The answer is APPLE DUMPLING
The corresponding square is house puzzle
by Brian Tivol
The devious part about this logic puzzle is that it admits two
different solutions:
Statement:
0000000001111111111222222222233333333334
1234567890123456789012345678901234567890
A: FTFFFFFTFTFTFFFFFTFTFFFFFTFFTTFFFTFFFTFT
B: FFTFFTFTFTFTTFTTFFFFFTTFFFTFFTFTTTFFFTTT
Therefore, pure deduction and constraint propagation will not solve
this puzzle-- you must make a guess at some point. Not only that, but
you must double-check your work and try the opposite guess.
Solution A gives the word "APPLE". Solution B gives the word
"DUMPLING". Either word alone is insufficient.
###
Here's one way to reason through this giant mess:
Statement 24 says 22 == 23.
Statement 31 says 32 == 33.
Statement 33 says 23 == 33.
Therefore, 22 == 23 == 32 == 33.
We can now argue that 34 is true. In any consistent solution to this
puzzle, we know that 32 == 33; so, when the statements 32 and 33 are
interchanged, the truth value of statement 32 is unchanged; thus the
number of statements numbered with a power of two which are true is
unchanged. Hence that that statement (formerly 32, now 33) is still
consistent with its truth value. Similarly, the statement (formerly
33, now 32) referring to the statement listed ten statements prior is
still consistent with its truth value.
Moreover, any other statement referring to numbers 32 and 33 refer
only to the truth values of the statements at 32 and 33, not what the
statements actually say. Therefore after the swap, the entire
solution is still consistent. This is what 34 says, and so 34 is true.
Statement 36 means 36 is false.
Statement 12 says 24 == 36.
Therefore, statement 24 is false.
Statement 13 says 26 != 39.
Statement 37 says 30 is true.
Therefore, two of {6,12,18,24,36} are true.
Since we know about 24 and 36, that means that two of {6,12,18} are
true.
What if 12 were false?
Then 6 and 18 would both be true, but 6 implies that 3 is true while
18 implies that 3 is false. This is a contradiction.
So 12 is true.
Therefore, 6 != 18.
Statement 17 says 'Every statement that begins "Exactly one-sixth" is
true' is true.
Therefore, 15 == 16.
Also, 8 is true.
Also, 38 is true.
Therefore, there are either six, twelve, or eighteen true statements.
We know that 8, 12, 30, 34, 38, one of {6,18}, and one of {26,39} are
all true, which makes at least seven. Therefore, there are either
twelve (we'll call that "A") or eighteen ("B") true statments.
At least one of 39 and 40 is true.
Given that, by 39, either
(A) 39 is false and 40 is true
or:
(B) 39 is true and 40 is true
Therefore, 40 is true.
Therefore, two of {5,10,15,20,25,35} are true.
At most two of 1, 2, 3, 4, 5, 6, and 7 are true.
At least one of 1, 2, 3, 4, 5, 6, and 7 is true.
By 4 and 7, we have 1 == 4 == 7.
Therefore, 1 is false, 4 is false, and 7 is false.
Therefore, 5 is false.
By 6, we have 3 == 6.
Statements 2 and 3 are mutually exclusive.
Therefore, either
(A) 2 is true, 3 is false, 6 is false
or:
(B) 2 is false, 3 is true, 6 is true
Here's an overview of where we stand so far:
1234567890123456789012345678901234567890
*: F--FF-FT---T-----------F-----T---T-F-T-T
###
Let's try option A: There are twelve true statements.
Therefore, 2 is true, 3 is false, 6 is false, 39 is false.
Since 26 != 39, we know 26 is true.
Since 26 is true, we know 25 is false, 27 is false, and 28 is false.
Since 6 != 18, we know 18 is true.
Since 18 is true, we know 9 is false, 15 is false, 21 is false, and 33
is false.
Since 15 == 16, we know 16 is false.
Since 22 == 23 == 32 == 33, we know 22 is false, 23 is false, and 32
is false.
Since 21 and 22 are false, we know that 20 is true.
Since 20 is true and 18 is true, we know that 19 is false.
Let's see how option A is going so far:
1234567890123456789012345678901234567890
A: FTFFFFFTF--T--FF-TFTFFFFFTFF-T-FFT-F-TFT
There is a sequence of five consecutive false answers (3 through 7),
and there is no room for a sequence of six consecutive false
answers. Therefore, 10 is true and 11 is false.
By 40, statement 35 is false.
Since 20 is true and since there are twelve true statements, it
follows that there are exactly six true statements between 1 and 20.
These must be 2, 8, 10, 12, 18, and 20. Therefore, 13 is false, 14
is false, and 17 is false.
There are three true statements in the first quarter and three true
statements in the second quarter. Since statements 22 and 23 are
false, it follows that there cannot be more than three true
statements in the last quarter. Therefore, 31 is false and 37 is
false.
Since 20 is true and since there are twelve true statements, it
follows that there are exactly six true statements between 21 and
40. These must be 26, 30, 34, 38, 40, and one other. The only
undetermined statement is 29; therefore 29 is true.
