Mr. and Mrs. Crazy: Squimes
Math brain teasers require computations to solve.
Mr. and Mrs. Crazy were just one step away from winning the $1,000,000 prize in the contest. In order for them to win, they must answer this question:
What are the only ten 8digit squimes?
In case you don't know what a squime is, it is any number that:
has no zero
has all different digits
starts and ends with a singledigit prime or singledigit square number
each pair of consecutive digits is a square or prime
Mr. and Mrs. Crazy solved it and won one million dollars. What were their answers?
HintA squime cannot contain the number 8 since no square or prime ends with 8.
The largest squime has at most eight digits, since it cannot contain
8 or 0. If we use 2, it must come first because no square or
2digit prime ends in 2. No 2digit prime ends in 5 and only 25 is
square. If, therefore, we use 5, it must come after 2. So we begin
with 25.
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Answer
The 8digit squimes are:
25316479
25316497
25364179
25364197
25364719
25364971
25371649
25971643
25973164
25973641
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Comments
princess2007
Sep 25, 2009
 huh? i still didn't get how you got the 25...neways... 
(user deleted)
Jan 10, 2010
 16432597 works as well. 1, while not considered a prime number, is a square so it fits the rule that the first digit must be either prime or a square.
16 is a square, 43 is prime, 25 is a square and 97 is prime. 7 is of course prime as well. 
jhosek
May 12, 2010
 16432597 will not work. The rule about each pair of consecutive digits will cause problems here.
When looking as each pair of digits you have to look at each digit and the one following it. so for 16432597 this would be
16, 64, 43, 32, 35, 97. In this list 32 and 35 are not prime. This is also the reason that 2 has to be the first digit.
I do think there is one additional squime: 25971364
 1st digit (2) is prime
 digit pairs
25, 59, 71, 13, 36 and 64 are primes or squares
 last digit (4) is a square 
RRAMMOHAN
Nov 08, 2012
 A great problem! I did solve this, though I didnt have the patience to work out all the 11 answers. Thanks a lot!
I have forwarded this to a few people who are interested in math puzzles. They have also appreciated this problem and have requested to me to convey their thanks to you. 
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