Brain Teasers
Mystery Number 2
There is a ten-digit mystery number, represented by ABCDEFGHIJ, where each numeral, 0 through 9, is used once. Given the following clues, what is the number?
1) A * F = CE
2) B * H = JI
3) D * G = BH
4) G * I = CF
1) A * F = CE
2) B * H = JI
3) D * G = BH
4) G * I = CF
Answer
5738069241A = 5, B = 7, C = 3, D = 8, E = 0, F = 6, G = 9, H = 2, I = 4, J = 1
1) A * F = CE, 5 * 6 = 30
2) B * H = JI, 7 * 2 = 14
3) D * G = BH, 8 * 9 = 72
4) G * I = CF, 9 * 4 = 36
A) If a number has 5 as a units digit, then one of its factors must also have 5 as a units digit. Therefore, the following can not be 5: E, F, H, I.
B) If any multiplier is 0, then the product is also 0. Therefore, the following can not be 0: A, B, D, F, G, H, I.
C) If any multiplier is 1, then the product equals the other multiplier. Therefore, the following can not be 1: A, B, D, F, G, H, I.
D) Through trial and error, it can be found that C and J can not be 0. Therefore, E = 0.
E) To satisfy equation #1, A = 5 and F must be an even number and C must be 1 through 4.
F) Through trial and error, it can be found that C can not be 1. Therefore, J = 1.
G) Through trial and error, it can be found that C can not be 2 or 4. Therefore, C = 3, which results in F = 6, G = 9, and I = 4.
H) To satisfy equation #2, B and H must 2 and 7 or 7 and 2. This results in D = 8.
I) To satisfy equation #3, B = 7 and H = 2.
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Comments
Excellent teaser
However, There is no need for any trial & error as mentioned in the solution steps D, F and G. Tough I have not tried/verified the answer provided , the solution can be easily achieved through iterative process of elimination.
However, There is no need for any trial & error as mentioned in the solution steps D, F and G. Tough I have not tried/verified the answer provided , the solution can be easily achieved through iterative process of elimination.
Man I LOOOOOOOVE these teasers! (Similar concept to a few that I made a while back - Crack the Code)
Please make more...
Please make more...
Man I LOOOOOOOVE these teasers! (Similar concept to a few that I made a while back - Crack the Code)
Please make more...
Please make more...
I did it a little more directly, I think.
X * Y = AB can't have X or Y be zero or one. Only C, E and J aren't on the left side, so two of them are 0 and 1. Zero can't be the first digit and only E is not a first digit, so E is zero.
If E is zero, then either A or F is five. Since F is the last digit of CF, F can't be five, so A is five and F is 2, 4, 6 or 8; C is 1, 2, 3 or 4; CF is 12, 24, 36 or 48; and G and I are (3,4), (3,, (4,9), (6,. Since only 6x8=48, then CF can't be 48 and G and I can't be (6,.
If CF is 12, then G and I are (3,4), leaving 6, 7, 8 and 9. The only possibility for B * H = JI is 9 * 7 = 63 or 7 * 9 = 63, but BH can't be 79 or 97.
Since C isn't one, then J is one; CF is 24 or 36; G and I are (3, or (4,9); and JI is 14 or 18. If JI is 18, then BH is 29 or 92, which can't be the product of two single-digit numbers. So JI is 14, CF is 36, and G is 9. This makes H = 2, leaving only 8 for D.
That's the complete logic with no guessing or trying.
X * Y = AB can't have X or Y be zero or one. Only C, E and J aren't on the left side, so two of them are 0 and 1. Zero can't be the first digit and only E is not a first digit, so E is zero.
If E is zero, then either A or F is five. Since F is the last digit of CF, F can't be five, so A is five and F is 2, 4, 6 or 8; C is 1, 2, 3 or 4; CF is 12, 24, 36 or 48; and G and I are (3,4), (3,, (4,9), (6,. Since only 6x8=48, then CF can't be 48 and G and I can't be (6,.
If CF is 12, then G and I are (3,4), leaving 6, 7, 8 and 9. The only possibility for B * H = JI is 9 * 7 = 63 or 7 * 9 = 63, but BH can't be 79 or 97.
Since C isn't one, then J is one; CF is 24 or 36; G and I are (3, or (4,9); and JI is 14 or 18. If JI is 18, then BH is 29 or 92, which can't be the product of two single-digit numbers. So JI is 14, CF is 36, and G is 9. This makes H = 2, leaving only 8 for D.
That's the complete logic with no guessing or trying.
The in the middle of the text above are actually 8 )
I was having trouble getting there logically (I hate trial and error) but I couldn't say that E didn't equal 0 because it was never stated that that was the case. It happens to work out that way, but, for example, if the numbers on the left were 2 and 3, or 2 and 4, it could be "06" or "08" and that would still be valid.
Tough, I'd like to try another one knowing all the contingencies necessary.
Tough, I'd like to try another one knowing all the contingencies necessary.
I really hate trial and error.. always thought there was a better way to solve it. But felt good when I saw the answer with explanation, those were the same steps I took. Good teaser but would have liked it better if so much trial and error weren't involved, that took the fun away a little.
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