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2n+1 Statements (Posted on 2005-06-13) Difficulty: 2 of 5
1. At least 1 statement among these 2n+1 are true.
2. At least 3 statements among these 2n+1 are false.
3. At least 5 statements among these 2n+1 are true.
...
2n. At least (4n-1) statments among these 2n+1 are false.
2n+1. At least (4n+1) statements among these 2n+1 are true.

How many statements are true? Which?

No Solution Yet Submitted by Dustin    
Rating: 4.0000 (7 votes)

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examples | Comment 8 of 15 |

Two examples I post directly to verify the expresions of my precedent mail (n=4 and n=11):

--------------------------------------------------

n=4  (number of Statements = 2n+1= 9)

St 6,7,8,9 are F (At least 11, 13, 15, 17 statements among these 9 are V/F). Because: There are only 9 statemens.

St 2 is V (At least 3 statements among these 9 are F) Because 6,7,8,9 are F.

St 1 is V (At least 1 statement among these 9 are V) Because St 2 is V.

St 5 is F (At least 9 statement among these 9 are V) Because St 5 and St 4 could not be V both, but if St 4 is F follows that also St 5 is F.

St 3 is F (At least 5 statement among these 9 are V) Because by now I've 5 F St. So just a maximum of 4 St can be V.

St 4 is perplexus... Because till now I've 6 F and 2 V St. St 7 say that: At least 7 statement among these 9 are F. If this is V the total is 6F and 3V so St 4 is F. But is St 4 is F the total is 7F and 2V, so St 4 is V.

The set of Statement for n=4 conferm the rule in my last mail: if n is a 8k-4 number there is no solution

Pattern is: VVF?FFFFF

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n=11  2n+1= 23

St 23,22,21,20,19,18,17,16,15,14,13, are F (At least 45,43,41,39,37,35,33,31,29,27,25, statments among these 23 are V/F). Because: There are only 23 statements.

St 2, 4, 6, is V (At least 3, 7, 11 statements among these 23 are F) Because precedent 11 St are F.

St 1 is V (At least 1 statement among these 23 are V) Because St 2, 4, 6 are V.

St 12 is F (At least 23 statements among these 23 are F) Because St 1 is V.

St 7, 9, 11 are F (At least 13, 17, 21 statements among these 23 are V) Because by now I've 12 F St. So just a maximum of 11 St can be V.

St 8 is V (At least 15 statements among these 23 are F). Because by now I've 15 F St.

St 3 is V (At least 5 statements among these 23 are V). Because precedent St 8 is V, and it is the fifth V term.

St 5 is F (At least 9 statements among these 23 are V). By now I've already 15 F and 6 V so just other two St remain and maximum V st will be 7.

St 10 is F (At least 19 statements among these 23 are F). Because this is the last St and the actual number of F is 16.

The set has this pattern

VVVVFVFVFFF FFFFFFFFFFFF with 6V and 17F

The set of Statement for n=11 conferm the formula in my last mail: if n is a 8k+3  number of V = (n+1)/2  F = (3n+1)/2,

V=6 F= 17

 

Edited on June 15, 2005, 9:24 pm
  Posted by armando on 2005-06-15 21:10:38

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