a pure AIC only imposes r4c7 = 1 but not that r4c9 = 6, therefore it is an AIC with memory (memory r5c1 = 2) which surely imposes r5c7 = 1 and r5c9 = 6 and consequently r4c789 = 16 false. When you say that in inference (16) r4c789 = 16r4c6 neither of the two hypotheses is true you forget that this is not an AIC but an AIC with memory and therefore you have to take into account the memory that one of the 2 present in r5c7 and r5c9 is certainly true.

Paolo
Ajò Dimonios

Posts: 201
Joined: 07 November 2019

### Re: 22 (Clues) / 7 (Columns)

Ajò Dimonios wrote:Nothing to complain about the beautiful resolution of Eleven. Only a clarification on the term "|" which I interpret as "xor", in Latin "aut" and not as "or". I guess you mean "|" be or (inclusive).

'|' is the inclusive OR just like in programming languages which use that symbol. There's not even a standard XOR-symbol for AICs, though ^ has been suggested if one was ever needed. David Bird has used XOR (^) in some very advanced chains (probably in the JExocet Compendium, if I remember correctly), but they're not AICs.

Normal AICs don't use XOR-logic, ever, because it's not needed and would only complicate things. Both the strong link symbol '=' and '|' mean OR (inclusive), though there's a subtle difference because '=' is obviously not commutative (unless the chain has no other nodes).

"|" or inclusive is not allowed in the step "(2 = 4,5,1,6) r5c3479- (1 | 6) r4c789" because (2 = 4,5,1,6) r5c3479 imposes three independent possibilities -1r4c789; -6r4c789 and -16r4c789 (all 3 can be true), to say that it is enough that only one is true ,you have to prove it. The Eleven chain, if it is a pure AIC only imposes r4c7 = 1 but not that r4c9 = 6, therefore it is an AIC with memory (memory r5c1 = 2) which surely imposes r5c7 = 1 and r5c9 = 6 and consequently r4c789 = 16 false. When you say that in inference (16) r4c789 = 16r4c6 neither of the two hypotheses is true you forget that this is not an AIC but an AIC with memory and therefore you have to take into account the memory that one of the 2 present in r5c7 and r5c9 is certainly true.

Nope, eleven's chain is a pure and perfectly valid AIC. No ugly memories anywhere. For obvious reasons I won't even try to explain why. In fact, I'll be following with great interest if eleven (or someone else) will be able to pull off what I never could

SpAce

Posts: 2573
Joined: 22 May 2017

### Re: 22 (Clues) / 7 (Columns)

The chain - (2 = 4,5,1,6) r5c3479 means that r5c79 ≠ 2 implies that 1,4,5,6 are present in cells r5c3479, if we want to be more precise the only certain information is that r5c7 = 1 . At this point if I want to go further and I don't want to keep the memory that 2 is false in r5c9 or that it is true in r5c1 I can deduce with a weak inference that 1 is false in r4c789 or that it is false in r5c3, any of these two inferences does not lead me to conclude that r4c56 = 16 if I do not use the memory of the previous inferences.
Paolo
Ajò Dimonios

Posts: 201
Joined: 07 November 2019

### Re: 22 (Clues) / 7 (Columns)

Ajò Dimonios wrote:The chain - (2 = 4,5,1,6) r5c3479 means that r5c79 ≠ 2 implies that 1,4,5,6 are present in cells r5c3479

Yes, but it doesn't mean just that. Didn't you stop to wonder why it was written with the commas instead of just (2=1456)r5c3479? The commas imply specific cell assignments for the digits in the listed order. In other words:

(4,5,1,6)r5c3479 <-> (4r5c3 & 5r5c4 & 1r5c7 & 6r5c9)

Without the commas the digit order is not explicitly specified, and the following weak link would be harder to understand. Technically it would still be valid in this case, because there's only one possible permutation for the digits (1456) in those four cells. However, it requires more effort from the reader to verify, so eleven's choice of using the commas was good for clarity. It's also safer, because in many other similar situations the weak link wouldn't even work without the commas.

if we want to be more precise the only certain information is that r5c7 = 1

No. Commas or not, all four digits get locked into specific cells, most crucially 1r5c7 and 6r5c9, which are both needed for the following weak link. It is commonly understood that ALS-terms imply such internal arrangement logic based on the possible permutations of the locked set. Thus, this is perfectly correct:

Code: Select all
2r5c1 - (2=4,5,1,6)r5c3479 - (1|6)r4c789

An alternate way to write that part:

Code: Select all
2r5c1 - 2r5c7|6r5c1 = 16r5c79 - (1|6)r4c789

Both ways 16 gets locked into r5c79 as needed (when read from left to right). The latter just makes it more explicit.

That enables the next weak link to erase both 1 and 6 from r4c789. If you thought it would only erase one or the other (because of the OR-symbol) you were wrong. I'm almost certain that you did, because it's the most common mistake. Like I said, some things aren't very intuitive, until you understand the reasons behind them. In this case I suggest you take a look at De Morgan's Laws.

