# Applied logic challenge problem #2

This is similar to my previous Applied Logic Challenge Problem but with a significant difference at the end which could make this problem somewhat more difficult. Again it is a problem of my own devising. You must determine what can be concluded from the 40 true statements below. The numbers in each statement represent binary values (true / false, 1 / 0), or attributes that either are present or are not present. Do not confuse statement numbers (like Statement 9:) with the numbers within each statement which refer to the binary attributes (like 10,9,13 and 16 within Statement 9). These attributes do not refer to statement numbers.

For example, in words, Statement 1 reads:
If 22 is true and 29 is true then 9 is false.
Statement 9 is unrelated to this and remains true (all the statements are true)

statement 1 : if 22 and 29 then not 9
statement 2 : if 21 and 10 then 29
statement 3 : if 20 and 32 and not 17 then 24
statement 4 : if 20 and 19 and not 5 then 18
statement 5 : if ( 9 and 16 ) or ( 20 and not 30 ) then 11
statement 6 : if ( 28 and 15 ) or ( 23 and not 8 ) then 21
statement 7 : if 7 or 35 then 3
statement 8 : if 19 and 11 then not 25
statement 9 : if 10 and 9 and not 13 then 16
statement 10 : if 29 or 27 then 7
statement 11 : if 34 or 1 then not 32
statement 12 : if 25 or 32 then 27
statement 13 : if 25 and 18 then not 22
statement 14 : if 5 or 14 then not 24
statement 15 : if 25 and 34 then not 29
statement 16 : if 36 or 27 then not 34
statement 17 : if 14 and 15 then not 13
statement 18 : if 20 or 12 then not 30
statement 19 : if 6 or 26 then 34
statement 20 : if 14 and 11 and not 24 then 16
statement 21 : if ( 5 and 7 ) or ( 14 and not 12 ) then 1
statement 22 : if 15 or 9 then 2
statement 23 : if 25 or 3 then 16
statement 24 : if ( 32 and 18 ) or ( 27 and not 26 ) then 11
statement 25 : if 22 and 9 and not 7 then 14
statement 26 : if 9 or 13 then 6
statement 27 : if 2 or 19 then 1
statement 28 : if 25 and 12 then not 9
statement 29 : if 21 or 11 then 27
statement 30 : if ( 17 and 1 ) or ( 24 and not 36 ) then 22
statement 31 : if 26 or 23 then 6
statement 32 : if 2 or 28 then 23
statement 33 : if 29 or 35 then 11
statement 34 : if 20 or 10 then 18
statement 35 : if 19 and 7 and not 10 then 1
statement 36 : if ( 27 and 22 ) or ( 2 and not 34 ) then 6
statement 37 : if 34 and 7 and not 4 then 29
statement 38 : if 29 or 5 then not 12
statement 39 : if 4 and 33 and not 1 then 14
statement 40 : if 31 or 15 then 32

• "statement 9 : if 10 and 9 and not 13 then 16" - That 9 means that they are self-referential? – Victor Stafusa Dec 11 '18 at 21:33
• @VictorStafusa, I made the same mistake in the previous puzzle. The statement numbers are unrelated to the numbers in the statement. Statement 9 is definitely true. Number 9 could be true or false. – Dr Xorile Dec 12 '18 at 0:17
• Yes, Dr. Xorile is right. Also, I added some further clarification in the problem description. Another way to easily visualize this is to consider the numbers within each statement (not the statement number) as letters instead of numbers. It just was much easier for me to generate this problem without using letters. – Bob Bixler Dec 12 '18 at 0:23
• Bob, if it's easy for you (and if you intend to post another one) to just use letters right from the beginning. You can go A-Z, then keep going by using greek letters. – Hugh Dec 12 '18 at 3:22

I believe the answer is:

15 is always False. Any other number can go either way.

The reason is:

if 15 is True then:
statement 22 : 15 or 09 implies 02
statement 27 : 02 or 19 implies 01
statement 11 : 34 or 01 implies not 32
and
statement 40 : 31 or 15 implies 32
Therefore, 15 is False

Apart from that:

There exist examples of all the other ones going either way.

Here is a more general set of conclusions to help anyone looking for the other conclusion the OP is looking for. In each case, if you set one of the values to 1, then there are certain conclusions that you can draw. In this case, "_" means that you can't draw any particular conclusion about that specific variable.

If 01 is 1 then 1______________________________0____
If 02 is 1 then 11___1________________1________0_1__
If 03 is 1 then __1____________1____________________
If 04 is 1 then ___1________________________________
If 05 is 1 then ____1______0___________0____________
If 06 is 1 then _____1_________________________0_1__
If 07 is 1 then __1___1________1____________________
If 08 is 1 then _______1____________________________
If 09 is 1 then 11___1__1_____________1________0_1__
If 10 is 1 then _________1_______1__________________
If 11 is 1 then __1___1___1____1__________1______0__
If 12 is 1 then ___________1_________________0______
If 13 is 1 then _____1______1__________________0_1__
If 14 is 1 then _____________1_________0____________
If 15 is 1 then no stable solution found
If 16 is 1 then _______________1____________________
If 17 is 1 then ________________1___________________
If 18 is 1 then _________________1__________________
If 19 is 1 then 1_________________1____________0____
If 20 is 1 then __1___1___1____1_1_1______1__0___0__
If 21 is 1 then __1___1________1____1_____1______0__
If 22 is 1 then _____________________1______________
If 23 is 1 then _____1________________1________0_1__
If 24 is 1 then _______________________1____________
If 25 is 1 then __1___1________1________1_1______0__
If 26 is 1 then _____1___________________1_____0_1__
If 27 is 1 then __1___1________1__________1______0__
If 28 is 1 then _____1________________1____1___0_1__
If 29 is 1 then __1___1___10___1__________1_1____0__
If 30 is 1 then _____________________________1______
If 31 is 1 then __1___1________1__________1___11_0__
If 32 is 1 then __1___1________1__________1____1_0__
If 33 is 1 then ________________________________1___
If 34 is 1 then _______________________________0_1__
If 35 is 1 then __1___1___1____1__________1______01_
If 36 is 1 then _________________________________0_1


So a possible conclusion could be something like:

34 and 32 are not both true.

