Re: Enigma 1395 - A matter of fives
- From: "Chappy" <petergregorychapman@xxxxxxxxxxx>
- Date: 26 Jul 2006 19:06:53 -0700
John R Jones wrote:
Chappy wrote:
Enigma 1395 - A matter of fives
New Scientist magazine, 10 June 2006.
by Adrian Somerfield.
I have in mind five five-digit numbers.
For each of them individually, five of
the following 10 statements are true.
(a) It contains two zeroes; (b) It is a
perfect cube; (c) It is the product of
five successive numbers; (d) It is a
perfect square; (e) The central digit is
a 5; (f) The sum of the digits is a
square or cube, or has both its digits
the same; (g) It is not prime but has
only one prime factor; (h) It is a power
greater than the third; (i) It is a
palindromic square of a palindrome;
(j) The first and last digits put
together in that order form a perfect
square.
In ascending order, what are my five
numbers?
Ciao,
Chappy.
that should be enough spoiler
10201
14641
28561
83521
65536
The key to these is to look for the highest constraints - here (c) is
highest with only four possible numbers, but none of them bear fruit
against the other criteria.
The next highest seems to be (g), powers of a prime. there are 57
five-digit such
and it is easy to see which are perfect squares or cubes etc. The other
criteria can be
slogged out for each and only five fit the bill as above.
They are respectively 101^2 (a d f g i)
11^4 (d f g h i)
13^4 (d e f g h)
17^4 (d e g h j )
and 2^16 (d e f g h)
Is there any deeper way?
Inspection (as you have done) or a computer program (as I did) wouldFrom the lack of response from the r.p. regulars, it would appear not.
appear
to be the go. BTW, a more pedantic person than I would probably have
claimed
that your answer was absolutely wrong, because it didn't list the
numbers
in ascending order (as requested), but common sense prevails.
Good work.
Ciao,
Chappy.
.
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- Enigma 1395 - A matter of fives
- From: Chappy
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