### Brain Teasers

# Mystery Number

There is a 10-digit number, represented by ABCDEFGHIJ, where each numeral, 0 through 9, is used once. Given the following clues, what is the number?

1) A + D = a square number

2) G + J = a triangle number

3) B + I = an even number

4) E * F = an odd number

5) C * H = a prime number

6) A / G = G / C

7) E + I = B + H

1) A + D = a square number

2) G + J = a triangle number

3) B + I = an even number

4) E * F = an odd number

5) C * H = a prime number

6) A / G = G / C

7) E + I = B + H

### Hint

A triangle number can be expressed as the sum of the numbers 1 to n, for any integer value of n. Therefore the triangle numbers are 1, 3, 6, 10, 15, 21, 28, ...### Answer

4615972308For the product of two numbers to be a prime number (clue #5), one of those two numbers must be 1. Therefore, either C or H must be 1.

To achieve the required proportion (clue #6), the numerals [AGC] must be one of the following combinations: 931 or 421 or 842.

If [AGC] = 842, then D would have to be 1 (clue #1). Since either C or H must be 1, then D can not be 1. Since D can not be 1, then A can not be 8. Therefore, [AGC] can not be 842. The only remaining possibilities for [AGC] are 931 and 421. In either case, C = 1.

If [AGC] = 931, then D must be either 0 or 7 (clue #1) and J must be either 0 or 7 (clue #2). Therefore, between them, D and J must use 0 and 7. E and F must both be odd (clue #4). However, four of the odd numerals would already be assigned (A = 9, G = 3, C = 1, either D or J = 7). Therefore, [AGC] can not be 931. This means that [AGC] must be 421. Therefore, A = 4 and G = 2.

Since G = 2, then J = 8 (clue #2). J can not be 1 or 4, since those two numerals are now assigned.

The only remaining letters that can be assigned to 6 are B and I. D can not be 6, as it would not produce a square number (clue #1). Neither E nor F can be 6, since that would not produce an odd number (clue #4). H can not be 6, as it would not produce a prime number (clue #5). Since either B or I must be 6, then 6 must be paired with 0 (clue #3), as the other even numerals have now been assigned.

Since B and I share 0 and 6, then D = 5 (clue #1).

Since B and I share 0 and 6, then E and H must also be separated by 6 (clue #7). The only remaining numerals that satisfy this requirement are 3 and 9. This means that F = 7. This sets H = 3 and E = 9, which then sets B = 6 and I = 0.

Hide Hint Show Hint Hide Answer Show Answer

## What Next?

**Solve a Similar Brain Teaser...**

Or, get a random brain teaser.

If you become a registered user you can vote on this brain teaser, keep track of

which ones you have seen, and even make your own.

## Follow Braingle!