Mystery Number
Logic puzzles require you to think. You will have to be logical in your reasoning.
There is a 10-digit 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).
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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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