11 of 100: 999 Sentences

1 says 2 is true, 2 says 3 is false, 3 says 4 is true. ( So 4 must be false as 2 said 3 is false ) 4 says 5 is false. So 5 must be independent of whatever other have said just like 1.

So, we can see a repetition after every 4 sentences. 998%4=2 This means the cycle has ended in 996th sentence.

997) the next is true. 998) the next is false. 999) Y>=5

So, Y<5.

1 tell us 2 is true.

2 tells us 3 is false, so 4 must also be false, meaning 5 is true again.

This means that after 4 steps, the same 4 lines emerge.

Since 996 is divisible by 4, we can consider 997 equivalent to 1, meaning 998 is also correct.

Hence 998 tells us 999 is false => Y < 5.

This solution is the same as the others, just dressed in fancy language. Let $a_{n}\in\{0,1\},\:1\leq n\leq 999$ denote the truth value of the $n^{\text{th}}$ statement ( $0$ : false, $1$ : true). Then we obtain the recursion $a_{n+1}=a_n\odot (n\!\!\!\!\mod {2}),\:1\leq n\leq 998,\:a_1=1$ where $\odot$ denotes the $\text{XNOR}$ operation. The validity of this statement can be easily verified by writing down the truth table. Computing the first few terms it is clear that $a_{n+4}=a_n$ and thus $a_{999}=a_3=0$ .

The truth value $\tau(n)$ of sentence $n$ is defined recursively as $\begin{cases} \tau_1 = \mathbb{T}, \\ \tau_{2n} = \tau_{2n-1}, && n \geq 1 \\ \tau_{2n+1} = \neg\tau_{2n}. \end{cases}$ For an arbitrary $n \geq 1$ , therefore, $\tau_{2n+1} = \underbrace{\neg\neg\cdots\neg}_{n\ \text{times}}\mathbb{T} = \begin{cases} \mathbb{T} & n\ \text{even} \\ \mathbb{F} & n\ \text{odd}\end{cases}$ If $2n + 1 = 999$ then $n = 499$ is odd, showing that $\tau_{999} = \mathbb{F}$ . The statement $Y \geq 5$ if false . Therefore $\boxed{Y < 5}$ .

That's the pattern, the first column is the number of the statement, the second is what the statement description is, and the final column is whether the statement itself is true. 0 is false and 1 represents true.

If the first sentence is True : $T;F;T;F;T;F;T;F;\ldots$ becomes $T;F;F;T;T;F;F;T;\ldots$ so we can see the pattern $T;F;F;T$ Since $\frac{998}{2} = 499$ is no more divisible by 2, it means we are on 2nd value of the pattern, therefore the next sentence is false, and $Y < 5$ .

If the first sentence is False : $T;F;T;F;T;F;T;F;\ldots$ becomes $F;T;T;F;F;T;T;F;\ldots$ so we can see the pattern $F;T;T;F$ Therefore the sentence after 998 is True and $Y \geq 5$ .

Sentence (1) is true (the question says).

Sentence (1) says "The next sentence is true." So, sentence (2) is true.

Sentence (2) says " The next sentence is false." So, sentence (3) is false. (because according to the TRUE statement (1), whatever (2) says is the truth)

So, sentence (3) "The next sentence is true" is false. Sentence (4) is not true.

And sentence (4) " the next sentence is false" is false. Sentence (5) is true.

And the cycle continues.

This goes on in a cycle for every 4 sentences -

Sentences (1), (5), (9), .... , (997) are true. (Numbers of the type 4n+1, where n is a whole number)

Sentences (2),(6),(10), .... , (998) are false. (Numbers of the type 4n+2)

Sentences (3),(7),(11), ... , (999) are also false, because of the respective previous statements. (Numbers of the type 4n+3)

Sentences (4),(8),(12), ... , (1000) are also false, because of their respective previous statements. (Numbers of the type 4n)

Since 1000 is divisible by 4, the last cycle begins at the 997th sentence.

In the cycle , the third sentence, sentence (999) " Y is greater than or equal to 5" is false.

So, Y < 5.

