Mystery Number
Logic puzzles require you to think. You will have to be logical in your reasoning.
There is a 10digit number, represented by ABCDEFGHIJ, where each numeral, 0 through 9, is used once. Given the following clues, what is the number?
1) A + D = a square number
2) G + J = a triangle number
3) B + I = an even number
4) E * F = an odd number
5) C * H = a prime number
6) A / G = G / C
7) E + I = B + H
HintA triangle number can be expressed as the sum of the numbers 1 to n, for any integer value of n. Therefore the triangle numbers are 1, 3, 6, 10, 15, 21, 28, ...
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Answer
4615972308
For the product of two numbers to be a prime number (clue #5), one of those two numbers must be 1. Therefore, either C or H must be 1.
To achieve the required proportion (clue #6), the numerals [AGC] must be one of the following combinations: 931 or 421 or 842.
If [AGC] = 842, then D would have to be 1 (clue #1). Since either C or H must be 1, then D can not be 1. Since D can not be 1, then A can not be 8. Therefore, [AGC] can not be 842. The only remaining possibilities for [AGC] are 931 and 421. In either case, C = 1.
If [AGC] = 931, then D must be either 0 or 7 (clue #1) and J must be either 0 or 7 (clue #2). Therefore, between them, D and J must use 0 and 7. E and F must both be odd (clue #4). However, four of the odd numerals would already be assigned (A = 9, G = 3, C = 1, either D or J = 7). Therefore, [AGC] can not be 931. This means that [AGC] must be 421. Therefore, A = 4 and G = 2.
Since G = 2, then J = 8 (clue #2). J can not be 1 or 4, since those two numerals are now assigned.
The only remaining letters that can be assigned to 6 are B and I. D can not be 6, as it would not produce a square number (clue #1). Neither E nor F can be 6, since that would not produce an odd number (clue #4). H can not be 6, as it would not produce a prime number (clue #5). Since either B or I must be 6, then 6 must be paired with 0 (clue #3), as the other even numerals have now been assigned.
Since B and I share 0 and 6, then D = 5 (clue #1).
Since B and I share 0 and 6, then E and H must also be separated by 6 (clue #7). The only remaining numerals that satisfy this requirement are 3 and 9. This means that F = 7. This sets H = 3 and E = 9, which then sets B = 6 and I = 0.
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Comments
vlerma
Feb 19, 2007
 Too invovled for this time of night and it is sort of out of my abilities. Hate to admit that, but it kind of went right over my head. 
scallio
Feb 20, 2007
 A most excellent teaser, Cinny! Very, very tough for me... took me a long time and I had to cheat a bit! Math is not my best subject. 
peanut
Feb 22, 2007
 Brilliant!! Took me close on forever but figured it out
More please..??

peanut
Feb 22, 2007
 Brilliant!! Took me close on forever but figured it out
More please..??

ChattyMonkey
Feb 24, 2007
 eeeeekkk!
I'm really bad at these kind!
I did really like it, though.
Good luck with 'em and keep them comin'!
xoxo,
Chatty 
Jimbo
Mar 24, 2007
 An excellent explanation. 
(user deleted)
Apr 09, 2007
 whatever! 
4demo
Oct 19, 2007
 I finally gave up, but a good teaser for anyone with time and who's unlazy enough to think it through. Difficult but fun! 
om123
Jul 12, 2008
 wow.. this took FOREVER to do.. i like it tho 
(user deleted)
Oct 11, 2008
 Dude there is one more answer to the puzzle...and that is 9817453260..
here A=9
B=8
C=1
D=7
E=4
F=5
G=3
H=2
I=6
J=0
A + D = 16, a square no
G + J = 3, a triangle no
B + I = 14, even
E + F = 9, odd
C * H = 2, prime
A/G=G/C=3
E + I = B + H = 10 
cnmne
Oct 11, 2008
 Clue 4 is E times F equals odd, not E plus F equals odd, so your solution does not work. 
javaguru
Jan 08, 2009
 Nice teaser and nice explanation, although I think it solves even easier than the explanation. The solution below doesn't require any presumptive assignments.
I started with the same first three observations involving ACDGH.
Then since B and I must either both be even or odd (#3), then H and E must also both be even or odd (#7). Since E and F are odd (#4), then H is also odd.
Now 4 of the 5 odd numbers are accounted for, so B and I are even and AG can't be 9,3 so must be 4,2 which forces the values for D (#1) and J (#2).
This leaves 2 even numbers to assign to B & I and 3 odd numbers to assign to E, F & H. Using the difference between the even numbers forces the choices for E and H (#7), leaving only one choice for F. The remaining prime number is assigned to H (#5) which forces the assignment of the remaining three numbers (#7).

Holografik
Dec 29, 2011
 I think I never heard of triangle numbers before so had to google that.. great teaser though, was quite fun 
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