### Brain Teasers

# Mystery Number 7

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) 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.

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.

### Answer

5038612947A = 5, B = 0, C = 3, D = 8, E = 6, F = 1, G = 2, H = 9, I = 4, J = 7

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.

A) Check each letter for any value that it can not be on the left side of an equation. For example, A can not be 0, as B would also have to be 0 or G would have to be negative. On the first pass through the letters, the following can be determined: A not 0, C not 0, C not 1, D not 0, D not 5, D not 6, D not 7, D not 9, E not 9, F not 7, F not 9, G not 8 (because F not 7), H not 0, H not 5, H not 7, H not 8, I not 0, I not 3 (because H not 0 and D not 6), I not 8 (because H not 5), J not 3 (because H not 5), J not 5 (because H not 7), and J not 9.

B) Make a second pass through the letters. The following can be determined: C not 5, C not 7, D not 3, E not 4, E not 8, and G not 0.

C) Make a third pass. The following can be determined: B not 8.

D) For D, only four numbers are still valid: 1, 2, 4, and 8. Plug each of these numbers into D and check the resulting flow of equations. If D = 1, then the resulting flow will lead to J having to be both 4 and 0. If D = 2, then the resulting flow will lead to 2 also being assigned to G. If D = 4, then the resulting flow will lead to 4 also being assigned to either A or I. If D = 8, then G = 2, F = 1, and A = 5.

E) For E, four numbers are still valid: 0, 3, 6, and 7. Plug each number into E and check the resulting flow. If E = 0, then the flow will lead to J being assigned 1, which has already been assigned to F. If E = 3, then the flow will lead to C being assigned 2, which has already been assigned to G. If E = 7, then the flow will lead to J being assigned 8, which has already been assigned to D. If E = 6, then J = 7, H = 9, and C = 3.

F) The only remaining value for I is 4, which leaves 0 being assigned to B.

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