If sentence 1 is true, then sentence 2 is also true. Since sentence 2 is true, sentence 3 must be false and what sentence 4 says "The next sentence is false" is false, so the next sentence is true. The pattern is True True False False, True True False False, ... Sentence 996 leads to sentence 997 being true and 998 also true, saying sentence 999 is false, so Y < 5

As the pattern goes, the first sentence: 1. The next sentence is true 2. The next sentence is false ................... and so on Now- 998. The next sentence is false. 999. Y ≥ 5 Since the pattern of the sentences are true, false, true...............(assuming that what they state is true) the 999 statement is false. Thus the correct option is B since it states the opposite of what the sentence says- Y < 5

If the first one is true, then the second one must be too. The third one has to be false and, since the one that is false says that the next one is true, #4 must be false. A lier is saying that the next one is false, so the next one, number 5 must true. This ABBA pattern goes on until the end. This is what it looks like.

Now, divide 999 by 4 and you get 249 with a remainder of 3. Go 3 more in to the sequence and you find out that the 999th sentence is FALSE. So, Y<5

What color is snow? What color are clouds? What color is chalk? What do cows drink???

As this quickly of 99 people and most say milk. The same pattern happens in this puzzle. You are told sentence 1 is true, and it tells you sentence 2 is true. True true true almost a thousand times, so you drink the milk and forget that sentence 998 clearly tells you that 999 is false. True true true... true true but alas 999 is false and thus Y < 5.

$1$ is true.
$2$ is true.
$\therefore 3$ is false.
$\therefore 4$ is false.
$\therefore 5$ is true.
$\therefore 6$ is true.
$...$

There is a pattern:
$1$ and $2 \mod 4$ are true .
$3$ and $0 \mod 4$ are false .
$998 \mod 4 = 2$
$\therefore 999$ is false.
$\therefore Y \lt 5$
Q.E.D

You can make a pattern of T,T,F,F,T and use this to find that 999 is a false statement.

$\$\text{since 1 says that 2 is true, and that 2 says that 3 is false, and 4 says that the next sentence is true, and 998 is also true, 999 is false, so }Y<5.$

First of all we see a pattern that goes: 1. True 2. True 3. False 4. False That repeats indefinitely (if you would like clarification on how to determine this let me know!) We can simplify this pattern by making each two adjacent statements (1&2, 3&4 etc.) into pairs. Now we have 998/2=499 pairs of statements that alternate True, False, True, False - starting with the first pair (odd numbered pairs) always being true.

Since, ultimately, we are looking for the last statements 997 and 998 and want to find if they are true or false we look at the last pair, 499, which is odd numbered. We can thus conclude the last pair of statements are True and thus 997 and 998 are true.

We can use this to deduce that the Final statement, 999 is false and thus Y < 5

statement 1 is true, statement 2 is true, statement 3 is false, statement 4 is false, statement 5 is true, statement 6 is true, statement 7 is false statement 8 is false... this pattern continues. Splitting the sentences into sequences where each term in the given sequence represents the sentence number gives:

(A): 1,5,9,13,... Gives us a sequence of true statements; (B): 2,6,10,14,... Gives us a sequence of true statements; (C): 3,7,11,15,... Gives us a sequence of false statements; (D): 4,8,12,16,... Gives us a sequence of false statements.

Notice that the terms in sequences (A) and (C) are odd and that the terms in sequences (B) and (D) are even. The statement we're interested in, statement 999, is an odd number, therefore we can eliminate sequences (B) and (D). Now, The n'th term for sequences (A) and (C) is as follows:

(A): 4n-3; (C): 4n-1. For n in the natural numbers

We will now check to see if the term 999 will either be in sequence (A) or sequence (C).

Claim 1: 4n-3=999, we have

n=501/2, (which is not an integer, and hence the term 999 is not in this sequence).

Claim 2: It follows that the sequence (C) must contain the term 999, we have

4n-1=999 => n=250.

Finally, since sequence (C) contains term 999, statement 999 is false.

So, Y<5.

Sentence 998 says that sentence 999 is false! So the answer is y<5 !

