Demystifying The Third Mystery Number

What was the challenge?

Taking a look at the given clues first for identifying the mystery number ABCDEFGHIJ.

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1) Digit A is either a square number or a triangle number, but not both.

2) Digit B is either an even number or a cube number, but not both.

3) Digit C is either a cube number or a triangle number, but not both.

4) Digit D is either an odd number or a square number, but not both.

5) Digit E is either an odd number or a cube number, but not both.

6) Digit F is either an odd number or a triangle number, but not both.

7) Digit G is either an odd number or a prime number, but not both.

8) Digit H is either an even number or a square number, but not both.

9) Digit I is either a square number or a cube number, but not both.

10) Digit J is either a prime number or a triangle number, but not both.

11) A < B, C < D, E < F, G < H, I < J

12) A + B + C + D + E < F + G + H + I + J  

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STEPS  :

Even numbers between 0 to 9 - 0, 2, 4, 6, 8

Odd numbers between 0 to 9 - 1, 3, 5, 7, 9

Square numbers between 0 to 9 - 0, 1, 4, 9 

Cube numbers between 0 to 9 - 8.

Prime numbers between 0 to 9 - 2, 3, 5, 7

Triangle numbers between 0 to 9 - 0, 1, 3, 6

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STEP 1 : 

A can't be 0 as no leading 0's in the mystery number ABCDEFGHIJ.

Remember both of hints can't be true at the same time. That is Digit A is either a square or a triangle number but not both. Since, 0 and 1 are square as well as triangle numbers, A can't be 0 or 1. 

Similarly, 8 is even as well as cube, B can't take 8. And so on.

Possible values of A -  3, 4, 6, 9. (Square/Triangle)

Possible Values of B - 0, 2, 4, 6. (Even/Cube).

Possible Values of C - 0, 3, 6, 8. (Cube/Triangle)

Possible Values of D - 3, 4, 5, 7. (Odd/Square)

Possible Values of E - 3, 5, 7, 8. (Odd/Cube)

Possible Values of F - 0, 5, 6, 7, 9 (Odd/Triangle)

Possible Values of G - 1, 2, 9 (Odd/Prime)

Possible Values of H - 0, 2, 6, 8, 9 (Even/Square)

Possible Values of I -  4, 8, 9 (Square/Cube)

Possible Values of J -  0, 1, 2, 5, 6, 7  (Prime/Triangle) 

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STEP 2 :

The value of I can't be 8 or 9 as there will be no digit left for J which is greater than I. 

Hence, I = 4.
 
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STEP 3 : 

Digit A can't be 6 or 9 since no digit will be left for B>A. 

As we have, I = 4, the digit A = 3 and hence B = 6.

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STEP 4 :

The only digit left for C is 0, i.e. C = 0 as C can't be 8 for C<D to be true.

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STEP 5 :

  Since I = 4 itself, so digit J can't be 0, 1, 2 for I<J to be true.

Digit D and digit J take either take 5 or 7. With that the only valid digit left for E is 8 to satisfy the condition E<F. 

Hence E = 8.

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STEP 6 :

If E = 8, then F has to be 9. So, F = 9.
 
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STEP 7 : 

With 6, 8, 9 already taken up by other letters, the only digit left for H is 2 for G<H to be true. So, H = 3 and hence G = 1.

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STEP 8 :

As per condition 12, 

A + B + C + D + E < F + G + H + I + J 

3 + 6 + 0 + D + 8 < 9 + 1 + 2 + 4 + J

17 + D < 16 + J 

As deduced in STEP 4, D must be 5 or 7 and J must be 7 or 5.

For above equation to be true, D = 5 and J = 7.


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Conclusion :

A = 3, B = 6, C = 0, D = 5, E = 8, F = 9, G = 1, H = 2, I = 4, J = 7

Hence, the number ABCDEFGHIJ must be 3605891247.

Verifying the given hints - 

1) 3 is a triangle number, but not a square number.
2) 6 is an even number, but not a cube number.
3) 0 is a cube number, but not a triangle number.
4) 5 is an odd number, but not a square number.
5) 8 is a cube number, but not an odd number.
6) 9 is an odd number, but not a triangle number.
7) 1 is an odd number, but not a prime number.
8) 2 is an even number, but not a square number.
9) 4 is a square number, but not a cube number.
10) 7 is a prime number, but not a triangle number.
11) 3 < 6, 0 < 5, 8 < 9, 1 < 2, 4 < 7
12) 3 + 6 + 0 + 5 + 8 = 22, 9 + 1 + 2 + 4 + 7 = 23, 22 < 23