### Brain Teasers

# Mystery Number 6

There is a ten-digit mystery number (no leading 0), represented by ABCDEFGHIJ, where each numeral, 0 through 9, is used once. Given the following clues, what is the number?

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

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

### Answer

3605891247A = 3, B = 6, C = 0, D = 5, E = 8, F = 9, G = 1, H = 2, I = 4, J = 7

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

A) First, determine what two subsets of numbers qualify for each digit. However, each digit can only be in one of the subsets, not both.

B) Digit A can not be 0 (a leading zero is not allowed).

C) Next, eliminate those numbers that do not satisfy Condition 11. Digit I can only be 4. Digit A can only be 3. Digit B can only be 6. Digit C can only be 0.

D) Digit D and Digit J must be some combination of 5 and 7. Digit E can only be 8. Digit F can only be 9. Digit G can only be 1. Digit H can only be 2.

E) To satisfy Condition 12, Digit D is 5 and Digit J is 7.

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