sentences c & e are true.
start from the bottom up. assume e is true. this means three of a,b,c,d are false and only one of a,b,c,d is true. if d is true than e or c cannot be true. *so d is false as long a e is true. if c is true than abc, bcd, or cde cannot be false. if e and c are true than neither of the sentences are broken.
but let's continue. if b is true than at least 3 of the sentences must be true. so if at least three sentences are true and three sentences are false than there are more than five sentences. so b & e cannot both be true. if a is true than ab, bc, cd, de cannot both be false. if a and e are true than c must be false but then bcd must be false making c true. so a and e cannot be true.
-so far if e is true than c must be true as well.
now we start again and assume that d is true. this means two sentences in a row cannot be true. if d is true than c & e cannot be true so are thereforth false. Now a & b cannot both be true or they would voliate d. if a is true than ab, bc, cd, de cannot be true. this means a & b both can't be true and they both can't be false. so one must be false and one must be true. if either statement a or b is true, this makes statement e true which they nullifies statement e making it a paradox. so d cannot be true.
-so far only statements e & c can be true together.
Let's now assume statement c is true. c states in plain words that 3 stenences in a row are not false, so abc, bcd, cde cannot be false. if e is true than exactly three statements must be false. if three are false than two must be true. we have designated statements c and e as true. now lets check. a says that 2 sentences in a row cannot be false. c and e are not in a row. this makes a true. thereforth e cannot be true because less than three sentences are false. we do not need to check the other ones to know because we have already roven that c cannot be true.
-statements e and c true together are still the only statements that work.
now let's assume that statement b is true. b states there are fewer false sentences than true sentences. so at least three of the sentence must be true in order for b to be true. if both b and a are true than two sentences in a row cannot be false and three sentences can be true. so ab, bc, cd, de cannot be false and three a,b,c,d,e must be true. alright, since a and c bare already true(according to us) than either d must be true with c & e being false, d & e must be true with d being false, or c,d,e must all be true. so far so good. lets take the first way first. c & e must be false with d being true. if d is false than two sentences in a row cannot be true. oops. we have both a & b as being true. gues that didn't work. no solution there.
-still only e & c are true together, but lets that jump ahead. we still have one more to go.
Now lets assume that statement a is true. two sentences in a row cannot be false. alright, that means bc, cd, de cannot be false. lets try b now. there are fewer false sentences than tre sentences. alright. two or less false sentences. good so far. onto c. three sentence false in a row. nothing to worry about so far. so c must be true is a & b are true as well. d- two sentence in a row cannot be true. whoops. a, b, & c are all true, so d cannot be true. only e left, getting really close to another solution. alright, e-three false sentences. yep that's false. wait... a was true so two in a row cannot be false so both e & d cannot be false yet are so a cannot be true. there goes that.
-now we've made it all the way through if you followed my very long-winded response (even though I'm typing) then you might get what i said. the only solution was sentence e and c being true.
hear that: SENTENCES E & C ARE TRUE.
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Posted by kat
on 2004-12-14 04:15:36 |
solution