1234567890123456789012345678901234567890
A: FTFFFFFTFTFTFFFFFTFTFFFFFTFFTTFFFTFFFTFT
###
Let's now try option B: There are eighteen true statements. (There is
no reason not to double-check our work done with option A.)
Therefore, 2 is false, 3 is true, 6 is true, 39 is true.
Since 26 != 39, we know 26 is false.
Since 6 != 18, we know 18 is false.
By 19, since 18 is false, 19 == 20.
Here's a quick overview of option B so far:
1234567890123456789012345678901234567890
B: FFTFFTFT---T-----F-----F-F---T---T-F-TTT
It's tricky but useful at this point to prove that, if 31 were false,
then 29 would also be false. Since there are already three
consecutive true answers (38 through 40), if 29 were true, then it
would have to be part of a series of four consecutive true answers.
If 31 were false, then 29 would have to be true along with 27, 28,
and 30. However, 27 and 28 cannot both be true at the same time.
Therefore, 29 would have to be false. We'll use this lemma below,
twice.
Now, ask what if 22 were false?
In the first quarter, there are three known true statements (3, 6,
and 8) and at most one more (9 or 10). In the last quarter, there
are four known true statements (34, 38, 39, and 40). If 22 were
false, then there would have to be exactly four true statements in
the first quarter and exactly four true statements in the last
quarter. Therefore, 31, 32, 33, 35, and 37 would be false; exactly
one 9 or 10 would be true, and therefore 11 would be false. Since
22 == 23, it would also be the case that 23 were false. By our
earlier lemma, with 31 false, it follows that 29 is false.
Here's how our hypothetical situation looks:
1234567890123456789012345678901234567890
?: FFTFFTFT--FT-----F---FFF-F--FTFFFTFFFTTT
Yet another consequence of 22 being false is that, by 21, we would
have 20 != 21. If 20 were true, there would have to be nine true
statements between 21 and 40. At most, there could only be six: 40;
39; 38; 34; 30; and one of 25, 27 or 28. Therefore 20 would be
false and 21 true. Since 19 == 20, 19 would be false.
This leaves us with the following known and possible true
statements: 3; 6; 8; one of 9 or 10; 12; possibly 13; possibly 14;
possibly 15; possibly 16; possibly 17; 21; at most one of 25, 27,
and 28; 30; 34; 38; 39; and 40. There are only seventeen of these,
which is not enough.
That means that 22 cannot be false.
Therefore, 22 is true.
Therefore, 23 is true, 32 is true, and 33 is true.
Therefore, 31 is false.
Therefore, by our lemma, 29 is false.
Because 32 is true, since 1, 2, and 4 are false and since 8 is true,
it follows that 16 is true.
Therefore 15 is true.
Therefore 13 != 14.
Since 22 is true, by 21, 20 == 21.
Therefore, 19 == 20 == 21.
If 20 were true, then so would be 21, 22, 23, 30, 32, 33, 34, 38, 39,
and 40, making at least ten true statements between 21 and 40.
Therefore, 20 is false.
Therefore 19 is false and 21 is false.
Here's another overview.
1234567890123456789012345678901234567890
B: FFTFFTFT---T--TT-FFFFTTF-F--FTFTTT-F-TTT
There is a series of four consecutive false answers (18 through 21)
and a series of seven consecutive false answers is impossible.
Therefore, one of 9, 10, and 11 is true.
This gives us the following known true statements: 3; 6; 8; one of 9,
10, and 11; 12; one of 13 and 14; 15; 16; 22; 23; 30; 32; 33; 34;
38; 39; 40. That makes seventeen, so exactly one of 17, 25, 27, 28,
35, and 37 must be true.
If 17 is false, then the five statements from 17 through 21 are all
false; if 17 is true, then 25, 27, and 28 must be false, making the
six statements from 24 through 29 are all false. Either way, this
means that either 10 or 11 is true, and therefore 9 must be false.
Since 36 is false, as are 9 and 18, it follows that 27 must be true.
Therefore, 17, 25, 28, 35, and 37 are false.
There is a series of five consecutive false answers (17 through 21)
and a series of six is impossible. Therefore 10 is true and 11 is
false.
Since 35 is false, it follows that 14 is false.
Therefore, 13 is true.
1234567890123456789012345678901234567890
B: FFTFFTFTFTFTTFTTFFFFFTTFFFTFFTFTTTFFFTTT
###
It still remains to verify that both final configurations are valid.
This is left as an exercise for the reader.
Now, how to find the magic words?
If we take true statements to be 1 and false statements to be 0, then,
for option A, the encoding of ASCII seems to yield an English word.
With the same encoding, option B seems to be numbers from 1 to 26.
These options look like the right "common encoding" to use.
A: 01000001 01010000 01010000 01001100 01000101 = APPLE
B: 00100 10101 01101 10000 01100 01001 01110 00111 = DUMPLING
(Personally, I'd've associated a circle with APPLE and a square with
DUMPLING, but that's not what happened with the people I polled.)