The questions repeat, starting with 1 stating the next statement being true, the 2nd statement stating the 3rd being false, etc... Notice, once you get to the 5th statement, which turns out to be true, it says the same statement as statement 1. Therefore, the puzzle cycles every 4 statements starting with statement one. Using what we know about statement 1-4, we know the cycle goes: True, True, False, False. Now, we subtract 1 from 999 and mod it by 4 to check if it is true: $(999-1)\mod4 = 998\mod4 = 2$ . Now, we subtract 999 by 2 to check for the other possibility: $(999-2)\mod4 = 997\mod4 = 1$ . Since none of these end up being a factor of 4, statement 999 is false, implying that $\boxed{Y < 5}$

Follow along the sequence assuming statement one (1) is "T" then the pattern T T F F T T F F emerges. Another way to say this is if statement (n-1)mod4 is less than 2 the statement is "T" else it is "F". Since (999-1) = 998 and 998 mod 4 is "2" then statement 999 is "F" so it is not the case that Y >= 5. This means that Y < 5

After working out the first four we see that it makes a cycle of true, true, false, false. We know that cycle repeats here because it begins with a true affirmative statement the same as the first true in the prior cycle.

The pattern is affirmative statement is true, negative statement true from prior statement, negative statement negates next statement, negated negative statement affirms next (affirmative) statement to repeat the cycle.

The formula for this is simply statement index mod 4. 0 is false, 1 is true, 2 is true, 3 is false.

With that in hand one can go directly to 999 = 249 + 3, that is, 3 mod 4 to determine that statement 999 is false, and Y < 5.

The first sentence is true and says the following is true .

The second sentence is true and says the following is false .

The third sentence is false and says the following is true (so the next one will be false) .

The fourth sentence is false and says the following is false (so the next one one will be true) .

Note that the next two sentences will be exactly the same as our original first two sentences, so a pattern has shown up. Since for every four sentences we get back to our starting state/point, we can find at what point will the $998^{th}$ sentence be by simply divinding 998 by 4 and getting the remainder (or finding 998 mod 4) which is equal to 2. So we discover that the $998^{th}$ sentence will be logically equivalent to the second sentence (true and implying that the next is false). Finally, we know that the $999^{th}$ sentence is false and so $Y<5$ .

The First sentence says that the Second sentence (ie the next sentence) is TRUE and the Second sentence says that the Third sentence(ie the next sentence) is FALSE. If we go on like this we can observe that the sentences with even numbering ( 2,4,6,8,...998) are TRUE while those with odd numbering( 1,3,5,7,9....999) are FALSE. So, Y >= 5 being the 999th sentence is a false sentence (999 is odd). Hence the correct answer is Y<5.

Since the $999$ sentence indicates that $Y\ge 5$ which is false because of the pattern; therefore, $Y<5$

It's not very scientific, but a True /False pattern quickly emerged about the sentences; all of the odd numbered sentences being false. 999 is an odd number, so Y< 5.

Saying the "true" any times, or saying "false" twice ( just like swiching a light twice), does no change at all. Because they makes no change, let's throw them away( they make no change, huh?). If the number of the "false" is EVEN, there remains nothing but the original "Y>5" --we throw the "false" twice at a time...no change. Else, when we have ODD( EVEN+1) number of the "false",it's like saying 1 "false",Y<5. Now review the problem, the number of the "false" is (998-2)/2 +1=499, ODD, solution completes.

The equation is about the arithmetic progression. How to understand: numbers are standing one by one with the same "gap",we have n-1 "gap"s in n numbers. Since we know the distance between the biggest and the smallest is the same as n-1 "gap"s, we can know n-1=(biggest-smallest)/"gap",n is easy to find then.In this problem,the gap between "false" is 2,the biggest "false" is 998...all in all, what we need is n.

Putting 4 sentences into a group is even better, it's logically the same:"false" is mentioned twice, no change. I found that in others solution--great idea!

Which we can also write as

Thus,

$999 \mod 4=3$ thus $999=249\cdot4+3$ hence 999 is of the form $4n+3$ and so the truthiness of the 999-th sentence is false. Hence $Y<5$

Here is some python code to solve this problem:

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truth = 'true'                  # Commences with the first statement as true
for n in range(1,1000):         # Performs the function for each sentence from 1-999
    if n % 2 != 0:              # When the statement says 'The next sentence is true' (odd numbered statements)
        if truth == 'true':     # If the statement is true
            truth == 'true'     # Therefore the next statement is true
            print (n, truth)    # Prints the statement number and its truth
        else:                   # If the statement is false
            truth = 'false'     # Therefore the next statement is false
            print (n, truth)    # Prints the statement number and its truth
    else:                       # If the statement says 'The next sentence is false' (even numbered statements)
        if truth == 'true':     # If the statement is true
            truth = 'false'     # Therefore the next statement is false
            print (n, truth)    # Prints the statement number and its truth
        else:                   # If the statement is false
            truth = 'true'      # Therefore the next statement is true
            print (n, truth)    # Prints the statement number and its truth

This finally prints to the console: '999 false'

Thus, Y<5 .