In fact, you can draw this kind of conclusion for a number of results:

1 and 31 are not both true
1 and 32 are not both true
2 and 11 are not both true
2 and 20 are not both true
2 and 21 are not both true
2 and 25 are not both true
2 and 27 are not both true
2 and 29 are not both true
2 and 31 are not both true
2 and 32 are not both true
2 and 35 are not both true
2 and 36 are not both true
3 and 9 are not both true
5 and 12 are not both true
5 and 24 are not both true
5 and 31 are not both true
5 and 32 are not both true
6 and 11 are not both true
6 and 20 are not both true
6 and 21 are not both true
6 and 25 are not both true
6 and 27 are not both true
6 and 29 are not both true
6 and 31 are not both true
6 and 32 are not both true
6 and 35 are not both true
6 and 36 are not both true
7 and 9 are not both true
9 and 11 are not both true
9 and 16 are not both true
9 and 20 are not both true
9 and 21 are not both true
9 and 25 are not both true
9 and 27 are not both true
9 and 29 are not both true
9 and 31 are not both true
9 and 32 are not both true
9 and 35 are not both true
9 and 36 are not both true
11 and 13 are not both true
11 and 22 are not both true
11 and 23 are not both true
11 and 26 are not both true
11 and 28 are not both true
11 and 34 are not both true
12 and 29 are not both true
12 and 30 are not both true
13 and 20 are not both true
13 and 21 are not both true
13 and 25 are not both true
13 and 27 are not both true
13 and 29 are not both true
13 and 31 are not both true
13 and 32 are not both true
13 and 35 are not both true
13 and 36 are not both true
14 and 24 are not both true
19 and 31 are not both true
19 and 32 are not both true
20 and 22 are not both true
20 and 23 are not both true
20 and 26 are not both true
20 and 28 are not both true
20 and 30 are not both true
20 and 34 are not both true
21 and 22 are not both true
21 and 23 are not both true
21 and 26 are not both true
21 and 28 are not both true
21 and 34 are not both true
22 and 25 are not both true
22 and 27 are not both true
22 and 29 are not both true
22 and 31 are not both true
22 and 32 are not both true
22 and 35 are not both true
23 and 25 are not both true
23 and 27 are not both true
23 and 29 are not both true
23 and 31 are not both true
23 and 32 are not both true
23 and 35 are not both true
23 and 36 are not both true
25 and 26 are not both true
25 and 28 are not both true
25 and 34 are not both true
26 and 27 are not both true
26 and 29 are not both true
26 and 31 are not both true
26 and 32 are not both true
26 and 35 are not both true
26 and 36 are not both true
27 and 28 are not both true
27 and 34 are not both true
28 and 29 are not both true
28 and 31 are not both true
28 and 32 are not both true
28 and 35 are not both true
28 and 36 are not both true
29 and 34 are not both true
31 and 34 are not both true
32 and 34 are not both true
34 and 35 are not both true
34 and 36 are not both true
Of course, it's always possible to have both be false, since all false is a valid solution.

Each of these statements can be proved in a similar way. For example:

If 1 and 31 are true then
statement 11 : 34 or 01 implies not 32
statement 40 : 31 or 15 implies 32
1 and 31 are not both true

Or, a bit more interesting:

If 2 and 11 are both true then:
statement 27 : 02 or 19 implies 01
statement 29 : 21 or 11 implies 27
statement 16 : 36 or 27 implies not 34
statement 32 : 02 or 28 implies 23
statement 31 : 26 or 23 implies 06
statement 19 : 06 or 26 implies 34
2 and 11 are not both true

Similar arguments can be made for each of the pairs above.

• Right again, Dr. Xorile, and much faster this time. I figured as much so this time as promised there is a twist ---- there are 2 conclusions to be reached. You got one of them. The hint I will give you is that everything you stated is true. – Bob Bixler Dec 12 '18 at 3:30
• First a minor point -- The conclusion "34 will be not 32 unless they are both false" is not strictly true because there is no solution (all statements true) when both 32 and 34 are true. Speaking in the general sense there are many valid conclusions of this general type involving different variables in groups of 2,3,4 or more. However what I'm looking for is shorter and simpler involving 2 variables. – Bob Bixler Dec 12 '18 at 16:42
• I don't understand your point. But if 32 is true then 34 is false. And if 34 is true then 32 is false. They cannot both be true. However, there are solutions in which 32 and 34 are both false. – Dr Xorile Dec 12 '18 at 18:07
• Yes, we are both correct depending on the universe of possible variable values to start with. If we consider only good solutions where all statements are true then you are correct. If we start with all possible variable values where the universe of combinations include outcomes where not all statements are true then I would be correct. In short I think this is not important and my bringing this up has only slowed down your path to the solution. I'll stick to directly relevant stuff from now on. – Bob Bixler Dec 12 '18 at 22:22
• But all statements are true, right? Do you mean values? I looked at inferences that can be drawn by considering all possible inferences for each statement and checking to see if they were all true. – Dr Xorile Dec 12 '18 at 23:18