--
Btw, you're insulting eleven if you're even suggesting he used a memory chain. I don't think I've ever seen him use a memory chain. Besides, memory chains can be easily recognized because they should be marked with memory markers (usually *, ^, %, etc). If a chain doesn't have those, it should be a valid AIC (no memories) unless otherwise specified.

SpAce

Posts: 2573
Joined: 22 May 2017

### Re: 22 (Clues) / 7 (Columns)

Hi Space

Space wrte:
Yes, but it doesn't mean just that. Didn't you stop to wonder why it was written with the commas instead of just (2=1456)r5c3479? The commas imply specific cell assignments for the digits in the listed order. In other words:

(4,5,1,6)r5c3479 <-> (4r5c3 & 5r5c4 & 1r5c7 & 6r5c9)

Without the commas the digit order is not explicitly specified, and the following weak link would be harder to understand. Technically it would still be valid in this case, because there's only one possible permutation for the digits (1456) in those four cells. However, it requires more effort from the reader to verify, so eleven's choice of using the commas was good for clarity. It's also safer, because in many other similar situations the weak link wouldn't even work without the commas.

If we want to consider a single logical step - (2 = 4,5,1,6) r5c3479 without using the single memories that lead to this result (r5c3 = 4; r5c4 = 5; r5c7 = 1; r5c9 = 6), which is perfectly , can you tell me why in your opinion - (2 = 4,5,1,6) r5c3479 - (16) r4c789 = 16r4c56 is not correct?
I didn't say that - (2 = 4,5,1,6) r5c3479 - (1 | 6) r4c789 = 16r4c56 is not correct if you mean by "|" an inclusive or but simply, given that the conclusion = 16r4c56 is obtained exclusively when 1 and 6 are simultaneously excluded by r4c789, it was more immediate to write - (2 = 4,5,1,6) r5c3479 - (16) r4c789 = 16r4c56, in this case the two possible logical truths XOR (1 true and 6 false) and (1 false and 6 true) are excluded in the cells r4c789 which certainly do not lead to the conclusion r4c56 = 16. It is simply a matter of simplification.

Paolo
Ajò Dimonios

Posts: 201
Joined: 07 November 2019

### Re: 22 (Clues) / 7 (Columns)

Ajò Dimonios wrote:If we want to consider a single logical step - (2 = 4,5,1,6) r5c3479 without using the single memories that lead to this result (r5c3 = 4; r5c4 = 5; r5c7 = 1; r5c9 = 6), which is perfectly

There are no memories in this chain. All links work independently.

can you tell me why in your opinion - (2 = 4,5,1,6) r5c3479 - (16) r4c789 = 16r4c56 is not correct?

I told you the reason right off the bat. The last strong link is not valid that way, which is easily verified because both sides are false in the solution. There are rarely situations where two ANDed terms are strongly linked anyway, certainly not here. Usually one side is ANDed and the other ORed. In this case both (1|6)r4c789 = 16r4c56 and 16r4c789 = (1|6)r4c56 are valid strong links, but 16r4c789 = 16r4c56 is not.

The weak link actually works with 16r4c789 too, but then the last node should be written (1|6)r4c56 to make the strong link valid. That obviously doesn't give us the conclusion we need, because both 1 AND 6 are needed in r4c56 to eliminate 7r5c6. Thus the only way to get the elimination and write the strong link correctly is to have the ANDed node last and the ORed node before it, just like eleven wrote it. Fortunately the weak link works with that arrangement as well.

If you're still wondering why the strong link is invalid, consider this. There are four ways to arrange the two digits between the two sides:

Code: Select all
L    R   L  R    L  R
c789  c56  16=16  1|6=16  Solution
-----------------------------------
16    -     T       T        F
1     6     F <-    T        T*    <- missing case
6     1     F <-    T        F     <- missing case
-     16    T       T        F

Your strong link (third column) only accounts for the first and the last option, ignoring the two others that could be true as well. A strong link (or more generally a strong inference set) must guarantee that at least one of the included options is true. Yours doesn't, and in fact it's not even lucky enough to include the true option (even if it did, it would still be invalid). On the other hand, eleven's strong link (fourth column) accounts for all of the options, guaranteeing that the true one is among them, which makes it valid.

I didn't say that - (2 = 4,5,1,6) r5c3479 - (1 | 6) r4c789 = 16r4c56 is not correct if you mean by "|" an inclusive or but simply, given that the conclusion = 16r4c56 is obtained exclusively when 1 and 6 are simultaneously excluded by r4c789, it was more immediate to write - (2 = 4,5,1,6) r5c3479 - (16) r4c789 = 16r4c56, in this case the two possible logical truths XOR (1 true and 6 false) and (1 false and 6 true) are excluded in the cells r4c789 which certainly do not lead to the conclusion r4c56 = 16. It is simply a matter of simplification.

I don't understand much of that. There's no XOR-logic in AICs, nor a simpler way to write the last strong link. It's perfect as it is.

--
Edit. Corrected the true option in the grid.
Last edited by SpAce on Thu Aug 27, 2020 10:14 pm, edited 1 time in total.

SpAce

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