However, this is somewhat unnecessary, as the statements cycle truth every 4 sentences (true, true, false, false), meaning that the 999th statement will be false (as 999 % 4 = 3, and the fourth statement in the cycle is false).

Another way of seeing it: imagine a long chain of statements saying "the next statement is true", and that we know that the first statement tells the truth. Then the second statement will be true, and thus the next, etc. Also, anything appearing immediately after this chain will also be true.

Now, imagine the same chain, but change one of the statements somewhere in the middle, say #17, to "this statement is false". Now, that means that statements 1-17 tell the truth, but that #18 is lying. Since #18 says "the next statement is true", we know that #19 is false. For the same reason, #20 is false. That is, for statements 1-17, we are in "truth world", but for 18 until the end, we are in "untruth world" (if you excuse the silly terminology), and the reason for the flip is the statement "the next sentence is false".

With similar reasoning, it is possible to conclude that the statement "the next sentence is false" flips both back and forth from "untruth world", and that this state remains until the next appearance of this statement (no matter how many "the next statement is true"-statements, whether 0 or more, are in between).

In the given example, there are 499 "the next statement is false"-statements, so starting from "truth world", we make the flip an odd number of times. So by the time we reach the unique statement, we are in "untruth world", so Y < 0.

Solving problem in general:

We notice that pattern emerges and there is repetition every 4 sentences - after every 4 sentence we are at the beginning (check out other solutions why it is like that).

We let n be the number of a final sentence. If:

• $n mod 4 = 1$ the sentence is true,

• $n mod 4 = 2$ the sentence is true,

• $n mod 4 = 3$ the sentence is false,

• $n mod 4 = 0$ the sentence is false.

Brilliant described this problem as the hardest so far and yet 74% of people managed to do it right. I guess leaving only 2 options to choose from significantly raise chances of ones who didn't actually know the correct answer (although the nature of this problem doesn't enable many other reasonable options).

Statements are cyclic. If the index of the statement is 2(2x - 1) or 4x - 3, the statement is true, which means 998 is true.

Let's see the first four sentences, knowing the first is true we have

1) the next sentence is true (True)

2) the next sentence is false (True)

3) the next sentence is true (False)

4) the next sentence is false (False)

5) the next sentence is true (True)

.....

All repeated with a period of 4 (T, T, F, F)

So we have only to see that 999 = 3 (mod4) and we know that the sentence is false.

when the first sentence is true it means that the second one is true, third is false, fourth is false and fifth one is true.Actually, there is a pattern that recurs in every 4 sentences.So, divide 999 by 4 the remainder is 3 and we know that the third sentence is wrong so the last sentence is wrong too.

Here given sentence 1 true therefore sentence 2 false , sentence 3 false , sentence 4 true and again sentence 5 true.

Therefore the sequence running with a period 4 as ' T F F T ' .

Now 998=2 (mod 4)

Therefore 998th sentence saying that 999 th sentence is false

Therefore , Y<5

If the first sentence is true, then the next sentence is true, which means that the next sentence is false and so on... Example: Whether the statement is true or false - According to whether it is true or false, what the statement actually means. 1. True - The next sentence is true 2. True - The next sentence is false 3. False - The next sentence is false 4. False - The next sentence is true 5. True - The next sentence is true 6. True - The next sentence is false At this point, you can see a pattern emerging. True, true, false, false, true, true, false, false... Every 4 true/false statements, the pattern repeats. 999/4=249 R3 The remainder of 3 tells us that the 999th statement is false by counting to the 3rd true/false statement: 1. True 2. True 3. False 4. False

So, Y>5 is false, meaning that it means Y<5.