Find the number

Assume that each statement is either true or false. What is the unknown (whole) number from the following possibly false statements:

1) At least one of the last two statements in this list is true. 2) This is either the first true or the first false statement in the list. 3) There exist three consecutive false statements. 4) The difference beween the numbers of the last true statement and the first true statement is a factor of the unknown number. 5) The sum of the numbers of the true statements is the unknown number. 6) This is not the last true statement. 7) Each true statement's number is a factor of the unknown number. 8) The unknown number equals the percentage of these statements which are true. 9) The number of different factors which the unknown number has (excluding 1 and itself) is more than the sum of the numbers of the true statements. 10) There are no three cosecutive true statements.

What is the number?

(recirculated problem, first found in New Scientist 11 years ago.)

SPOILER

Statement 1 must be false, otherwise statement 2 always causes a contradiction. Therefore statements 9 and 10 must be both false as well, and this means that there must be 3 consecutive true statements present. Trial and error from this point results in finding the statements to be F T T T F T T F F F, and the number must be 420.

Please reply to drgmayer at hotmail dot com

Is Informatics different from Computer Science?

* Jim Ward

In practice the difference is not great. However, our department's aim is to be a little bit more theoretical or conceptual if you like. We do not have a good translation of the term 'computer science' in Norwegian anyway. You'll find 'Informatics' in German and French universities as well. And I bet someone can trace this tendencies back to the divisions of philosophies that took place in the 18th

Here's the way I see it:

V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V

If 2 is true, then 1 is false. If 2 is false, then 1 is false. Hence:

STATEMENT 1 IS FALSE.

Therefore:

STATEMENT 9 IS FALSE. STATEMENT 10 IS FALSE.

Independently, if 6 were false, it would be the last true statement, so it would be true. Therefore:

STATEMENT 6 IS TRUE.

If 8 were true, the unknown number would be a multiple of 10 between 0 and 100 inclusive. In fact, it would be between 20 and 70 inclusive, since we would have at least two true statements (6,8) and at least three false statements (1,9,10). This means 7 would be false, because if 7 were true, the unknown number would have to be a multiple of both 6 and 7, which none of the numbers 20,30,40,50,60,70 are. Also, 3 would be false, since if 3 were true there would be no room anywhere for three consecutive false statements. This means 4 and 5 would be true, because of the three consecutive true statements guaranteed by the falsity of 10. By now we would have at least four true statements (4,5,6,8) and at least five false statements (1,3,7,9,10) so the unknown number would have to be either 40 or 50. But this would contradict the truth of 5, since there would no longer be any way the sum of the true statements could be as high as 40. This contradiction proves (whew!) that:

STATEMENT 8 IS FALSE.

Now, because 6 is not the last true statement:

STATEMENT 7 IS TRUE.

Also, because of 8-9-10:

STATEMENT 3 IS TRUE.

Now, if 5 were true, the unknown number would have to be 3+5+6+7, plus possibly 2, plus possibly 4. Thus the unknown number would be one of 21,23,25,27. But this would contradict the truth of 7, since 6 is not a factor of any of 21,23,25,27. This contradiction proves that:

STATEMENT 5 IS FALSE.

Now the falsity of 10 proves that:

STATEMENT 2 IS TRUE. STATEMENT 4 IS TRUE.

Because 7 is true, the unknown number must be a multiple of all of 2,3,4,6,7. Because 4 is true, the unknown number must also be a multiple of 5. The LCM of all these (2,3,4,5,6,7) is 420, so the unknown number must be 420 or a multiple thereof. 420 has 22 distinct non-trivial factors, which barely avoids contradicting the falsity of 9. If the unknown number were a non-trivial multiple of 420, it would have 420 as a 23rd non-trivial factor, which WOULD contradict the falsity of 9.

So the unknown number is 420 itself.

Bill Smythe

Using Google Groups (which naively groups threads by subject line) to view this thread shows this post as #4 in a thread which was started by you 11 years ago with a very similar post

Guilty!

