Demystifying The Second Mystery Number
What was the challenge?
Let's take a look at clues given once again to identify number ABCDEFGHIJ.
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1) Either A = B / 3 or A = G + 3.
2) Either B = I - 4 or B = E + 4.
3) Either C = J + 2 or C = F * 3.
4) Either D = G * 4 or D = E / 3.
5) Either E = J - 1 or E = D / 4.
6) Either F = B * 2 or F = A - 4.
7) Either G = F + 1 or G = I - 3.
8) Either H = A / 2 or H = C * 3.
9) Either I = H + 3 or I = D / 2.
10) Either J = H - 2 or J = C * 2.
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STEPS :
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STEP 1 :
Few digits can be eliminated for few numbers straightaway on the first go. Those digits just don't 'fit' into the both equations provided for the particular letter.
1) Either A = B / 3 or A = G + 3.
A - 0.
3) Either C = J + 2 or C = F * 3.
C - 0, 1.
4) Either D = G * 4 or D = E / 3.
D - 0, 5, 6, 7, 9
5) Either E = J - 1 or E = D / 4.
E - 9
6) Either F = B * 2 or F = A - 4.
F - 7, 9
7) Either G = F + 1 or G = I - 3.
G - 8 (since F can't be 7 & I can't be 11)
8) Either H = A / 2 or H = C * 3.
H - 0, 5, 7, 8
9) Either I = H + 3 or I = D / 2.
I - 8 (since H can't be 5 and obviously D can't be 16).
10) Either J = H - 2 or J = C * 2.
J - 3 (since H can't be 5 & C can't be 1.5), 5 (since H can't be 7 & C can't
be 2.5), 9.
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STEP 2 :
Now, after eliminating some digits for letters we can revise the list of digits a particular letter(s) on LHS of equation can't make because the letter(s) on RHS doesn't (don't) take some digits.
For example, if J can't be 3 or 5 then C = J + 2 can't be 5 or 7. The deduction is supported by the other equation as F can't be 5/3 or 7/3. Hence, C can't be 5 or 7 for sure.
3) Either C = J + 2 or C = F * 3.
C - 0, 1, 5, 7
4) Either D = G * 4 or D = E / 3.
D - 0, 5, 6, 7, 9, 3 ( since E can't be 9 & G can't be 9/4).
5) Either E = J - 1 or E = D / 4.
E - 9, 4 (J can't be 5 & D can't be 16), 8 (J can't be 9 & D can't be 32).
7) Either G = F + 1 or G = I - 3.
G - 8, 0 [since G can't be -1 and I can't be 3 (since if I = 3 then I = H +3
gives H = 0 but H doesn't take 0)].
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STEP 3 :
For B,
2) Either B = I - 4 or B = E + 4.
B - 8 (since I can't be 12 & E can't be 4)
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STEP 4 :
Possible values left for D are - 1, 2, 4, 8. Remember, only one of the two given hints for the particular letter has to be true.
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STEP 4.1 :
If D = 1, then D = E/3 (Hint 4) gives E = 3 & hence E = J - 1 (Hint 5)
gives J = 4.
Here, D = G*4 of Hint 4 must not be true as G would be 1/4.
Similarly, E = D/4 of Hint 5 must be false and hence other hint i.e.
E = J - 1 must be true.
Following the same logic as above -
If D = 1 ---> E = 3 ----> E = J - 1 ---> J = 4 ---->
---> J = H - 2 or J = C * 2.
Hence, H = 6 or C = 2.
If H = 6, then out of H = A / 2 or H = C * 3 only H = C*3 remains
valid which gives C = 2.
And if C = 2, then C = J + 2 (Hint 3) gives J = 0.
So, D = 1 produces 2 different values of J as 0 and 4.
Hence, this value of D is invalid.
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STEP 4.2 :
If D = 2, then (using Hint 4) E = 6 --> (Hint 5)---> J = 7 --->(Hint 10)
---> H = 9 ---> makes I = H + 3 invalid hence I = D/2 gives I = 1
I = 1 ---> leaves G = F + 1 (of Hint 7) valid.
Further, H = 9 ---> (Hint 8) ---> C = 3 ---> (Hint 3) --->
---> F = 1 (since C = J + 2 = 9 invalid as H = 9 already) ---> (Hint 7)
---> G = 2.
So if D = 2, the value of G will be also 2. That's against the rule.
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STEP 4.3 :
If D = 4, then (using Hint 4) G = 1 ----> (Hint 7) --->
---> F = 0 or I = 4 (invalid as D = 4 already) ---> (Hint 6) ---> A = 4.
Again, if D = 4, then A = 4 also hence this value of D is invalid.
Hence, the only valid value of D is 8.
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STEP 5 :
If D = 8, then using Hint 4 we have, G = 2 --->(Hint 7) ---> F = 1 or I =5
If I = 5, --->(Hint 9) ----> H = 2 but already we have G = 2. So,
G = 2 --->(Hint 7) ---> F = 1 ---> (Hint 6) ---> A = 5.
So far, we get, D = 8, G = 2, F = 1, A = 5 at this stage so far.
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STEP 6 :
Possible values left for E are - 0, 3, 6 and 7.
E = 0 ---> (Hint 5 ) ---> J = 1. But 1 is already taken by F.
E = 3 ---> (Hint 5 ) ---> J = 4 ---> (Hint 10) ---> H = 6 or invalid C = 2
H = 6 ---> (Hint 8 ) ---> invalid C = 2.
The digit 2 is already taken by G, so C = 2 is invalid. Hence, E = 3 is invalid.
E = 7 ---> (Hint 5 ) ---> J = 8. But 8 is already taken by G.
So, the only valid value of E = 6.
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STEP 7 :
E = 6 ---> (Hint 5 ) ---> J = 7 ---> (Hint 10) ---> H = 9 ---> (Hint 5) --->
H = 9 ---> (Hint 8) ---> C = 3
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STEP 8 :
The values we got so far, E = 6, J = 7, H = 9, C = 3, D = 8, G = 2, F = 1,
A = 5.
Only letters left are B and I while digits left are 0 and 4.
Correct hint out of "Either B = I - 4 or B = E + 4" must be B = I - 4 since
B = E + 4 = 6 + 4 = 10 must be invalid.
So, B = I - 4 gives us I = 4 and B = 0 .
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Conclusion :
A = 5, B = 0, C = 3, D = 8, E = 6, F = 1, G = 2, H = 9, I = 4, J = 7
all together gives us number
ABCDEFGHIJ as 5038612947.
Let's verify the above number as per given hints.
1) A = G + 3 = 2 + 3 = 5.
2) B = I - 4 = 4 - 4 = 0.
3) C = F * 3 = 1 * 3 = 3.
4) D = G * 4 = 2 * 4 = 8.
5) E = J - 1 = 7 - 1 = 6.
6) F = A - 4 = 5 - 4 = 1.
7) G = F + 1 = 1 + 1 = 2.
8) H = C * 3 = 3 * 3 = 9.
9) I = D / 2 = 8 / 2 = 4.
10) J = H - 2 = 9 - 2 = 7.