(I know I should have disguised the problem a little better. Have mercy with me, I thought the quality of the problems in rec.puzzle was

) Assume that each statement is either true or false. What is the ) unknown (whole) number from the following possibly ) false statements: ) ) 1) At least one of the last two statements in this list is true. ) 2) This is either the first true or the first false statement in the ) list. ) 3) There exist three consecutive false statements. ) 4) The difference beween the numbers of the last true statement and ) the first true statement is a factor of the unknown number. ) 5) The sum of the numbers of the true statements is the unknown ) number. ) 6) This is not the last true statement. ) 7) Each true statement's number is a factor of the unknown number. ) 8) The unknown number equals the percentage of these statements which ) are true. ) 9) The number of different factors which the unknown number has ) (excluding 1 and itself) is more than the sum of the numbers of the ) true statements. ) 10) There are no three cosecutive true statements.

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Statement 6 must be true; if it were false, that would directly lead to a contradiction.

Statement 1 must be false; if it were true, statement 2 would lead to a contradiction.

That implies that statements 9 and 10 are false.

Statements 5 and 8 cannot both be true (for N true statements, they give a different result for the unknown number.)

Assuming 5 is true, then 8 would be false, 7 would have to be true because of statement 6. Because of 7, the unknown number has 5 and 7 as factors, and is therefore at least 35. However, because of 5, the unknown number can at most be 27. Contradiction. Therefore, statement 5 is false.

Assuming 7 is false, then 8 must be true because of 6, and the only place for three consecutive true statements (that must occur because of 10) would be 2, 3 and 4. However, there would be no three consecutive false statements, meaning 3 must be false. Contradiction. Statement 7 is true.

Because of 6 and 7, the unknown number must be a multiple of 42, but 8 implies that it is a multiple of 10, between 0 and 100. Therefore, 8 must be false. Because of statement 10, there must be three consecutive true statements, which must be 2, 3 and 4. (This does not lead to any contradictions, by the way.)

So the truth values are: F, T, T, T, F, T, T, F, F, F

Because of statement 7, the number must have factors 2, 3, 4, 6 and 7, and because of statement 4 it must have a factor 5. Therefore, it is a multiple of 420.

Now, because of 9, it cannot have more than 22 different factors. 420 happens to have exactly 22 different factors, any multiple would have more, so the number has to be 420.

SaSW, Willem

If 'wissenschaft' means 'science' in German, I'd expect something like 'komputerwissenschaft'.

Jim Ward schrieb:

Historically, as a university department computer science was born either out of electrical engineering or mathematics. Calling the field 'computer science' stresses the EE approach, whilst calling it 'Informatique' avoids using an English term (always important for the French) and stresses the thereotical aspects of it. It allowed some people to be a computer scientist and to actually never use a computer!

If I had to translate 'computer science' literally into German, I'd use 'Rechnerkunde'.

The term 'cybernetics' is 1/12 as popular as 'computer science', but being of Greek origin, it has the advantage of sounding similar (and equally strange) in most languages.

ObPuzzle: Which key on my keyboard could be (logically) replaced with a key labeled 'Cyb' ?

[spoiler space]

If 2 is true, then it is the first true statement. If 2 is false, then it is not the first false statement. In either case, 1 is false, and so 9 and 10 are both false.

If 8 is true, then: * 6 is true. * X is one of (20, 30, 40, 50, 60, 70). * 7 is false (none of those X values have both 6 and 8 as factors). * X is not 70. * 5 is false (no combination of 8 + 6 + 5 + zero or more of 4, 3, 2 adds up to one of those X values). * X is not 60. * If 3 is true, then we arrive at a contradiction. 3 is false, so 2 and 4 are true, and X = 40. * 4's truth means 8 - 2 = 6 is a factor of X. Contradiction.

8 is false. 3 is true.

If 6 is false, then it is true (a false statement cannot be the last true statement), contradiction. So 6 and 7 are both true.

* 3, 6, 7 are all factors of X, so X is a multiple of 42. * 5 is false (highest possible sum, 7 + 6 + 5 + 4 + 3 + 2 = 27, is not high enough to be a multiple of 42). * 2 and 4 are both true (otherwise 10's falsity would be contradicted). * 4's truth means 7 - 2 = 5 is a factor of X, so X is a multiple of 210. * 7's truth means 2, 3, 4, 6, 7 are all factors of X, so X is a multiple of 420. * X = 420 = 2 * 2 * 3 * 5 * 7 has 22 divisors for clue 9. 22 >= 22 so no problem there. Any higher multiple of 420 would have more divisors, contradicting clue 9's falsity.

X = 420. 2, 3, 4, 6, and 7 are true; the rest are false.